text:
\begin{document}

\title{Weak compactness of simplified nematic liquid flows in 2D}

\author{
Hengrong Du\footnote{Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA}
\quad Tao Huang\footnote{Department of Mathematics, Wayne State University, Detroit, MI 48202, USA}  \quad Changyou Wang\footnote{Department of Mathematics, Purdue University, West Lafayette, IN 47906, USA}\\
}

\maketitle

\begin{abstract}
For any bounded, smooth domain $\Omega\subset \R^2$,
we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of  the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of  the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argument for the Ginzburg-Landau type
nematic liquid crystal flow.

\end{abstract}

\section{Introduction}

Let $\Omega\subset\R^2$ be a bounded domain with smooth boundary, and $\N\subset \R^L$ (for $L\geq 2$) be a smooth compact Riemannian manifold without boundary, and $0<T\le\infty$.

 We formulate a generalized form of simplified Ericksen-Leslie system of nematic liquid crystals in which the director
 field takes values in $\N$:
\begin{equation}
\begin{cases}
 u_t+u\cdot\nabla u-\Delta u+\nabla P=-\nabla\cdot\big(\nabla v\odot\nabla v\big),\\
\nabla \cdot u=0, \\
 v_t+u\cdot\nabla v=\Delta v+A(v)(\nabla v, \nabla v),
\end{cases}
\ {\rm{in}}\ \Omega\times (0, T),
\label{NEL}
\end{equation}
where $(u(\x,t), v(\x,t), P(\x,t)):\Omega\times (0,T)\rightarrow \R^2\times\N\times \R$ represents the fluid velocity field, the orientation director field of nematic material (into a general Riemannian manifold), and the pressure function respectively, $\big(\nabla v\odot\nabla v\big)_{ij}=\nabla_{\x_i}v\cdot\nabla_{\x_j} v$ for $i,j=1, 2$ represents the Ericksen-Leslie stress tensor, and $A(y)(\cdot, \cdot)$ is the second fundamental form of $\N$ at the point
$y\in\N$.

The generalized system \eqref{NEL} covers the two important cases in nematic liquid crystals:
\begin{enumerate}
\item[(1)] For $\N=\mathbb S^2$, the system \eqref{NEL} becomes the simplified, uniaxial Ericksen-Leslie system first proposed by \cite{lin1989nonlinear}
\begin{equation}\label{ELlc}
\begin{cases}
\pa_t u+u\cdot\nabla u-\Delta u+\nabla P=-\nabla\cdot\big(\nabla d\odot\nabla d\big),\\
\nabla \cdot u=0, \\
\pa_t d+u\cdot\nabla d=\Delta d+|\nabla d|^2d,
\end{cases}
\end{equation}
 for $(u(\x,t), d(\x,t), P(\x,t)):\Omega\times (0,T)\rightarrow \R^2\times\mathbb S^2\times \R$.

In dimension two, the existence of a unique global weak solution has been proved in \cite{lin2010liquid} \cite{lin2010uniqueness}, which satisfies the energy inequality and has at most finitely many singular times,
see also \cite{Hong2011}. Very recently, the authors in \cite{LLWWZ19} have constructed example of singularity at finite time. In dimension three, a global weak solution has been constructed in \cite{linwang2016weakthreedim} with initial data $d_0\in \mathbb S^2_+$. Examples of finite time singularity have been constructed by \cite{Huang2016FiniteTS}. Interested readers
can consult the survey article \cite{lin2014recent} and the references therein.

\item[(2)]
For
$$
\mathcal N=\big\{({\bf y}_1, {\bf y}_2)\in \mathbb S^2\times \mathbb S^2~\big|~{\bf y}_1\cdot{\bf y}_2=0\big\}
\subset \mathbb R^6,
$$
let $v(\x, t)=(n(\x, t), m(\x, t)):\Omega\times (0,T)\rightarrow\mathbb S^2\times\mathbb S^2$ with $n\cdot m=0$. Then the system \eqref{NEL} becomes the biaxial, Ericksen-Leslie system
\begin{equation}\label{blc}
\begin{cases}
\pa_t u+u\cdot\nabla u-\Delta u+\nabla P=-\nabla\cdot\big(\nabla n\odot\nabla n+\nabla m\odot\nabla m\big),\\
\nabla \cdot u=0, \\
\pa_t n+u\cdot\nabla n=\Delta n+|\nabla n|^2n+\langle\nabla n,\nabla m\rangle m\\
\pa_t m+u\cdot\nabla m=\Delta m+|\nabla m|^2m+\langle\nabla m,\nabla n\rangle n\\
n\cdot m=0
\end{cases}
\ {\rm{in}}\ \Omega\times (0,T).
\end{equation}
This is a simplified version of the hydrodynamics of biaxial nematics model proposed by Grovers and
Vertogen \cite{GoversVertogen1984a, GoversVertogen1984b, GoversVertogen1985}.
In  dimensional two, the existence of a unique global weak solution has recently been shown in \cite{LinLiWang20},
which is smooth off at most finitely many singular times.

\end{enumerate}

A strategy to construct a weak solution of \eqref{NEL} and \eqref{inbdy} is to consider a Ginzburg-Landau approximated system (cf. \cite{linliu1995nonparabolic},\cite{linliu1996partialregularity}).
More precisely,  for any $\delta>0$ set the $\delta$-neighborhood of $\N$ by
$$
\mathcal N_\delta=\Big\{{\bf y}\in \mathbb R^L~\big|~ \dist({\bf y}, \mathcal N)<\delta\Big\},
$$
where $\dist (\y, \mathcal N)$ is the distance from $\y$ to $\mathcal N$. Let $\Pi_{\mathcal N}:\mathcal N_\delta\rightarrow \mathcal N$ be the nearest point projection map. There exists $\delta_{\N}=\delta(\mathcal N)>0$ such that $\dist (\y, \mathcal N)$ and
$\Pi_{\N}$ are smooth in $\N_{2\delta_{\N}}$. Let $\chi(s)\in C^\infty([0,\infty))$ be a monotone increasing function such that
$$
\chi(s)=
\left\{
\begin{array}{ll}
s, \quad & \mbox{if } 0\leq s
\leq \delta_{\N}^2,\\
4 \delta_{\N}^2, \quad &\mbox{if }  s
\geq 4\delta_{\N}^2.
\end{array}
\right.
$$
Consider the following Ginzburg-Landau energy functional for the director $v$
$$
E_{\varepsilon}(u, {v})
=\int_{\Omega}\big(\frac12|\nabla {v}|^2+\frac{1}{\varepsilon^2}\chi\big(\dist^2({v}, \mathcal N)\big)\big).
$$

Then the corresponding Ginzburg-Landau approximated system of \eqref{NEL} can be written as
\begin{equation}\label{NGL}
\begin{cases}
u_t+u\cdot\nabla u-\Delta u+\nabla P=-\nabla\cdot\big(\nabla {v}\odot\nabla {v}\big),\\
\nabla \cdot u=0, \\
\displaystyle{v}_t+u\cdot\nabla {v}=\Delta {v}-\frac{1}{\ve^2}\chi'\big(\dist^2({v}, \mathcal N)\big)\frac{d}{d{v}}\big(\dist^2({v}, \mathcal N)\big).
\end{cases}
\end{equation}

The main purpose of this paper is to study the weak compactness of  solutions to the simplified Ericksen-Leslie system \eqref{NEL} and convergence of solutions of the Ginzburg-Landau  approximation \eqref{NGL} to the
simplified Ericksen-Leslie system \eqref{NEL}.  For this purpose, we will consider
the following initial and boundary condition
\beq\label{inbdy}
(u, v)\,|\,_{\partial_p Q_T}=(u_0, v_0)
\eeq
where $Q_T=\Omega\times(0,T)$ and $\partial_p Q_T=\big(\Omega\times\{t=0\}\big)\cup\big(\partial \Omega\times[0,T]\big)$ is the parabolic boundary of $Q_T$.

We assume that
\beq\label{uoc}
u_0\,\big|\,_{\partial\Omega}=0,\quad v_0(x)\in \N\ \ \mbox{for  a.e. }\ x\in \Omega,
\eeq
and  introduce the notations
\begin{equation*}
  \begin{split}
  {\bf H}=\text{closure of }C_0^\infty\left(\Omega, \R^2\right)\cap \left\{ f~\big|~\nabla\cdot f=0\right\} \text{ in }L^2\left(\Omega,\R^2\right),\\
  {\bf J}=\text{closure of }C_0^\infty\left(\Omega, \R^2\right)\cap \left\{ f~\big|~\nabla\cdot f=0 \right\}\text{ in }H_0^1\left(\Omega, \R^2\right), \\
  H^1(\Omega, \N)=\left\{ f\in H^1(\Omega, \R^L)~\big|~ f(x)\in \N \text{ a.e. }x\in \Omega \right\}.
  \end{split}
\end{equation*}
We also assume that
\begin{equation}  \label{initialRegularity}
  u_0\in {\bf H},\ \ v_0\in H^1(\Omega, \N).

\end{equation}
Recall the definition of weak solutions of \eqref{NEL}.

\begin{definition}\label{defwsol}
A pair of maps $u\in L^{\infty}([0,T], {\bf H})\cap L^2([0,T], {\bf J})$ and $v\in L^2([0,T], H^1(\Omega, \N))$ is called a weak solution to initial and boundary problem \eqref{NEL}, \eqref{inbdy}-\eqref{initialRegularity}, if
  \begin{equation}
  \begin{split}
    &-\int_{Q_T}\left\langle u, \xi' \varphi  \right\rangle+\int_{Q_T} \left\langle u\cdot \nabla u, \xi\varphi \right\rangle+\left\langle \nabla u, \xi \nabla\varphi \right\rangle\\
    &\ =-\xi(0)\int_\Omega\left\langle u_0, \varphi \right\rangle+\int_{Q_T} \left\langle \nabla v\odot\nabla v, \xi \nabla \varphi \right\rangle, \\
    &-\int_{Q_T} \left\langle  v, \xi'\phi\right\rangle+\int_{Q_T}\left\langle u\nabla v, \xi\phi \right\rangle
    +\left\langle \nabla v, \xi \nabla\phi \right\rangle\\
    &\ =-\xi(0)\int_\Omega\left\langle v_0, \phi \right\rangle
    +\int_{Q_T} \left\langle A(v)(\nabla v, \nabla v), \xi\phi \right\rangle,
    \label{weakdef}
    \end{split}
  \end{equation}
  for any $\xi\in C^\infty([0,T])$ with $\xi(T)=0$, $\varphi\in {\bf J}$ and $\phi \in H_0^1(\Omega, \R^3)$. Moreover, $(u, v)|_{\partial \Omega}=(u_0, v_0)$ in the sense of trace. The notion of a weak solution to
the system \eqref{NGL} can be defined similarly.
\end{definition}

Our first main theorem concerns the convergence of weak solutions of the system \eqref{NGL}
to the system \eqref{NEL} as $\varepsilon\rightarrow 0$. We remark that the existence of weak solutions
to \eqref{NGL} has been established by \cite{linliu1995nonparabolic, linliu1996partialregularity} for $\N=\mathbb S^2$ by the Galerkin method, which can be easily adapted to handle the case that $\N$ is a compact Riemannian manifold.

\begin{theorem}  \label{thm3}
  For $\ve>0$, let $(u^\ve, {v}^\ve)$ be a sequence of weak solutions to the Ginzburg-Landau approximated system \eqref{NGL} with the initial and boundary condition \eqref{inbdy}-\eqref{initialRegularity}. Then there exists a weak solution $(u, v)$ of \eqref{NEL} with the initial and boundary condition \eqref{inbdy}-\eqref{initialRegularity} such that, after passing to subsequences,
\begin{equation}\notag
 u^\ve\rightharpoonup u\ \text{ in }\ L^2 ([0,T], H^1(\Omega)) , \quad
      {v}^\ve \rightharpoonup v\ \text{ in }\ L^2 ([0,T], H^1(\Omega)) .
  \end{equation}
In particular, the initial and boundary problem \eqref{NEL} and \eqref{inbdy}-\eqref{initialRegularity} admits at least one weak solution $u\in L^{\infty}([0,T], {\bf H})\cap L^2([0,T], {\bf J})$ and $v\in L^2([0,T], H^1(\Omega, \N))$.
\end{theorem}

We would like to mention that when $\N=\mathbb S^2$, the convergence of solutions of system \eqref{NGL} to the system \eqref{ELlc} has recently been proved in two dimensional torus $T^2$ by Kortum in an interesting article \cite{kortum2019concentrationcancellation}. In order to deal with convergence of the most difficult terms $\nabla d_\varepsilon\odot\nabla d_\varepsilon$ in the limit process, Kortum employed the concentration-cancellation method for the Euler equation developed by DiPerna and Majda \cite{DipernaMajda1988concentration} (see also \cite{MajdaBertozzi2002Vorticity}).  Thanks to the rotational covariance of $\nabla d_\varepsilon\odot\nabla d_\varepsilon$, the test functions can be taken to a function of periodic one spatial variable
ensuring the weak convergence of $\nabla d_\varepsilon\odot\nabla d_\varepsilon$ to $\nabla d\odot\nabla d$.

In this paper, we make some new observations on the Ericksen stress tensor $\nabla v\odot \nabla v$, which is flexible enough to handle any smooth domain $\Omega\subset \R^2$.  Namely, by adding $-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2$ to $\nabla v^\ve\odot \nabla v^\ve$, where $\mathbb I_2$ is the $2\times2$ identity matrix,
we have
$$
\nabla v^\ve\odot\nabla v^\ve-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2
=\frac12\left(
\begin{array}{ll}
|\partial_x v^\ve|^2-|\partial_y v^\ve|^2, &2\langle\partial_x v^\ve,\partial_y v^\ve\rangle\\
2\langle\partial_x v^\ve,\partial_y v^\ve\rangle, &|\partial_y v^\ve|^2-|\partial_x v^\ve|^2
\end{array}
\right).
$$
This is a matrix whose components constitute the Hopf differential of map $v^\ve$, which are
$|\partial_x v^\ve|^2-|\partial_y v^\ve|^2$ and $\langle\partial_x v^\ve,\partial_y v^\ve\rangle$.
Since $v^\epsilon$ is either an approximated harmonic map to $\N$ or a Ginzburg-Landau
approximated harmonic map, we can develop its compensated compactness property
by the Pohozaev type argument.

\medskip
As a byproduct of the proof of Theorem \ref{thm3}, we obtain the following compactness for a sequence of weak solutions to the system \eqref{NEL}.

\begin{theorem}\label{thm1}
 Let $(u^k, v^k):\Omega\times (0,T)\rightarrow \R^2\times\N$ be a sequence of weak solutions to \eqref{NEL},
 along with the initial and boundary condition $(u^k_0,v^k_0)$ satisfying \eqref{uoc}, such that
 \beq\label{kengbd}
\begin{split}
\sup\limits_{k\geq 1} \left\{\int_{Q_t}\big(|u^k|^2+|\nabla v^k|^2\big)+\int_{Q_t}\big(|\nabla u^k|^2+|v^k_t+u^k\cdot\nabla v^k|^2\big)\right\}< \infty,
\end{split}
\eeq

Furthermore, if we assume that
\beq\notag
(u^k_0,v^k_0)\rightharpoonup (u_0, v_0)\ \ \mbox{in }\ \ L^2(\Omega)\times H^1(\Omega),
\eeq
then there exists a weak solution $(u, v)$ of \eqref{NEL} with the initial and boundary condition $(u_0, v_0)$
such that, after passing to subsequences,
\begin{equation}\notag
  u^k\rightharpoonup u \text{ in } L^2 ([0,T], H^1(\Omega)),  \qquad
      {v}^k \rightharpoonup v \text{ in } L^2 ([0,T], H^1(\Omega)).
\end{equation}
\end{theorem}

Since the system \eqref{NEL} possesses the geometric structure, i.e.,
$$
A(v^k)(\cdot, \cdot) \perp  T_{v^k}\N
$$
where $T_{v^k}\N$ is the tangent space of $\N$ at $v^k$,
we can show the weak convergence of $|\partial_x v^k|^2-|\partial_y v^k|^2$ and $\langle\partial_x v^k,\partial_y v^k\rangle$ by utilizing the $L^p$-estimate, $1<p<2$, of the Hopf differential of $v^k$.

\section{Estimates on inhomogeneous Ginzburg-Landau equations}

In this section, we will consider the inhomogeneous Ginzburg-Landau equation
\beq\label{appGLb}
\Delta {v}^\ve-\frac{1}{\ve^2}\chi'\big(\dist^2({v}^\ve, \mathcal N)\big)\frac{d}{d{v}}\big(\dist^2({v}^\ve, \mathcal N)\big)=\tau_\ve
\quad \mbox{in }\ \ \Omega.
\eeq
Suppose
\begin{equation}
  \sup_{0<\ve\le 1} \mathcal{E}_\ve({v}^\ve)=\int_\Omega\left( \frac{1}{2}|\nabla {v}^\ve|^2+\frac{1}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N})) \right)\le \Lambda_1<\infty,
  \label{eqn:Lambda1}
\end{equation}
and
\begin{equation}
  \sup_{0<\ve\le 1}\left\|\tau^\ve\right\|_{L^2(\Omega)}\le \Lambda_2<\infty.
  \label{eqn:Lambda2}
\end{equation}
Assume that there exist ${v}\in H^1(\Omega, \mathcal{N})$ and $\tau\in L^2(\Omega,\mathbb R^L)$
such that
\begin{equation}\notag
    \tau^\ve\rightarrow \tau \text{ in }L^2(\Omega), \quad
    \vv^\ve\rightharpoonup \vv  \text{ in }H^1(\Omega).
\end{equation}
Then we have
\begin{lemma}\label{lemma:biGL}
  There exists $\delta_0>0$  such that if $\vv^\ve\in H^1(\Omega, \R^L)$ is a family of solutions to \eqref{appGLb} satisfying \eqref{eqn:Lambda1} and \eqref{eqn:Lambda2}, and for $\x_0\in\Omega$
  and $0<r_0<\dist(x_0,\partial\Omega)$,
\beq\label{engdelta}
\sup_{0<\ve\le 1}\int_{B_{r_0}(\x_0)}\big(\frac12|\nabla {v}^\ve|^2+\frac{1}{\varepsilon^2}\chi\big(\dist^2({v}^\ve, \mathcal N)\big)\big)\leq \delta_0^2,
\eeq
then there exists an approximated harmonic map ${v}\in H^1(B_{\frac{r_0}{4}}(\x_0),\mathcal N)$ with tension filed $\tau$, i.e,
\beq
\Delta {v}+A({v})(\nabla {v},\nabla {v})=\tau,
\eeq
such that as $\epsilon\rightarrow 0$,
\beq
{v}^\ve\rightarrow {v}\quad \mbox{in }\ H^1(B_{\frac{r_0}4}(\x_0)),\ \ {\rm{and}}\
\ \frac{1}{\varepsilon^2}\chi\big(\dist^2({v}^\ve, \mathcal N)\big)\rightarrow 0\ \ {\rm{in}}\ \
L^1(B_{\frac{r_0}4}(\x_0)).
\eeq
\end{lemma}

\pf For any fixed $\x_1 \in B_{\frac{r_0}2}(\x_0)$
and $0<\ve\le \frac{r_0}{2}$, define $\hv^\ve(\x)=\vv^\ve(\x_1+\ve \x):B_1(0)\to \R^L$. Then we have
\begin{equation*}
  \Delta \hv^\ve=\chi'(\dist^2(\hv^\ve, \mathcal{N}))\frac{d}{d\vv}(\dist^2(\hv^\ve, \mathcal{N}))+\widehat{\tau}^\ve \text{ in }B_1(0),
\end{equation*}
where $\widehat{\tau}^\ve(\x)=\ve^2 \tau^\ve(\x_1+\ve \x)$. Since
\begin{align*}
  \left\|\Delta \hv^\ve\right\|_{L^2(B_1(0))}&\le \big\| \chi'(\dist(\hv^\ve, \mathcal{N}))\frac{d}{d\vv}(\dist^2(\hv^\ve, \mathcal{N})) \big\|_{L^2(B_1(0))}+\left\|\widehat{\tau}^\ve\right\|_{L^2(B_1(0))}\\
  &\le C\Big(\int_{\Omega\cap \{\dist(v^\epsilon, \N)\le 2\delta_{\N}\}}
  |\dist(v^\epsilon, \N)|^2\Big)^\frac12+\ve\left\|\tau^\ve\right\|_{L^2(\Omega)}\le C+\Lambda_2.
\end{align*}
Thus $\hv^{\ve}\in H^2(B_\frac12)$ and $\left\|\hv^\ve\right\|_{H^2(B_\frac12)}\le C(1+\Lambda_2)$. By Morrey's inequality, we conclude that $\hv^\ve\in C^{\frac12}(B_\frac12)$ and
\begin{equation*}
  [\hv^{\ve}]_{C^{\frac12}(B_\frac12)}\le C\left\|\hv^\ve\right\|_{H^2(B_\frac12)}\le C(1+\Lambda_2).
\end{equation*}
By rescaling, we get
\begin{equation*}
  |\hv^\ve(\x)-\hv^\ve(\y)|\le C(1+\Lambda_2)\Big( \frac{|\x-\y|}{\ve} \Big)^{\frac{1}{2}}, \qquad \forall \x, \y\in B_\ve(\x_1).
\end{equation*}
We claim that $\dist(\vv^\ve, \mathcal{N})\le \delta_{\mathcal{N}}$ on $B_{\frac{r_0}2}(\x_0)$. Suppose it were false. Then there exists $\x_1\in B_{\frac{r_0}2}(\x_0)$ such that $\dist(\vv^\ve(\x_1), \mathcal{N})>\delta_{\mathcal{N}}$. Then for any $\theta_0\in (0, 1)$ and $\x\in B_{\theta_0 \ve}(\x_1)$, it holds
\begin{equation*}
  |\vv^\ve(\x)-\vv^\ve(\x_1)|\le C\Big( \frac{|\x-\x_1|}{\ve} \Big)^{\frac12}\le C \theta_0^{\frac12}\le \frac{1}{2}\delta_{\mathcal{N}},
\end{equation*}
provided $\theta_0\le \frac{\delta_{\mathcal{N}}^2}{4C^2}$. It follows that
\begin{equation*}
  \dist(\vv^\ve(\x), \mathcal{N})\ge \frac{1}{2}\delta_{\mathcal{N}}, \ \forall\x\in B_{\theta_0\ve}(\x_1),
\end{equation*}
so that
\begin{equation*}
  \int_{B_{\theta_0 \ve}(\x_1)}\frac{1}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\ge \pi \delta_{\mathcal{N}}^2 \theta_0^2.
\end{equation*}
which contradicts to the assumption that
\begin{equation*}
  \int_{B_{\theta_0 \ve}(\x_1)}\frac{1}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\le \int_{B_{r_1}(0)}\Big( \frac{1}{2}|\nabla\vv^\ve|^2+\frac{1}{\ve^2}\chi\left(\dist^2(\vv^\ve, \mathcal{N})\right) \Big)\le \delta_0^2
\end{equation*}
for a sufficiently small $\delta_0>0$.

From $\dist(\vv^\ve, \mathcal{N})\le \delta_{\mathcal{N}}$ in $B_{\frac{r_0}2}(\x_0)$,
we may decompose ${v}^\ve $ into
$$
{v}^\ve=\Pi_{\mathcal N}({v}^\ve)
+\dist({v}^\ve, \mathcal N)\nu\big(\Pi_{\mathcal N}({v}^\ve)\big):=\omega_\ve+\zeta_\ve\nu_\ve,
$$
so that the equation of ${v_\ve}$ becomes
\beq\label{eqnwdnu}
\Delta \omega_\ve+\Delta \zeta_\ve\nu_\ve+ 2\nabla \zeta_\ve\nabla\nu_\ve+\zeta_\ve\Delta\nu_\ve-\frac{1}{\ve^2}\chi'(\zeta_\ve^2)\nabla_{{v_\ve}}\zeta_\ve^2=\tau_\ve.
\eeq
Multiplying \eqref{eqnwdnu} by $\nu_\ve$, we get
\beq\label{eqnd}
\Delta \zeta_\ve=\langle \nabla \omega_\ve, \nabla\nu_\ve\rangle+\zeta_\ve|\nabla \nu_\ve|^2+\frac{1}{\ve^2}\chi'(\zeta_\ve^2)\langle\nabla_{{v_\ve}}\zeta_\ve^2,\nu_\ve\rangle+\tau^\perp_\ve,
\eeq
where $\tau^\perp_\ve=\langle\tau_\ve, \nu_\ve\rangle$. Plugging $\Delta \zeta_\ve$ into \eqref{eqnwdnu}, we obtain
\beq\label{eqnwnu}
\Delta \omega_\ve+\langle \nabla \omega_\ve,\nabla \nu_\ve\rangle\nu_\ve+\zeta_\ve\big(\Delta \nu_\ve+|\nabla \nu_\ve|^2\nu_\ve\big)+2\langle\nabla\nu_\ve,\nabla \zeta_\ve\rangle=\tau''_\ve,
\eeq
where $\tau''_\ve=\tau_\ve-\tau^\perp_\ve\nu_\ve$. Here we have used the fact
$$
\langle\nabla_{{v_\ve}}\zeta_\ve^2,\nu_\ve\rangle\nu_\ve
=\nabla_{{v_\ve}}\zeta_\ve^2.
$$

Let $\eta\in C^{\infty}_0(B_{\frac{r_0}2}(\x_0), \R)$ be a standard cutoff function of $B_{\frac{3r_0}8}(\x_0)$.
Since $\dist(\vv^\ve, \mathcal{N})\le \delta_{\mathcal{N}}$, we have that $\chi'(\zeta_\ve^2)=1$ and hence

\beq\label{eqndeta}
\begin{split}
\left(-\Delta+\frac{2}{\ve^2}\right) (\zeta_\ve\eta^2)=&-\zeta_\ve\Delta(\eta^2)-2\nabla \zeta_\ve\nabla(\eta^2)+\langle \nabla \omega_\ve, \nabla(\nu_\ve\eta^2)\rangle-\langle \nabla \omega_\ve, \nu_\ve\nabla(\eta^2)\rangle\\
&+\zeta_\ve\left(|\nabla (\nu_\ve\eta^2)|^2-|\nu_\ve\nabla(\eta^2)|^2\right)+\tau^\perp_\ve\eta^2.
\end{split}
\eeq
Applying the $W^{2,\frac43}$-estimate for $(-\Delta+\frac2{\ve^{2}})$ (see \cite{Krylov2008Sobolev}), we obtain
\beq
\begin{split}
&\|\nabla^2(\zeta_\ve\eta^2)\|_{L^{\frac43}}\\
\les&\|\zeta_\ve\Delta (\eta^2)\|_{L^{\frac43}}+\|\nabla  \zeta_\ve\nabla(\eta^2)\|_{L^{\frac43}}
+\|\nabla \omega_\ve\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}\\
&+\|\nabla \omega_\ve\|_{L^{2}}+\|\zeta_\ve\|_{L^{\infty}}\|\nabla \nu_\ve\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}+\|\tau^\perp_\ve\|_{L^{\frac43}}\\
\les& \|\zeta_\ve\|_{L^{\infty}}+\|\nabla \zeta_\ve\|_{L^{2}}+\|\nabla \omega_\ve\|_{L^{2}}(\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}+1)\\
&+\|\zeta_\ve\|_{L^{\infty}}\|\nabla\nu_\ve\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}+\|\tau_\ve\|_{L^{2}},
\end{split}
\eeq
where $A\les B$ stands for $A\leq CB$ for some universal positive constant $C$.

For $\omega_\ve$, by a similar calculation we obtain
\beq\label{eqnweta}
\begin{split}
\Delta (\omega_\ve\eta^2)=&-\langle \nabla \omega_\ve,\nabla (\nu_\ve\eta^2)\rangle\nu_\ve+\langle \nabla \omega_\ve, \nu_\ve\nabla(\eta^2)\rangle\nu_\ve\\
&-\zeta_\ve\big[\Delta (\nu_\ve\eta^2)-\nu_\ve\Delta(\eta^2)-2\nabla \nu_\ve\nabla(\eta^2)\big]+\zeta_\ve\big[|\nabla \nu_\ve\eta^2|^2-|\nu_\ve\nabla(\eta^2)|^2\big]\nu_\ve\\
&-2\big[\langle\nabla(\nu_\ve\eta^2),\nabla \zeta_\ve\rangle-\langle\nabla(\eta^2),\nabla \zeta_\ve\rangle\nu_\ve\big]+\tau''_\ve\eta^2+\omega_\ve\Delta(\eta^2)+2\nabla \omega_\ve\nabla (\eta^2).
\end{split}
\eeq
Applying the $W^{2,\frac43}$-estimate, we obtain
\beq
\begin{split}
&\|\nabla^2(\omega_\ve\eta^2)\|_{L^{\frac43}}\\
\les& \|\nabla \omega_\ve\|_{L^{2}}\|\nabla (\nu_\ve\eta^2)\|_{L^{4}}+\|\nabla \omega_\ve\|_{L^{\frac43}}+\|\zeta_\ve\|_{L^{\infty}}\|\Delta(\nu_\ve\eta^2)\|_{L^{\frac43}}\\
&+\|\zeta_\ve\|_{L^{\infty}}\left(1+\|\nabla\nu_\ve\|_{L^{\frac43}}\right)+\|\zeta_\ve\|_{L^{\infty}}\|\nabla(\nu_\ve\eta^2)\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}\\
&+\|\nabla \zeta_\ve\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}+\|\nabla \zeta_\ve\|_{L^{\frac43}}+\|\tau_\ve\|_{L^{2}}.
\end{split}
\eeq
Therefore, we conclude that
\beq
\begin{split}
&\|\nabla^2(\zeta_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla^2(\omega_\ve\eta^2)\|_{L^{\frac43}}\\
\les&\|\nabla {v}^\ve\|_{L^{2}}\|\nabla(\nu_\ve\eta^2)\|_{L^{4}}+\| \zeta_\ve\|_{L^{\infty}}\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla {v}^\ve\|_{L^{2}}+\|\tau_\ve\|_{L^{2}}.
\end{split}
\eeq
Since
$$
{v}^\ve\eta^2=\omega_\ve\eta^2+\zeta_\ve\nu_\ve\eta^2
$$
we have
\beq
\begin{split}
&\|\nabla^2({v}^\ve\eta^2)\|_{L^{\frac43}}\\
\les&\|\nabla^2(\zeta_\ve\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla^2(\omega_\ve\eta^2)\|_{L^{\frac43}}\\
\les&\| \zeta_\ve\|_{L^{\infty}}\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla \zeta_\ve\|_{L^{2}}\|\nabla (\nu_\ve\eta^2)\|_{L^{4}}+\|\nabla^2\zeta_\ve\nu_\ve\eta^2\|_{L^{\frac43}}+\|\nabla^2(\omega_\ve\eta^2)\|_{L^{\frac43}}\\

\les&\| \zeta_\ve\|_{L^{\infty}}\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla \zeta_\ve\|_{L^{2}}\|\nabla (\nu_\ve\eta^2)\|_{L^{4}}+\|\nabla^2(\zeta_\ve\eta^2)\|_{L^{\frac43}}\\
&+\|\nabla \zeta_\ve\|_{L^{2}}+\|\nabla^2(w_\ve\eta^2)\|_{L^{\frac43}}+1.
\end{split}
\eeq
Therefore, we have
\beq
\begin{split}
&\|\nabla^2({v}^\ve\eta^2)\|_{L^{\frac43}}\\
\les&\| \zeta_\ve\|_{L^{\infty}}\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla {v}^\ve\|_{L^{2}}\left[1+\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}+\|\nabla (\nu_\ve\eta^2)\|_{L^{4}}\right]+\|\tau_\ve\|_{L^{2}}+1\\
\les&\| \zeta_\ve\|_{L^{\infty}}\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}+\|\nabla {v}^\ve\|_{L^{2}}\left[1+\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}\right]+\|\tau_\ve\|_{L^{2}}+1.
\end{split}
\eeq
Since $\nu_\ve=\nu_\ve({v}^\ve)$, we can directly calculate and show that
\beq
\begin{split}
\|\nabla^2(\nu_\ve\eta^2)\|_{L^{\frac43}}
\les\|\nabla^2({v}^\ve\eta^2)\|_{L^{\frac43}}+\|\nabla {v}^\ve\|_{L^{2}}\left[1+\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}\right]+1.
\end{split}
\eeq
Therefore, we can conclude that
\beq
\begin{split}
\left(1-C\|\zeta_\ve\|_{L^\infty}\right)\|\nabla^2({v}^\ve\eta^2)\|_{L^{\frac43}}
\les\|\nabla {v}^\ve\|_{L^{2}}\left[1+\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}\right]+\|\tau_\ve\|_{L^{2}}+1.
\end{split}
\eeq
By Sobolev's embedding, we have
\beq
\begin{split}
\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}
\les\|\nabla {v}^\ve\|_{L^{2}}\left[1+\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}\right]+\|\tau_\ve\|_{L^{2}}+1.
\end{split}
\eeq
Taking $\delta_0$ small enough in the assumption \eqref{engdelta}, we conclude that
\beq
\begin{split}
\|\nabla ({v}^\ve\eta^2)\|_{L^{4}}
\les\|\nabla {v}^\ve\|_{L^{2}}+\|\tau_\ve\|_{L^{2}}+1\leq C(\delta_0, \Lambda_2).
\end{split}
\eeq
Substituting this into (2.19), we obtain
which implies that
\beq
\begin{split}
\|\nabla^2 (v_\ve\eta^2)\|_{L^{\frac43}}
\leq C(\delta_0, \Lambda_2).
\end{split}
\eeq
Hence $v^\ve \rightarrow v$ in $H^1(B_{\frac{r_0}3}(\x_0))$.

By Fubini's theorem, there exists $r_1\in[\frac{r_0}4, \frac{r_0}3]$
\beq\label{estpgs}
\int_{\partial B_{r_1}(\x_0)}|\nabla \zeta_\ve|^2\leq C\int_{B_{\frac{r_0}3}(\x_0)}|\nabla \zeta_\ve|^2\leq C,
\quad \int_{\partial B_{r_1}(\x_0)}|\zeta_\ve|^2\leq C\int_{B_{\frac{r_0}3}(\x_0)}|\zeta_\ve|^2\leq C\ve^2.
\eeq
Multiplying the equation of $\zeta_\ve$ by $\zeta_\ve$ and integrating by parts over $B_{r_2}$, we obtain
\beq
\int_{B_{r_1}(\x_0)}\big(|\nabla \zeta_\ve|^2+\frac{2}{\ve^2}\chi'(\zeta_\ve)\zeta_\ve^2+|\nabla \nu_\ve|^2\zeta_\ve^2+\nabla \omega_\ve\nabla \nu_\ve \cdot \zeta_\ve\big)-\int_{\partial B_{r_1}(\x_0)}\frac{\partial \zeta_\ve}{\partial\nu} \zeta_\ve=\int_{B_{r_1}(\x_0)}\tau_\ve^{\perp}\zeta_\ve
\eeq
Then we have
\beq
\begin{split}
&\int_{B_{r_1}(\x_0)}\big(|\nabla \zeta_\ve|^2+\frac{2}{\ve^2}\zeta_\ve^2\big)\\
\leq &C\big(\int_{\partial B_{r_1}(\x_0)}|\nabla \zeta_\ve|^2\big)^{\frac12}
\big(\int_{\partial B_{r_1}(\x_0)}|\zeta_\ve|^2\big)^{\frac12}
+C\big(\int_{B_{r_1}(\x_0)}|\nabla \omega_\ve|^{4}\big)^{\frac 12}
\big(\int_{B_{r_1}(\x_0)}|\zeta_\ve|^{4}\big)^{\frac12}\\
&+C\big(\int_{ B_{r_1}(\x_0)}|\tau_\ve|^2\big)^{\frac12}\big(\int_{B_{r_1}(\x_0)}|\zeta_\ve|^2\big)^{\frac12}\leq C\ve.
\end{split}
\eeq
Therefore we have that
\beq
\frac{\zeta_\ve^2}{\ve^2}\rightarrow 0\ \ \mbox{in }\ \ L^1(B_{r_1}(\x_0)).
\eeq
This completes the proof.
\endpf

\medskip
Now we define the concentration set  by
\begin{equation}
  \Sigma:=\bigcap_{r>0}\big\{ x\in \Omega:\liminf_{k\to \infty}\int_{B_{r}(\x)}\big(\frac12|\nabla {v}^\ve|^2+\frac{1}{\varepsilon^2}\chi\big(\dist^2({v}^\ve, \mathcal N)\big)\big)> \delta_0^2 \big\},
  \label{}
\end{equation}
where $\delta_0>0$ is  given in Lemma \ref{lemma:biGL}.  We have

\begin{lemma}\label{lemma2.2}  $\Sigma$ is a finite set, and
\beq
v^\ve\rightarrow v\quad\mbox{in }\ \ H^1_{\loc} (\Omega\setminus\Sigma).
\eeq
\end{lemma}
The finiteness of $\Sigma$ follows from a simple covering argument, see also \cite{linwang2016weakthreedim} and \cite{kortum2019concentrationcancellation}.

\section{Convergence of Ginzburg-Landau approximation}

The section is devoted to the proof of Theorem \ref{thm3}. First, recall from the  global energy inequality for \eqref{NGL} that for almost every $t\in(0,T)$,
\begin{equation}
\begin{split}
  \int_{\Omega\times\left\{ t \right\}}\big( |u^\varepsilon|^2+|\nabla v^\varepsilon|^2+\frac{1}{\varepsilon^2}\chi\big(\dist^2({v}^\ve, \mathcal N)\big)\big)
  +2\int_{Q_t}\big( |\nabla u^\varepsilon|^2+\left| v^\varepsilon_t+u^\varepsilon\cdot \nabla v^\varepsilon\right|^2
  \big)\le E_0.
\end{split}
  \label{GLglobalenergy}
\end{equation}
This, combined with the equation \eqref{NGL}, implies that there exists $p>2$ such that
\begin{equation}
  \begin{split}
    \sup_{\ve>0}\left[ \left\|u^\ve_t\right\|_{L_t^2 H_x^{-1}+L_t^2 W^{-2, p}}+\left\| v^\ve_t\right\|_{L^{4/3}_t L_x^{4/3}} \right]<\infty.
  \end{split}
  \label{estut}
\end{equation}
Hence, by Aubin-Lions' Lemma,  there exists $u\in L_t^\infty L_x^2 \cap L_t^2 H_x^1(\Omega\times(0,T), \R^2)$ and $v\in L_t^\infty H_x^1\cap L_t^\infty H_x^1(\Omega\times (0,T), \N)$ such that after taking a subsequence,
\begin{equation}\notag
    (u^\ve, v^\ve)\rightarrow (u, v) \text{ in }L^2(\Omega\times(0, T)), \quad
    (\nabla u^\ve, \nabla v^\ve)\rightharpoonup (\nabla u, \nabla v)\text{ in  }L^2(\Omega\times (0,T)).
\end{equation}
Combining this with \eqref{GLglobalenergy}, we obtain
 $$
 v^\ve_t+u^\ve\cdot \nabla v^\ve\rightharpoonup v_t+u\cdot \nabla v\ \ \text{in }\ \ L^2(\Omega\times (0,T)).
 $$

By the lower semi-continuity, we have
\begin{equation}
  \begin{split}
    \int_{Q_t}(|\nabla u|^2+|v_t+ u\cdot \nabla v|^2)\le \liminf_{\ve\to 0}\int_{Q_t}(|\nabla u^\ve|^2+|v^\ve_t+u^\ve\cdot \nabla v^\ve|^2)<\infty.
  \end{split}
  \label{}
\end{equation}
By Fatou's Lemma, we have
\begin{equation}
  \begin{split}
    \int_{0}^{t}\liminf_{\ve\to0}\int_\Omega (|\nabla u^\ve|^2+|v^\ve_t+u^\ve\cdot \nabla v^\ve|^2)\le \liminf_{\ve\to 0}\int_{0}^{t}\int_\Omega
    (|\nabla u^\ve|^2+| v^\ve_t+u^\ve\cdot \nabla v^\ve|^2)\leq E_0.
  \end{split}
  \label{}
\end{equation}
Hence there exists $A\subset[0,T]$ with full Lebesgue measure $T$ such that for any $t\in A$
\beq
\big(u^\ve(t), v^\ve(t)\big)\rightharpoonup (u(t),v(t))\ \quad\mbox{in }\ \ L^2\times H^1
\eeq
and
\beq
\liminf\limits_{\ve\rightarrow 0^+}\int_{\Omega}\left(|\nabla u^\ve|^2+|v^\ve_t+u^\ve\cdot\nabla v^\ve|^2\right)(t)<\infty.
\eeq
Now we define the concentration set at $t$ by
\begin{equation}\label{sigma}
  \Sigma_t:=\bigcap_{r>0}\Big\{ \x\in \Omega:\liminf_{\ve\to 0}\int_{B_r(\x)\times\left\{ t \right\}}\frac{1}{2}|\nabla v^\ve|^2+\frac{1}{\varepsilon^2}\chi\big(\dist^2({v}^\ve, \mathcal N)\big)>\delta_0^2 \Big\},
\end{equation}
where $\delta_0$ is given by Lemma \eqref{lemma:biGL}.
By Lemma \ref{lemma2.2}, it holds $\#(\Sigma_t)\le C(E_0)$ and
\begin{equation*}
  v^\ve(t)\to v(t) \text{ in }H_{\loc}^1(\Omega\setminus \Sigma(t)).
\end{equation*}

We would first show that $v$ is a weak solution of \eqref{NEL}$_3$ by utilizing the geometric structure as in \cite{ChenStruwe89} (see also \cite{lin2008analysis}). First notice that
there exists a unit vector $\nu_{\N}^\ve\perp T_{\Pi_{\N}(v^\ve)}\N$ such that
\beq\notag
\frac{d}{d{v}}\chi\big(\dist^2({v}^\ve, \N)\big)=2\chi'(\dist^2(v^\ve, \mathcal{N}))\dist({v}^\ve, \N)\nu_{\N}^\ve.
\eeq
Thus for any $\phi\in C^\infty_0(\Omega,\R^L)$ and a.e. $t\in (0,\infty)$ it holds
\beq\notag
\int_{\Omega\times\{t\}}\langle v^\varepsilon_t+u^\varepsilon\cdot\nabla v^\varepsilon-\Delta v^\varepsilon,
D\Pi_{\N}(\Pi_{\N}(v^\varepsilon))\phi\rangle=0.
\eeq
If we choose $\phi\in C^\infty_0(\Omega\setminus\Sigma_t)$,
then it follows from
$\nabla v^\varepsilon\rightarrow \nabla v$ in $H^1_{\rm{loc}}(\Omega\setminus\Sigma_t)$
that, after passing to the limit of the above equation,
\beq\notag
\int_{\Omega\times\{t\}}\langle v_t+u\cdot\nabla v, D\Pi_{\N}(v)\phi\rangle=
-\int_{\Omega\times\{t\}}\langle\nabla v, \nabla(D\Pi_{\N}(v))\phi\rangle.
\eeq
This implies that
$$
v_t+u\cdot \nabla v-\Delta v=A_{\N}(v)(\nabla v,\nabla v)
$$
holds weakly in $\Omega\setminus\Sigma_t$. Since $\Sigma_t$ is a finite set, it also holds
weakly in $\Omega$ so that \eqref{NEL}$_3$ holds.

Now, we proceed to verify $u$ satisfies \eqref{NEL}$_1$. First by the estimate \eqref{estut}, we have
$$u^\ve_t\rightharpoonup u_t,\quad \mbox{in }
L^2([0,T], H^{-1})\cap L^2([0,T], W^{-2, p})$$
for some $p>2$.
For any $\xi\in C^\infty([0,T])$ with $\xi(T)=0$, $\varphi\in {\bf J}$, since
\beq\notag
\int_{Q_T}u^\ve_t\xi\varphi=-\int_{\Omega}u_0\xi(0)\varphi-\int_{Q_T}u^\ve\xi'\varphi,
\eeq
which, after taking $\epsilon\rightarrow 0$,  implies that
 $$
 \label{ibpvt}
\int_{Q_T}u_t\xi\varphi=-\int_{\Omega}u_0\xi(0)\varphi-\int_{Q_T}u\xi'\varphi.
$$

\medskip
\noindent{\it Claim:  For any $t\in A$, it holds
\begin{equation}
  \begin{split}
    &0=\int_{\Omega\times\left\{ t \right\}}\left\langle  \pa_t u^\ve, \varphi \right\rangle+\int_{\Omega\times\left\{ t \right\}}\left\langle u^\ve\cdot \nabla u^\ve, \varphi \right\rangle+\int_{\Omega\times \left\{ t \right\}}\left\langle \nabla u^\ve, \nabla \varphi \right\rangle+\int_{\Omega\times\{t\}} (\nabla v^\ve\odot \nabla v^\ve):\nabla \varphi\\
    &\ \ \to \int_{\Omega\times \left\{ t \right\}}\left\langle  u_t, \varphi \right\rangle+\int_{\Omega\times\left\{ t \right\}}\left\langle u\cdot \nabla u, \varphi \right\rangle+\int_{\Omega\times \left\{ t \right\}}\left\langle \nabla u, \nabla \varphi \right\rangle+\int_{\Omega\times\left\{ t \right\}}(\nabla v\odot \nabla v):\nabla \varphi,
  \end{split}
  \label{weakueqn}
\end{equation}
for any $\varphi\in {\bf J}$.
}
\medskip

 \noindent For this claim, it suffices to show the convergence of  Ericksen stress tensors, i.e.,
\begin{equation}\notag
  \int_{\Omega\times\left\{ t \right\}} (\nabla v^\ve\odot \nabla v^\ve):\nabla \varphi=\int_{\Omega\times\left\{ t \right\}} (\nabla v\odot \nabla v):\nabla \varphi.
  \label{weakcondd}
\end{equation}
For simplicity, we assume $\Sigma_t=\{(0,0)\}\subset\Omega$ consists of a single point at zero.
Let $\varphi\in C^\infty(\Omega, \R^2)$ be such that
$\dv \varphi=0$ and $(0,0)\in \mbox{spt}(\varphi)$.  Then
we observe that  by adding $-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2$, we have
\begin{equation}\notag
  \int_{\Omega\times\left\{ t \right\}} (\nabla v^\ve\odot \nabla v^\ve):\nabla \varphi=\int_{\Omega\times\left\{ t \right\}} \big(\nabla v^\ve\odot \nabla v^\ve-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2\big):\nabla \varphi.
  \label{weakcondd}
\end{equation}
While by direct computations, we have
\beq\label{2dsplitting}
\nabla v^\ve\odot\nabla v^\ve-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2
=\frac12\left(
\begin{array}{ll}
|\partial_x v^\ve|^2-|\partial_y v^\ve|^2, &2\langle\partial_x v^\ve,\partial_y v^\ve\rangle\\
2\langle\partial_x v^\ve,\partial_y v^\ve\rangle, &|\partial_y v^\ve|^2-|\partial_x v^\ve|^2
\end{array}
\right).
\eeq
We can assume that there are two real numbers $\alpha, \beta$ such that
\begin{equation}
  (|\pa_x\vv^\ve|^2-|\pa_y\vv^\ve|^2)dxdy\rightharpoonup (|\pa_x\vv|^2-|\pa_y\vv|^2)dxdy+\alpha \delta_{(0, 0)},
\label{convalpha}
\end{equation}
\begin{equation}
  \left\langle \pa_x\vv^\ve, \pa_y\vv^\ve \right\rangle dxdy\rightharpoonup \left\langle \pa_x\vv, \pa_y\vv \right\rangle dxdy+\beta \delta_{(0, 0)},
\label{convbeta}
\end{equation}
hold as convergence of Radon measures. Next we want to show
\begin{equation}\label{vanish}
\alpha=\beta=0.
\end{equation}

Denote
 \begin{equation}
   \Delta\vv^\ve-\frac{1}{\ve^2}\chi'\left( \dist^2(\vv^\ve, \mathcal{N})\right)\frac{d}{d\vv}\left( \dist^2(\vv^\ve, \mathcal{N}) \right)=\f^\ve:= \pa_t\vv^\ve+u^\ve\cdot \nabla\vv^\ve
   \label{}
 \end{equation}
 and
 \begin{equation*}
   e_\ve(\vv^\ve)=\frac{1}{2}|\nabla\vv^\ve|^2+\frac{1}{\ve^2}\chi\left( \dist^2(\vv^\ve, \mathcal{N}) \right).
 \end{equation*}

Now we derive the Pohozaev identity for $v^\epsilon$. For any $X\in C_0^\infty(\Omega,\mathbb R^2)$,
by multiplying the $\vv^\ve$ equation by $X\cdot \nabla \vv^\varepsilon$ and
integrating over  $B_r(0)$ we get
\begin{eqnarray}
&&\int_{\pa B_r(0)}(X^j \vv_j^\varepsilon)\cdot\big( \vv_i^\varepsilon \frac{\x^i}{|\x|} \big)-\int_{B_r(0)}X_i^j \vv_j^\varepsilon \cdot\vv_i^\varepsilon
+\int_{B_r(0)}{\rm{div}} X e_\varepsilon(v^\varepsilon)-\int_{\partial B_r(0)} e_\varepsilon(v^\varepsilon) (X\cdot
\frac{\x}{|\x|})
\nonumber\\
&&=\int_{B_r(0)}(X\cdot \nabla \vv^\varepsilon)\cdot \f^\varepsilon.
\label{poho}
\end{eqnarray}
If we choose $X(\x)=\x$, then we have
\begin{equation}
\begin{split}
r\int_{\pa B_r(0)} \left|\frac{\pa \vv^\varepsilon}{\pa r}\right|^2+\int_{B_r(0)}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))
-r\int_{\pa B_r(0)}e_\ve(\vv^\ve)=\int_{B_r(0)}|\x| \frac{\pa \vv^\varepsilon}{\pa r}\cdot \f^\varepsilon .
\end{split}
\label{}
\end{equation}
Then
\begin{equation}
\begin{split}
\int_{\pa B_r(0)}e_\ve(\vv^\ve)=\int_{\pa B_r(0)}
\left|\frac{\pa \vv^\varepsilon}{\pa r}\right|^2+\frac{1}{r}\int_{B_r(0)}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N})) +
O\big (\int_{B_r(0)}|\nabla \vv^\varepsilon||\f^\varepsilon| \big).
\end{split}
\label{}
\end{equation}
Integrating from $r$ to $R$, we have
\begin{equation}
\begin{split}
\int_{B_R(0)}e_\ve(\vv^\ve)-\int_{B_r(0)}e_\ve(\vv^\ve)&=\int_{B_R(0)\setminus B_r(0)} \left|\frac{\pa \vv^\varepsilon}{\pa r}\right|^2+\int_{r}^{R}\frac{1}{\tau}\int_{B_\tau(0)}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))d\tau\\
&+\int_{r}^{R}{O}\big( \int_{B_\tau(0)}|\nabla \vv^\varepsilon||\f^\varepsilon| \big)d\tau.
\end{split}
\label{5.10}
\end{equation}
Since $\Sigma_t=\{(0,0)\}$, we can assume that
\begin{equation}
e_\ve(\vv^\ve)d\x\rightharpoonup \frac12|\nabla\vv|^2 d\x+\gamma \delta_{(0, 0)}, \qquad \text{ in }\ \ B_\delta(0)
\label{}
\end{equation}
as convergence of Radon measures,  where $\gamma\ge 0$. Since $t\in A$,
\begin{align*}
\lim_{\varepsilon\to 0}\int_{B_\tau(0)}|\f^\varepsilon||\nabla \vv^\varepsilon|&\le \lim_{\varepsilon\to 0}\big( \int_{B_\tau(0)}|\f^\varepsilon|^2 \big)^{\frac{1}{2}}\big( \int_{B_\tau(0)}|\nabla \vv^\varepsilon|^2 \big)^\frac{1}{2}\le C E_0,
\end{align*}
Hence, by sending $\varepsilon\to 0$ we obtain from \eqref{5.10} that
\begin{align*}
\int_{B_R(0)\setminus B_r(0)} \frac12|\nabla\vv|^2&\ge \int_{B_R(0)\setminus B_r(0)}\left|\frac{\pa\vv}{\pa r}\right|^2
+\int_{r}^{R}\frac{1}{\tau}\lim_{\ve\to0}\int_{B_\tau(0)}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N})) d\tau
+{O}(R).
\end{align*}
Sending $r\to 0$, we have
\begin{align*}
\int_{B_R(0)} \frac{1}{2}|\nabla\vv|^2&\ge \int_{B_R(0)} \left|\frac{\pa \vv}{\pa r}\right|^2
+\int_{0}^{R}\frac{1}{\tau}
\lim_{\ve\to0}\int_{B_\tau(0)}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N})) d\tau+{O}(R).
\end{align*}
From this, we claim that
\begin{equation}
\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\to 0\ \  \text{ in }\ \ \ L^1(B_\delta).
\label{biaxialL1strongconv}
\end{equation}
For, otherwise,
\begin{equation*}
\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\,dx\rightharpoonup  \kappa \delta_{(0,0)}
\end{equation*}
for some $\kappa>0$, this implies
\begin{equation*}
\int_{0}^{R}\frac{1}{\tau}\lim_{\ve\to0}\int_{B_\tau}\frac{2}{\ve^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))=\int_{0}^{R}\frac{\kappa}{\tau}d\tau=\infty,
\end{equation*}
which is impossible.

Choosing $X(\x)=(x, 0)$ in \eqref{poho}, we obtain that
\begin{eqnarray}\label{hopf1}
&&\frac12\int_{B_r(0)}\Big( \big|{\pa_y \vv^\ve}\big|^2-\big|{\pa_x \vv^\ve}\big|^2 \Big)
+\int_{B_r(0)} \frac{1}{\varepsilon^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\nonumber\\
&&=\int_{B_r(0)} x\langle{\partial_x v^\ve}, \f^\ve\rangle +\int_{\partial B_r(0)} \frac{x^2}{r} e_\ve(v^\ve)
-\int_{\partial B_r(0)} x\langle{\partial_x v^\ve}, \frac{\pa v^\ve}{\pa r}\rangle.
\end{eqnarray}
Observe that by Fubini's theorem, for a.e. $r>0$ it holds that
\begin{equation*}
\begin{split}
\int_{\partial B_r(0)} x\langle{\partial_x v^\ve}, \frac{\pa v^\ve}{\pa r}\rangle
&\to \int_{\pa B_r(0)} x\langle{\pa_x \vv}, \frac{\pa \vv}{\pa r}\rangle, \\
\int_{\pa B_r(0)}\frac{x^2}{r}e_\varepsilon(\vv^\ve) &\to \frac{1}{2}\int_{\pa B_r}\frac{x^2}{r}|\nabla \vv|^2,
\end{split}
\end{equation*}
and by \eqref{biaxialL1strongconv},
\begin{equation*}
\int_{B_r(0)}\frac{1}{\varepsilon^2}\chi(\dist^2(\vv^\ve, \mathcal{N}))\to 0.
\end{equation*}
Furthermore,
\begin{align*}
\big|\int_{B_r(0)}x\langle{\pa_x \vv^\ve},  \f^\ve\rangle\big|
&\le Cr\left\|\f^\ve\right\|_{L^2}\left\|\nabla \vv^\ve\right\|_{L^2}={O}(r).
\end{align*}
Hence, by sending $\varepsilon\to 0$ in \eqref{hopf1}, we obtain
\begin{equation*}
\int_{B_r(0)} \big( \big|{\pa_y \vv}\big|^2-\big|{\pa_x \vv}\big|^2 \big)+\alpha={O}(r),
\end{equation*}
this further implies $\alpha=0$ after sending $r\to 0$.

Similarly, if we choose $X(\x)=(0, x)$ in \eqref{poho} and pass the limit in
 the resulting equation, we can  get that
\begin{equation*}
\int_{B_r(0)}\big\langle {\pa_x \vv}, {\pa_y \vv}\big\rangle+\beta ={O}(r).
\end{equation*}
Hence $\beta=0$. This proves \eqref{vanish} and hence completes the proof of Claim.

\medskip
Multiplying \eqref{weakueqn} by $\xi\in C^\infty([0,T])$ with $\xi(T)=0$ and integrating over $[0,T]$, we conclude that $u$ satisfies the \eqref{NEL}$_1$ on $Q_T$.
The proof of Theorem \ref{thm3} is complete.

\section{Compactness of simplified Ericksen-Leslie system}

This section is devoted to prove Theorem \ref{thm1}. First notice that since the sequence of weak solutions $(u^k, v^k)$
satisfies the assumption \eqref{kengbd}, and
\beq\notag
(u^k_0,v^k_0)\rightharpoonup (u_0, v_0),\quad \mbox{in } L^2(\Omega)\times H^1(\Omega),
\eeq
 there exists $(u(\x,t), v(\x,t)):\Omega\times (0,T)\rightarrow \R^2\times\N$ such that
\beq\label{engcon1}
(u^k, v^k)\rightharpoonup (u,v) \ \mbox{in }\ L^2([0,T],H^1(\Omega)),
\eeq
\beq\label{engcon2}
v^k_t+u^k\cdot \nabla v^k\rightharpoonup v_t+u\cdot \nabla v\ \ \text{in }\ L^2([0,T], L^2(\Omega)).
\eeq
Also it follows from  \eqref{NEL} and \eqref{kengbd} that there exists $p>2$ that
\begin{equation}
  \begin{split}
    &\sup_{k} \Big[\big\|u^k_t\big\|_{L_t^2 H_x^{-1}+L_t^2 W^{-2, p}_x}+\big\|v^k_t\big\|_{L_t^{2}H_x^{-1}}\Big]<\infty.
  \end{split}
  \label{}
\end{equation}
Hence, by Aubin-Lions' Lemma we have that
\begin{equation*}
  (u^k, v^k)\to (u, v) \text{ in } L^2(Q_T)\times L^2(Q_T).
\end{equation*}

By the lower semi-continuity, we have
\beq\notag
\begin{split}
\int_{Q_t}\left(|\nabla u|^2+|v_t+u\cdot\nabla v|^2\right)
\leq \liminf\limits_{k\rightarrow\infty}\int_{Q_t}\left(|\nabla u^k|^2+|v^k_t+u^k\cdot\nabla v^k|^2\right)\leq C_0.
\end{split}
\eeq
By Fatou's Lemma and \eqref{kengbd}, we have
\beq\notag
\begin{split}
\int_0^t\liminf\limits_{k\rightarrow\infty}\int_{\Omega}\left(|\nabla u^k|^2+|v^k_t+u^k\cdot\nabla v^k|^2\right)
\leq \liminf\limits_{k\rightarrow\infty}\int_{Q_t}\left(|\nabla u^k|^2+|v^k_t+u^k\cdot\nabla v^k|^2\right)\leq C_0.
\end{split}
\eeq
Hence, there exists $A\subset[0,T]$ with full Lebesgue measure $T$, such that for all $t\in A$
\beq
\big(u^k(t), v^k(t)\big)\rightharpoonup (u(t),v(t)),\quad\mbox{in }L^2(\Omega)\times H^1(\Omega)
\eeq
and
\beq
\liminf\limits_{k\rightarrow\infty}\int_{\Omega}\left(|\nabla u^k|^2+|v^k_t+u^k\cdot\nabla v^k|^2\right)(t)<\infty.
\eeq
Now we define the concentration set  at time $t\in (0,T]$ by
\begin{equation}
  \Sigma_t:=\bigcap_{r>0}\Big\{ x\in \Omega:\liminf_{k\to \infty}\int_{B_r(x)}|\nabla v^k|^2> \delta_0^2 \Big\}
  \label{}
\end{equation}
where $\delta_0$ is small constant given by Theorem 1.2 in \cite{SharpTopping2013}.
As in \cite{SharpTopping2013} (see also \cite{linwang2016weakthreedim}, \cite{wang1999remark}),
we can show that for any $t\in A$, it holds that $\#(\Sigma_t)\leq C(E_0)$ and
\beq
v^k(t)\rightarrow v\ \quad\mbox{in }\ \ H^1_{\loc} (\Omega\setminus\Sigma_t).
\eeq

Similar to the proof of Theorem \ref{thm3}, we can show the weak limit $(u,v)$ satisfies the third equation of \eqref{NEL} in the weak sense. It remains to show that the first equation of \eqref{NEL} is also valid in the weak sense.

Similar to the proof of Theorem \ref{thm3}, to complete the proof of Theorem \ref{thm1}, it is suffices to show
\beq\label{H1conv}
\lim\limits_{k\rightarrow\infty}\int_{\Omega\times\{t\}}\left(\nabla v^k\odot\nabla v^k\right):\nabla \varphi=\int_{\Omega\times\{t\}}\left(\nabla v\odot\nabla v\right):\nabla \varphi, \ \forall\varphi\in \mathbf{J}.
\eeq
For simplicity, assume $\Sigma_t=\{(0,0)\}\subset\Omega$.
Let $\varphi\in C^\infty(\Omega, \R^2)$ be such that $\dv \varphi=0$ and $(0,0)\in \mbox{spt}(\varphi)$.
By the same calculation as in \eqref{2dsplitting}, we have
\beq\notag
\nabla v^\ve\odot\nabla v^\ve-\frac{1}{2}|\nabla v^\ve|^2\mathbb I_2
=\frac12\left(
\begin{array}{ll}
|\partial_x v^\ve|^2-|\partial_y v^\ve|^2, &2\langle\partial_x v^\ve,\partial_y v^\ve\rangle\\
2\langle\partial_x v^\ve,\partial_y v^\ve\rangle, &|\partial_y v^\ve|^2-|\partial_x v^\ve|^2
\end{array}
\right).
\eeq

For any $t\in A$, $v^k(t)$ is an approximated harmonic maps from $\Omega$ to $\N$:
\begin{equation}
\Delta v^k(t)+A(v^k)(\nabla v^k,\nabla v^k)=g^k(t):=v^k_t(t)+u^k\cdot\nabla v^k(t)\in L^2(\Omega).
\end{equation}
Recall the Hopf differential of $v^k$ is defined by
\beq
\mathcal H^k=\big(\frac{\partial v^k}{\partial z}\big)^2
=\big|{\partial _xv^k}\big|^2-\big|{\partial_y v^k}\big|^2
+2i\big\langle{\partial_x v^k}, {\partial_y v^k}\big\rangle,
\eeq
where $z=x+iy\in \mathbb C$.
Then
\beq
\frac{\partial {{\mathcal{H}}}^k}{\partial\bar z}=2\frac{\partial_x v^k}{\partial z}\frac{\partial^2 v^k}{\partial \bar z\partial z}
=2\Delta v^k\frac{\partial v^k}{\partial z}
=2g^k(t)\cdot\frac{\partial v^k}{\partial z}:=G^k.
\eeq
It is clear that
\beq
\|G^k\|_{L^1(B_r)}\leq 2\|g^k(t)\|_{L^2(\Omega)}\big\|\frac{\partial v^k}{\partial z}\big\|_{L^2(\Omega)}
\leq 2C_0.
\eeq
Therefore, for any $z\in B_r(0)$
\beq
{{\mathcal{H}}}^k(z)=\int_{\partial B_{2r}(0)}\frac{{{\mathcal{H}}}^k(\omega)}{z-\omega}\,d\sigma+\int_{B_{2r}(0)}\frac{G^k(\omega)}{z-\omega}\,d\omega.
\eeq
By the Young inequality of convolutions, we obtain
\beq
\|{{\mathcal{H}}}^k\|_{L^p(B_r)}
\leq C(r, p) \|\mathcal{H}^k\|_{L^1(\partial B_{2r})}+\big\|\frac{1}{z}\big\|_{L^p}\|G^k\|_{L^1( B_{2r})}\leq C(r, p).
\eeq
for any $1<p<2$.
From this, we immediately conclude that
\beq\notag
|\partial_x v^k|^2-|\partial_yv^k|^2\rightharpoonup
|\partial_x v|^2-|\partial_yv|^2, \quad \langle\partial_x v^k,\partial_yv^k\rangle\rightharpoonup  \langle\partial_x v,\partial_y v\rangle\ \quad \mbox{in }\ \ L^p(B_r(0))
\eeq
for any $1<p<2$, which implies \eqref{H1conv}.
This completes the proof of Theorem \ref{thm1}.

\bigskip
\bigskip
\noindent{\bf Acknowledgments.} The first and third author are partially supported by NSF DMS 1764417.

config: arxiv


TITLE: What is the series to converge with $1/x$ from $(1,\infty)$?
QUESTION [1 upvotes]: I'm trying to find an alternative series of polynomials that can pssibly converge with $\frac{1}{x}$.
So far I know that the taylor series for $\frac{1}{x}$ is, as  should be known, $$\sum_{n=0}^{\infty} (-1)^n(-1+x)^n$$
However this only converges from $-1<x<1$
Perhaps there is a series of polynomials that can converge with $1/x$ from $(1,\infty)$. If so what could it be.

REPLY [3 votes]: If you can find a sequence of polynomials $q_n(x)$ that converge to $1/x$ on $(1,\infty)$, you can take the series $\sum_{n=1}^\infty p_n(x)$ where
$p_1(x) = q_1(x)$ and $p_n(x) = q_n(x) - q_{n-1}(x)$ for $n \ge 2$.
One possibility is to take $q_n$ to be a suitable Taylor polynomial for
$1/x$ centred at $x=n$, so that $|q_n(x) - 1/x|$ is small for $1 < x < 2n-1$.


TITLE: $A^n=0$ if $a_{ij}=0$ for all $i\geq j$.
QUESTION [0 upvotes]: Prove: $A^n=0$ if $a_{ij}=0$ for all $i\geq j$ where $A\in M_n(K)$.
I tested it for a few matrices and I guess I have to do it by induction.  How can I do that?

REPLY [1 votes]: $\newcommand{\R}{\mathbb{R}}\newcommand{\N}{\mathbb{N}}$Let me extend my comment into an answer.
As you may have noticed in your example calculations, for every power of the original matrix $A$ another off-diagonal full of zeros appears.
So here is the claim which we will prove by fighting against a battalion of indices (the argument given in the other two answers is much more elegant!).

Lemma:
Suppose $A\in\R^{n\times n}$ is such that $A_{ij}=0$ for all $i\geq j$.
Then for every $k\in \N$ the powers $A^k$ have entries $(A^k)_{ij}=0$ for all $i+k-1\geq j$.
Proof: The case $k=1$ is covered by the assumption. So suppose the claim is true for $k$ and lets show it for $k+1$. Let $B=A^k$ and calculate for $i+(k+1)-1\geq j$
$$(A^{k+1})_{ij}=(BA)_{ij}=\sum_{l=1}^nB_{il}A_{lj}.$$
Now we know that $A_{lj}=0$ if $l\geq j$ so the sum actually only runs until $l=j-1$. On the other hand we know by induction that $B_{il}=0$ if $i+k-1\geq l$ so the sum actually only starts at $l=i+k$. Hence
$$(A^{k+1})_{ij}=\sum_{l=i+k}^{j-1}B_{il}A_{lj}.$$
But by assumption $i+k\geq j$ and so this is actually an empty sum and equals $0$.
I'm sure one could write this down more elegantly without having to use an empty sum.

Since $i+n-1\geq j$ is true for all $i,j\in\{1,\ldots,n\}$ we have the result you actually wanted to prove.
Corollary:
If $A\in\R^{n\times n}$ is as above then $A^n=0$.


text:
\section{Introduction}

The problem of interest of this paper is the convex optimization problem
\begin{align}
	\label{eq:problem}
	f^*:=\min_{x\in \cX}f(x)
\end{align}
where $\cX\in\R^n$ is a convex compact set and $f:\R^n\to\R$ is a convex differentiable function such that
\begin{align}
	\label{eq:L}
	\|\nabla f(x) - \nabla f(y)\|\le L\|x - y\|,\ \forall x,y\in\cX.
\end{align}
Here $\|\cdot\|$ is the Euclidean norm.
Our goal is to compute an $\varepsilon$-approximate solution $y$ such that $f(y) - f^*\le\varepsilon$ using first-order information, namely, the function and gradient values $f$ and $\nabla f$.

When designing numerical methods that uses first-order information, the structures of the objective function and the compact feasible set $\cX$ have significant impact on the theoretical convergence theory and practical computing performance. As an example, consider the performance of two classical iterative numerical methods on a simple instance of problem \eqref{eq:problem} in which $\cX:=\{(x^{(1)},x^{(2)})\in\R^2|x^{(1)}+x^{(2)}=1,x^{(1)},x^{(2)}\ge 0\}$ is a line segment in $\R^2$ and $f(x)=\|x\|^2/2$. The first numerical method is the projected gradient method with iterates
\begin{align}
	x_k = y\km - (1/\beta_k)\nabla f(y\km),\ y_k=\text{Proj}_{\cX}(x_k).
\end{align}
In each iteration, the projected gradient method moves along the negative gradient direction with a pre-specified stepsize $(1/\beta_k)$ to obtain an updated iterate $x_k$. Such update $x_k$ may fall outside of the feasible region, so we maintain feasibility by projecting $x_k$ onto the feasible set to obtain a new approximate solution $y_k$. The second numerical method is the conditional gradient method \cite{frank1956algorithm}
\begin{align}
	\label{eq:cg}
	x_k = \min_{x\in\cX}\langle\nabla f(x\km),x\rangle,\ y_k = \sum_{i=1}^{k}{\lambda^i_k}x_i.
\end{align}
Here we compute an update $x_k$ from a linear optimization over the line segment $\cX$. Note that $x_k$ will always be selected from the two extreme points $(1,0)^\top$ and $(0,1)^\top$. To avoid oscillating between the extreme points, we select the new approximate solution $y_k$ to a convex combination of all previous updates $x_i$'s. Both the aforementioned methods have been extensively studied in the literature. See, e.g., the monograph \cite{bubeck2015convex} or the book \cite{lan2020first} for the survey of both methods.

\begin{figure}[!h]
	\centering
	\scalebox{.72}{
		\begin{tikzpicture}[>=latex]
		\begin{axis}[
		unit vector ratio*=1 1 1,
		axis x line=center,
		axis y line=center,
		xtick={0,.5,1},
		ytick={0,.5,1},
		xlabel={$x^{(1)}$},
		ylabel={$x^{(2)}$},
		xlabel style={below right},
		ylabel style={above left},
		xmin=-.1,
		xmax=1.1,
		ymin=-0.1,
		ymax=1.1]
		\addplot [mark=none,domain=0:1] {1-x};
		\addplot [mark=*, ultra thick] coordinates {(.5, .5)};

		\addplot[color=red,-stealth, thick] coordinates {
			(0, 1)
			(0, .5)
		};
		\addplot[color=red,-stealth, dashed, thick] coordinates {
			(0, .5)
			(.25, .75)
		};
		\addplot[color=red,-stealth, thick] coordinates {
			(.25, .75)
			(0.125, .375)
		};
		\addplot[color=red,-stealth, dashed, thick] coordinates {
			(0.125, .375)
			(.375, .625)
		};
		\addplot[color=red,-stealth, thick] coordinates {
			(.375, .625)
			(.1875, .3125)
		};
		\addplot[color=red,-stealth, dashed, thick] coordinates {
			(.1875, .3125)
			(.4375, .5625)
		};

		\addplot[color=blue,-stealth, ultra thick, mark=*] coordinates {
			(0, 1)
			(.25, .75)
		};
		\addplot[color=blue,-stealth, ultra thick, mark=*] coordinates {
			(.25, .75)
			(.375, .625)
		};
		\addplot[color=blue,-stealth, ultra thick, mark=*] coordinates {
			(.375, .625)
			(.4375, .5625)
		};

		\end{axis}
		\end{tikzpicture}
		\begin{tikzpicture}[>=latex]
		\begin{axis}[
		unit vector ratio*=1 1 1,
		axis x line=center,
		axis y line=center,
		xtick={0,.5,1},
		ytick={0,.5,1},
		xlabel={$x^{(1)}$},
		ylabel={$x^{(2)}$},
		xlabel style={below right},
		ylabel style={above left},
		xmin=-.1,
		xmax=1.1,
		ymin=-0.1,
		ymax=1.1]
		\addplot [mark=none,domain=0:1] {1-x};
		\addplot [mark=*, ultra thick] coordinates {(.5, .5)};

		\addplot[color=blue,-stealth, ultra thick, mark=none] coordinates {
			(0, 1)
			(1, 0)
		};
		\addplot[color=blue,-stealth, ultra thick, mark=none, smooth] coordinates {
			(1.02, 0.02)
			(0.7367, 0.4033)
			(.3533, .6867)
		};
		\addplot[color=blue,-stealth, ultra thick, mark=none, smooth] coordinates {
			(.3133, .6467)
			(.44, .44)
			(.6467, 0.3133)
		};

		\addplot[color=red,-stealth, thick, mark=none, smooth] coordinates {
			(1.02, 0.02)
			(0.7667, 0.4333)
			(0.02, 1.02)
		};
		\addplot[color=red,dashed,-stealth, thick, mark=none] coordinates {
			(0.02, 1.02)
			(0.3533, 0.6867)
		};
		\addplot[color=red,-stealth, thick, mark=none, smooth] coordinates {
			(.3133, .6467)
			(.41, .41)
			(.98, -.02)
		};
		\addplot[color=red,dashed,-stealth, thick, mark=none] coordinates {
			(.98, -.02)
			(0.6467, 0.3133)
		};

		\addplot [mark=*, ultra thick, color=blue] coordinates {(1, 0)};
		\addplot [mark=*, ultra thick, color=blue] coordinates {(.3333, .6667)};
		\addplot [mark=*, ultra thick, color=blue] coordinates {(.6667, 0.3333)};

		\end{axis}
		\end{tikzpicture}
	}
	\caption{\label{fig:simple_plot} \footnotesize Left: Project gradient method iterations. Blue arrows describe update of approximate solutions, red solid arrows describe gradient descent moves, and red dashed arrows describe projections onto $\cX$.
		Right: Conditional gradient method iterations. Blue arrows describe update of approximate solutions, red solid arrows describe the updates $x_k$ (always extreme points), and red dashed arrows describe the convex combination weighting process.
	}
\end{figure}

The performance of the projected gradient and conditional gradient methods are illustrated in Figure \ref{fig:simple_plot}. Both methods start at point $(0,1)^\top$. In the projected gradient method we select the constant stepsize $1/\beta_k\equiv 0.5$ and in the conditional gradient method we select the weights $\lambda_i = 2i/(k(k+1))$. From Figure \ref{fig:simple_plot} we can observe that approximate solutions of both methods are gradually approaching the optimal solution. The ones produced by the projected gradient method converge faster comparing to the ``back and forth'' behavior (see the blue arrows in the right plot) from the conditional gradient method; this is because the projected gradient method benefits from the structure of the objective function (i.e., smoothness and strong convexity). However, in the conditional gradient method iterations we only need to solve a linear program, while the projected gradient method needs to compute  projections onto $\cX$. If we extend the aforementioned simple example to the general case in where $\cX$ is a high dimensional general convex polytope, then solving linear programs would likely be more preferable than solving projections onto general convex polytopes.

In fact, the above simple example reflects the difference between the projected gradient and conditional gradient methods in terms of their theoretical convergence theory. For our problem of interest \eqref{eq:problem}, the accelerated version of projected gradient method (also known as Nesterov's optimal method; see \cite{nesterov1983method,nesterov2004introductory}) is able to compute an $\varepsilon$-approximate solution with $\MO(\sqrt{L/\varepsilon})$ gradient evaluations and projections, and the conditional gradient method will need $\MO(L/\varepsilon)$ gradient evaluations and linear objective optimizations \cite{harchaoui2015conditional,jaggi2013revisiting}. Here $L$ is the Lipschitz constant defined in \eqref{eq:L}. It should be noted that both convergence properties are not improvable in the following sense: for any numerical method that uses only first-order information for solving problem \eqref{eq:problem} with large number of decision variables $n$, it requires at least $\MO(\sqrt{L/\varepsilon})$ gradient computation to compute an $\varepsilon$-solution \cite{nemirovski1992information} (see also \cite{nemirovski1994efficient,nesterov2004introductory}); for any numerical method that defines a linear objective function and solve a linear objective optimization in each iteration, it requires at least $\MO(L/\varepsilon)$ iterations to compute an $\varepsilon$-solution \cite{lan2013complexity} to problem \eqref{eq:problem} with large $n$.

The conditional gradient methods are still preferred in many practices due to its advantage of not requiring the projection computation, even though its requirement of $\MO(L/\varepsilon)$ gradient computations is not as good as the theoretical limit $\MO(\sqrt{L/\varepsilon})$. For problems with sophisticated feasible set $\cX$, the possibly expensive computational time of projection operator can significantly outweigh the theoretical advantage of the smaller $\MO(\sqrt{L/\varepsilon})$ gradient evaluation of any projection-based methods.

Recently, there has been a breakthrough in the theory of conditional gradient methods. A condition gradient sliding (CGS) method is proposed in \cite{lan2016conditional} that is able to solve an $\varepsilon$-approximate solution of problem \eqref{eq:problem} with $\MO(\sqrt{L/\varepsilon})$ gradient evaluations and $\MO(L/\varepsilon)$ linear objective optimization subproblems. It should be pointed out that the CGS method still obeys the aforementioned theoretical performance limits of numerical methods that uses first-order information. In fact, it is the first conditional gradient-type method that reaches the theoretical performance limit. The key concept of the CGS method is to use the conditional gradient method to approximately solve the projection subproblem in the accelerated projected gradient method. With proper precision on solving the projection subproblem, the CGS method is able to reduce the number of gradient evaluations from the required $\MO(L/\varepsilon)$ of the original conditional gradient method to a improve order of $\MO(\sqrt{L/\varepsilon})$.

In the original CGS method in \cite{lan2016conditional}, the knowledge of the Lipschitz constant $L$ and the number of gradient evaluations $N$ are required for implementation\footnote{Some parameter settings of the CGS method does not require $N$; see, e.g., Corollary 2.3 in \cite{lan2016conditional}. However, since no termination criterion is proposed in \cite{lan2016conditional}, to terminate we still need to specify the total number of iterations $N$. It should be noted that we can use some termination criterion for the CGS method, e.g., the Wolfe gap, which we will use for CGS in the numerical experiments of our paper; however, the theoretical convergence property in terms of the number of iterations needed to achieve small Wolfe gap is different from the properties of the CGS method, and may deteriorate its practical performance significantly.}. Such requirements lead to two disadvantages in practice. First, in order to make sure that a constant $L$ satisfies the Lipschitz condition \eqref{eq:L}, we will need to choose a constant $L$ that satisfies the Lipschitz condition \eqref{eq:L} for all pairs $x$ and $y$ in $\cX$. Computing such $L$ can be difficult; the computed $L$ can also be too conservative and lead to worse practical performance. Second, in order to compute an $\varepsilon$-approximate solution, we need to either tune the number of gradient evaluations $N$ in practice or follow its theoretical property and specify a possibly conservative $N=\MO(L/\sqrt{\varepsilon})$. While the CGS method reaches the theoretical performance limits, such disadvantages may deteriorate its practical performance significantly.

In this paper, we propose a modification of the CGS method that allows for its practical implementation. Our proposed method, called the CGS with linesearch (CGS-ls), performs a backtracking linesearch strategy to gradually increase the initial guess of Lipschitz constant $L_0$ to values that satisfy the convergence condition. The initial guess $L_0$ does not need to satisfy the Lipschitz condition \eqref{eq:L} and can be significantly smaller than the actual Lipschitz constant. We also maintains the estimate of a lower bound of $f^*$ that certificates the achievement of an $\varepsilon$-approximate solution. Consequently, our propose method does not require the knowledge of either $L$ and $N$ and is able to stop before the theoretical $\MO(\sqrt{L/\varepsilon})$ bound of required gradient evaluations or the $\MO(L/\varepsilon)$ bound of linear objective optimization subproblems. It should be noted that our theoretical analysis of the backtracking linesearch is non-trivial. In order to improve the practical implementation, add proper termination criterion, and maintain the same theoretical convergence properties as the CGS method, we need to modify some theoretical analysis in the original CGS results in \cite{lan2016conditional}. We demonstrate through numerical experiments the advantages of our proposed CGS-ls method in implementation.


text:
\begin{document}

	\title{Critical Percolation and the Incipient Infinite Cluster on Galton-Watson Trees}
	\author{Marcus Michelen\thanks{University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab. 209 South 33rd Street Philadelphia, PA
			19104-6395
}}
	\date{\vspace{-5ex}}
\maketitle

\begin{abstract}We consider critical percolation on Galton-Watson trees and prove
	quenched analogues of classical theorems of critical branching
	processes. We
	show that the probability critical percolation reaches depth $n$ is
	asymptotic to a tree-dependent constant times $n^{-1}$. Similarly,
	conditioned on critical percolation reaching depth $n$, the number of
	vertices at depth $n$ in the critical percolation cluster almost surely
	converges in distribution to an exponential random variable with mean
	depending only on the offspring distribution. The incipient infinite cluster
	(IIC) is constructed for a.e. Galton-Watson tree and we prove a limit
	law for
	the number of vertices in the IIC at depth $n$, again depending only on the
	offspring distribution. Provided the offspring distribution used to generate
	these Galton-Watson trees has all finite moments, each of these results holds
	almost-surely.
\end{abstract}

\section{Introduction} \label{sec:intro}

We consider percolation on a locally finite rooted tree $T$: each edge
is open with probability $p \in(0,1)$, independently of all others.
Let $\rtt$ denote the root of $T$ and $\CC_p$ be the open
$p$-percolation cluster of the root. We may consider the \emph
{survival probability} $\theta_T(p) := \P[|\CC_p| = +\infty]$ and
note that $\theta_T$ is an increasing function of $p$. There thus
exists a \emph{critical percolation parameter} $p_c \in[0,1]$ so that
$\theta_T(p) = 0$ for all $p \in[0,p_c)$ and $\theta_T(p) > 0$ for
$p \in(p_c,1]$. If $T$ is a regular tree where each non-root vertex
has degree $d + 1$---i.e. each vertex has $d$ children---then the
classical theory of branching processes shows that $p_c = \frac{1}{d}$
and $\theta_T(p_c) = 0$ (see, for instance, \cite{athreya-ney}).
Since critical percolation does not occur, we may consider the \emph
{incipient infinite cluster} (IIC), in which we condition on critical
percolation reaching depth $M$ of $T$ and take $M$ to infinity.

The IIC for regular trees was first constructed and considered by
Kesten in \cite{kesten-subdiffusive}. In that work, along with \cite
{barlow-kumagai}, the primary focus was on simple random walk on the
IIC for regular trees. Our focus is on three elementary quantities for
random $T$: the probability that critical percolation reaches depth
$n$; the number of vertices of $\mathcal{C}_p$ at depth $n$
conditioned on percolation reaching depth $n$; and the number of
vertices in the IIC at depth $n$. For regular trees, these questions
were answered in the study of critical branching processes. In fact,
these classical results apply to \emph{annealed} critical percolation
on Galton-Watson trees. If we generate a Galton-Watson tree $T$ with
progeny distribution $Z \geq1$ with $\E[Z] > 1$, we may perform $p_c
= 1/\E[Z]$ percolation at the same time as we generate $T$; this is
known at the \emph{annealed} process---in which we generate $T$ and
percolate simultaneously---and is equivalent to generating a
Galton-Watson tree with offspring distribution $\Zt:= \Bin(Z,p_c)$.
Since $\E[\Zt] = 1$, this is a critical branching process and thus
the classical theory can be used:

\begin{thm}[\cite{kesten-ney-spitzer}] \label{thm:annealed}
	Suppose $\E[Z^2] < \infty$, and set $Y_n$ to be the set of vertices
	at depth $n$ of $T$ connected to the root in $p_c = 1/\E[Z]$
	percolation. Then

	\begin{enumerate}

		\item[$(a)$] The annealed probability of surviving to depth $n$
		satisfies

		\[
		n \cdot\P[|Y_n| > 0] \to\frac{2}{\Var[\widetilde{Z}]} = \frac{2
			\E[Z]^2}{\E[Z(Z-1)]}\,.
		\]

		\item[$(b)$] The annealed conditional distribution of $|Y_n|/n$ given
		$|Y_n| > 0$ converges in distribution to an exponential law with mean
		$\frac{\E[Z(Z-1)]}{2\E[Z]^2}$ as $n\to\infty$.
	\end{enumerate}

\end{thm}

Under the additional assumption of $\E[Z^3] < \infty$, parts $(a)$
and $(b)$ are due to Kolmogorov \cite{kolmorogov1938} and Yaglom \cite
{yaglom} respectively; as such, they are commonly referred to as
Kolmogorov's estimate and Yaglom's limit law. For a modern treatment of
these classical results, see \cite{LPP-95} or \cite[Section
$12.4$]{LP-book}. Although less widely known, Theorem \ref
{thm:annealed} quickly gives a limit law for the size of the annealed IIC.

\begin{corollary} \label{cor:IIC-annealed}
	If $\E[Z^2] < \infty$, let $C_n$ denote the number of vertices at
	depth $n$ in the annealed incipient infinite cluster. Then $C_n / n$
	converges in distribution to the random variable with density $\lambda
	^2 x e^{-\lambda x }$ with $\lambda:=\frac{2 \E[Z]^2}{\E[Z(Z-1)]}$
	on $[0,\infty)$. In other words,

	\[
	\lim_{n \to\infty}\left(\lim_{M\to\infty}\P[ |Y_n|/n \in(a,b)
	\| |Y_M| > 0 ] \right)= \int_{a}^b \lambda^2x e^{-\lambda x}\,dx
	\]

	for each $a < b.$
\end{corollary}

This can be easily proven from Theorem \ref{thm:annealed} using an
argument similar to the proof of Theorem \ref{thm:IIC}, and thus the
details are omitted.

Our goal is to upgrade Theorem \ref{thm:annealed} and Corollary \ref
{cor:IIC-annealed} to hold for the \emph{quenched} process; that is,
rather than generate $T$ and perform percolation at the same time as in
the annealed case, we generate $T$ and then perform percolation on each
resulting $T$. Before stating the quenched results, we recall some
notation and facts from the theory of branching processes. If we allow
$\P[Z = 0] > 0$ and condition on the resulting tree being infinite, we
may pass to the reduced tree as in \cite[Chapter $5.7$]{LP-book} in
which we remove all vertices that have finitely many descendants; this
results in a new Galton-Watson process with some offspring distribution
$\tilde{Z} \geq1$. We therefore assume without loss of generality
that $Z \geq1$. For a Galton-Watson tree $T$, let $Z_n$ denote the
number of vertices at distance of $n$ from the root; then the process
$W_n = Z_n / (\E[Z])^n$ converges almost-surely to some random
variable $W$.

A first quenched result is that of \cite{lyons90}, which states that
for a.e.\ supercritical Galton-Watson tree with progeny distribution
$Z$, we have that the critical percolation probability is $p_c = 1/\E
[Z]$; furthermore, for almost every Galton-Watson tree $\TT$, $\theta
_\TT(p) = 0$ for $p \in[0,p_c]$ and $\theta_\TT(p)>0$ for $p \in
(p_c,1]$. For a fixed tree $T$, let $\P_T[\cdot]$ be the probability
measure induced by performing $p_c$ percolation on $T$. When $T$ is
random, this is a random variable and we may ask about the almost sure
behavior of certain probabilities. Our main results are summarized in
the following theorem:

\begin{thm}\label{thm:main}
	Let $\TT$ be a Galton-Watson tree with progeny distribution $Z \geq1$
	with $\E[Z] > 1$. Suppose $\E[Z^p] < \infty$ for each $p \geq1$.
	Set $\lambda:= \frac{2 \E[Z]^2}{\E[Z(Z-1)]}$ and let $Y_n$ be the
	set of vertices in depth $n$ of $\TT$ connected to the root in $p_c =
	1/\E[Z]$ percolation. Then for a.e. $\TT$ we have

	\begin{enumerate}

		\item[$(a)$] $n\cdot\P_\TT[|Y_n| > 0] \to W \lambda$ a.s.

		\item[$(b)$] The conditioned variable $(|Y_n| / n \| |Y_n| > 0)$
		converges in distribution to an exponential random variable with mean
		$\lambda^{-1}$ a.s.

		\item[$(c)$] Let $\mathbf{C}_n$ denote the number of vertices in the
		quenched IIC of $\TT$ at depth $n$. Then $\mathbf{C}_n / n$ converges
		in distribution to the random variable with density $\lambda^2 x
		e^{-\lambda x}$ a.s.
	\end{enumerate}

\end{thm}

Note that, surprisingly, the limit laws of parts $(b)$ and $(c)$ of
Theorem \ref{thm:main} do not depend at all on $\TT$ itself but just
on the distribution of $Z$. This is in sharp contrast to the case of
near-critical and supercritical percolation on Galton-Watson trees, in
which the behavior is dependent on the tree itself \cite
{mpr-quenched}. One possible justification for this lack of dependence
on $W$, for instance, is that conditioning on $|Y_n| > 0$ forces
certain structure of the percolation cluster near the root; since $W$
is mostly determined by the levels of $\TT$ near the root, the
behavior when conditioned on $|Y_n| > 0$ for large $n$ does not depend
on $W$. Part $(a)$ of Proposition \ref{pr:spread} corroborates this
heuristic explanation.

The three parts of Theorem \ref{thm:main} are Theorems \ref
{thm:surv-prob}, \ref{thm:cond-surv} and \ref{thm:IIC} respectively.
The proof of part $(a)$ utilizes its annealed analogue, Theorem \ref
{thm:annealed}$(a)$, along with a law of large numbers argument. Part
$(b)$ is proven by the method of moments building on the work of \cite
{mpr-quenched}. Part $(c)$ follows from there with a similar law of
large numbers argument combined with two short facts about the
structure of the percolation cluster conditioned on $|Y_n| > 0$ (this
is Proposition \ref{pr:spread}).

\begin{remark} Theorem \ref{thm:main} assumes that $\E[Z^p] < \infty
	$ for each $p \geq1$, and we suspect that this condition is an
	artifact of the proof. Since we use the method of moments, it is
	natural that we require all moments of the underlying distribution to
	be finite. We suspect that less rigid conditions are sufficient, but
	this would require a different proof strategy than the method of
	moments, perhaps utilizing a stronger anti-concentration statement in
	the vein of Proposition \ref{pr:spread}.
\end{remark}

\section{Set-up and notation}

We begin with some notation and a brief description of the probability
space on which we will work. Let $Z$ be a random variable taking values
in $\{1,2,\ldots,\}$ with $\mu:= \E[Z] > 1$ and $\P[Z = 0] = 0$.
Define its \emph{probability generating function} to be $\phi(z):=
\sum\P[Z = k] z^k$. Let $\TT$ be a random locally finite rooted tree
with law equal to that of a Galton-Watson tree with progeny
distribution $Z$ and let $(\Omega_1,\T,\GW)$ be the probability
space on which it is defined. Since we will perform percolation on
these trees, we also use variables $\{U_i\}_{i = 1}^\infty$ where the
$U_i$ are i.i.d. random variables uniform on $[0,1]$; let $(\Omega
_2,\F_2, \P_2)$ be the corresponding probability space. Our canonical
probability space will be $(\Omega,\F,\P)$ with $\Omega:= \Omega_1
\times\Omega_2$, $\F:= \T\otimes\F_2$ and $\P:= \GW\times\P
_2$. We interpret an element $\omega=(T,\omega_2) \in\Omega$ as the
tree $T$ with edge weights given by the $U_i$ random variables. To
obtain $p$ percolation, we restrict to the subtree of edges with weight
at most $p$. Since we are concerned with quenched probabilities, we
define the measure $\P_\TT[\cdot] := \P[\cdot\| \TT] = \P[\cdot
\| \T]$. Since this is a random variable, our goal is to prove
theorems $\GW$-a.s.

We employ the usual notation for a rooted tree $T$, Galton-Watson or
otherwise: $\rtt$ denotes the root; $T_n$ is the set of vertices at
depth $n$; and $Z_n := |T_n|$. In the case of a Galton-Watson tree $\TT
$, we define $W_n := Z_n / \mu^n$ and recall that $W_n \to W$ almost
surely. Furthermore, if $\E[Z^p] < \infty$ for some $p \in[1,\infty
)$, we in fact have $W_n \to W$ in $L^p$ \cite[Theorems $0$ and
$5$]{bingham-doney74}. In the Galton-Watson case, define $\T_n :=
\sigma(\TT_n)$; then $(\T_n)_{n = 0}^\infty$ is a filtration that
increases to $\T$. For a vertex $v$ of $T$, define $T(v)$ to be the
descendant tree of $v$ and extend our notation to include $T_n(v),
Z_n(v), W_n(v)$ and $W(v)$. For vertices $v$ and $w$, write $v \leq w$
if $v$ is an ancestor of $w$.

For percolation, recall that the critical percolation probability for
$\GW$-a.e. $\TT$ is $p_c:= 1/\mu$ and that percolation does not
occur at criticality \cite{lyons90}. For vertices $v$ and $w$ with $v
\leq w$, let $\{v \conn w\}$ denote the event that there is an open
path from $v$ to $w$ in $p_c$ percolation; let $\{v \conn(u,w)\}$ be
the event that $v$ is connected to both $u$ and $w$ in $p_c$
percolation; for a subset $S$ of $\TT$, let $\{v \conn S \}$ denote
the event that $v$ is connected to some element of $S$ in $p_c$
percolation; lastly, let $Y_n$ be the set of vertices in $\TT_n$ that
are connected to $\rtt$ in $p_c$ percolation.

\section{Quenched results}
\subsection{Moments}

For $k \geq j$, let $\mathcal{C}_j(k)$ denote the set of
$j$-compositions of $k$, i.e.\ ordered $j$-tuples of positive integers
that sum to $k$. Define

\[
c_{k,j} := p_c^k \sum_{a \in\mathcal{C}_j(k)} m_{a_1} m_{a_2}\cdots m_{a_j}
\]

where $m_r:= \E[\binom{Z}{r}]$. We use the following result from
\cite{mpr-quenched}:

\begin{thm}[\cite{mpr-quenched}] \label{thm:martingales}
	Define

	\[
	M_n^{(k)} := \E_\TT\left[\binom{|Y_n|}{k}\right] - \sum_{i =
		1}^{k-1}c_{k,i} \sum_{j = 0}^{n-1} \E_\TT\left[ \binom{|Y_j|}{i}
	\right] \,.
	\]

	If $\E[Z^{2k}]< \infty$, then $M_n^{(k)}$ is a martingale with
	respect to the filtration $(\T_n)$, and there exist constants $C_k$
	and $c_k$ so that

	\[
	\Vert M_{n+1}^{(k)} - M_n^{(k)} \Vert_{L^2} \leq C_k e^{-c_k n}\,.
	\]

\end{thm}

While Theorem \ref{thm:martingales} is not stated precisely this way
in \cite{mpr-quenched}, the martingale property follows from \cite
[Lemma $4.1$]{mpr-quenched}, while the $L^2$ bound on the increments is
given in \cite[Theorem $4.4$]{mpr-quenched}. This gives us the leading
term of each $\E_\TT\left[|Y_n|^k\right]$.

\begin{proposition}\label{pr:factorial-moments}
	For each $k$,

	\[
	\E_\TT\left[|Y_n|^k\right]n^{-(k-1)} \to k! \left(\frac{p_c^2
		\phi''(1)}{2} \right)^{k-1} W
	\]

	almost surely and in $L^2$.
\end{proposition}

\begin{proof}
	By Theorem \ref{thm:martingales}, $M_n^{(k)}$ is a martingale with
	uniformly bounded $L^2$ norm for each $k$. By the $L^p$ martingale
	convergence theorem, $M_n^{(k)}$ converges in $L^{2}$ and almost
	surely. We now proceed by induction on $k$. For $k = 1$, $\E_\TT
	[|Y_n|] = W_n$ which converges to $W$. Suppose that the proposition
	holds for all $j < k$. Then by convergence of $M_n^{(k)}$,

	\[
	\E_\TT\left[\binom{|Y_n|}{k} \right]n^{-(k-1)} = \sum_{i
		=1}^{k-1} c_{k,i} n^{-(k-1)}\sum_{j = 0}^{n-1} \E_\TT\left[\binom
	{|Y_j|}{i} \right] + o(1)
	\]

	where the $o(1)$ term is both in $L^2$ and almost surely.

	By induction, the leading term is the contribution from $i = k-1$.
	Noting that $c_{k,k-1} = (k-1)p_c^2 \frac{\phi''(1)}{2}$ and the fact
	that $\sum_{j = 0}^{n-1} j^d \sim\frac{1}{d+1}n^{d+1}$ completes the proof.
\end{proof}

\subsection{Survival probabilities}

Throughout, define $\lambda:= \frac{2}{p_c^2 \phi''(1)}$. Our first
task is to find a quenched analogue of Kolmogorov's estimate:

\begin{thm} \label{thm:surv-prob}
	If $\E[Z^4] < \infty$, then

	\[
	n\cdot\P_\TT[|Y_n| > 0] \to W \lambda
	\]

	almost surely.
\end{thm}

The proof utilizes the Bonferroni inequalities. In order to control the
second-order term, the variance of a sum of pairs is calculated,
thereby introducing the requirement of $\E[Z^4] < \infty$. We begin
first by proving upper and lower bounds:

\begin{lemma} \label{lem:prop-sandwich}
	For each $n$,

	\begin{equation*}
	\frac{n\cdot\E_\TT[|Y_n|]^2}{\E_\TT[|Y_n|^2]}\leq n\cdot\P_\TT
	[|Y_n| > 0 ] \leq\frac{2 \overline{W}}{1 - p_c}
	\end{equation*}

	where, $\overline{W} = \sup_n W_n$.
\end{lemma}

\begin{proof}
	The lower bound is the Paley-Zygmund inequality. For the upper bound,
	we use \cite[Theorem 5.24]{LP-book}:

	\[
	\P_\TT[|Y_n| > 0] \leq\frac{2}{\mathscr{R}(\rtt\conn\TT_n)}
	\]

	where $\mathscr{R}(\rtt\conn\TT_n)$ is the equivalent resistance
	between the root and $\TT_n$ when all of $\TT_n$ is shorted to a
	single vertex and each edge branching from depth $k-1$ to $k$ has
	resistance $\frac{1 - p_c}{p_c^k}$. Shorting together all vertices at
	depth $k$ for each $k$ gives the lower bound

	\begin{equation*}
	\mathscr{R}(\rtt\conn\TT_n) \geq\sum_{k = 1}^{n}\frac{1 -
		p_c}{Z_k p_c^k} = \sum_{k = 1}^{n}\frac{1 - p_c}{W_k}\geq(1 -
	p_c)\frac{n}{\overline{W}}\,.\qedhere
	\end{equation*}

\end{proof}

\noindent{\emph{Proof of Theorem }\ref{thm:surv-prob}: } For each
fixed $m < n$, the Bonferroni inequalities imply

\begin{equation} \label{eq:bonf}
\left|n\P_\TT[\rtt\conn\TT_n] - n\sum_{v \in\TT_m}\P_\TT
[\rtt\conn v \conn\TT_n]\right| \leq n\sum_{u,v \in\binom{\TT
		_m}{2}} \P_\TT[\rtt\conn(u,v) \conn\TT_n]\,.
\end{equation}

If we can show that the right-hand side of \eqref{eq:bonf} converges
a.s. to zero for some choice of $m = m(n)$, then the survival
probability is sufficiently close to a sum of i.i.d. random variables.
The random variables $\P_\TT[\rtt\conn v \conn\TT_n]$ are i.i.d.\
with mean $p_c^m \P[\rtt\conn\TT_{n - m}]$, implying that the sum
is close to $W_m \P[\rtt\conn\TT_{n - m}]$. Applying the annealed
result Theorem \ref{thm:annealed} would then complete the proof after
noting that $W_m \to W$ almost surely provided $m \to\infty$. The
remainder of the proof follows this sketch.

Set $m = \lceil n^{1/4} \rceil$; we then bound the second moment

\begin{align*}
\E&\left[ \left(\sum_{u,v \in\binom{\TT_m}{2}} \P_\TT[\rtt
\conn(u,v) \conn\TT_n]\right)^2\right] \\
&\qquad=
\E\left[\left(\sum_{u,v \in\binom{\TT_m}{2}}\P_\TT[\rtt\conn
(u,v)]\P_\TT[u\conn\TT_n]\P_\TT[v \conn\TT_n] \right)^2 \right]
\\
&\qquad= \E\left[ \E\left[\left(\sum_{u,v \in\binom{\TT
		_m}{2}}\P_\TT[\rtt\conn(u,v)]\P_\TT[u\conn\TT_n]\P_\TT[v
\conn\TT_n] \right)^2 \, \Bigg| \, \T_m \right]\right] \\
&\qquad= \E\left[ \E\left[\left(\sum_{u,v \in\binom{\TT
		_m}{2}}\P_\TT[\rtt\conn(u,v)]\P_\TT[u\conn\TT_n]\P_\TT[v
\conn\TT_n] \right)^2 \, \Bigg| \, \T_m \right]^{(1/2)\cdot
	2}\right] \\
&\qquad\leq\E\left[ \left(\sum_{u,v \in\binom{\TT_m}{2}} \P
_\TT[\rtt\conn(u,v)] \left\Vert\P_\TT[u \conn\TT_n] \P_\TT[v
\conn\TT_n] \right\Vert_{L^2} \right)^2\right] &&\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{ by the triangle inequality}& \\
&\qquad\leq\left(\frac{2}{1 - p_c}\right)^4 \E[\overline{W}^2]^2
\cdot(n-m)^{-4} \E\left[\binom{|Y_m|}{2}^2 \right]\qquad\qquad\quad\text{ by
	Lemma }\ref{lem:prop-sandwich}&&\\
&\qquad\leq C m^2 n^{-4}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \  \text{ by Theorem }\ref{thm:martingales}\,.&&
\end{align*}

Multiplying by $n$, the second moment of the right-hand side of \eqref
{eq:bonf} is bounded above by $C m^2 n^{-2} = O(n^{-3/2})$ which is
summable in $n$. By Chebyshev's Inequality together with the
Borel-Cantelli Lemma, the right-hand side of \eqref{eq:bonf} converges
to zero almost surely. This implies

\begin{equation} \label{eq:surv-prob-avg}
n\P_\TT[\rtt\conn\TT_n] = n \sum_{v \in\TT_m} \P_\TT[\rtt
\conn v \conn\TT_n] +o(1) = \sum_{v \in\TT_m} \frac{n \P_\TT[v
	\conn\TT_n]}{\mu^m }+ o(1)\,.
\end{equation}

We want to show that the right-hand side of \eqref{eq:surv-prob-avg}
converges to $W \lambda$, so we first calculate

\begin{align*}
\Var\bigg[ \sum_{v \in\TT_m} &\frac{n \P_\TT[v \conn\TT_n] -
	n \P[\rtt\conn\TT_{n - m}]}{\mu^m} \bigg] \\
&= \E\left[\Var\left[ \sum_{v \in\TT_m} \frac{n \P_\TT[v
	\conn\TT_n] - n \P[\rtt\conn\TT_{n - m}]}{\mu^m} \, \bigg| \,
\T_m\right] \right] \\
&= \E\left[\frac{1}{\mu^{2m}} \sum_{v \in\TT_m} \Var[n \P_\TT
[v \conn\TT_n]] \right] \\
&\leq\frac{C}{\mu^m}
\end{align*}

where the last inequality is via Lemma \ref{lem:prop-sandwich}. Since
this is summable in $n$, Chebyshev's Inequality and the Borel-Cantelli
Lemma again imply

\[
\sum_{v \in\TT_m} \frac{n \P_\TT[v \conn\TT_n]}{\mu^m} = \sum
_{v \in\TT_m} \frac{n \P[\rtt\conn\TT_{n - m}]}{\mu^m} + o(1) =
W_m (n \cdot\P[\rtt\conn\TT_{n - m}]) + o(1)\,.
\]

Taking $n \to\infty$ and utilizing Theorem \ref{thm:annealed}
together with \eqref{eq:surv-prob-avg} completes the proof. $\Cox$

\subsection{Conditioned survival}

\begin{thm} \label{thm:cond-surv}
	Suppose $\E[Z^p] < \infty$ for all $p \geq1$. Then the conditional
	variable $(|Y_n|/n \, | \, |Y_n| > 0)$ converges in distribution to an
	exponential random variable with mean $\lambda^{-1}$ for $\GW$-almost
	every $\TT$.
\end{thm}

By conditional random variable $(|Y_n|/n \, | \, |Y_n| > 0)$, we mean
the random variable with law $\P_\TT[|Y_n|/n \in\cdot\, | \, |Y_n|
> 0 ]$.

\begin{proof} The proof is via the method of moments. In particular,
	since the moment generating function of an exponential random variable
	has a positive radius of convergence, its distribution is uniquely
	determined by its moments. Thus, any sequence of random variables with
	each moment converging to the moment of an exponential random variable
	must converge in distribution to that exponential random variable \cite
	[Theorems $30.1$ and $30.2$]{billingsley}.

	Let $X_n$ be a random variable with distribution $(|Y_n|/n \, | \,
	|Y_n| > 0)$. It is sufficient to show $\E_\TT[X_n^k] \to k! \lambda
	^{-k}$ $\GW$-a.s. since $k! \lambda^{-k}$ is the $k$th moment of an
	exponential random variable. Proposition \ref{pr:factorial-moments}
	and Theorem \ref{thm:surv-prob} imply

	\begin{align*}
	\E_\TT[X_n^k] &= \frac{\E_\TT[|Y_n|^k]}{n^k \P_\TT[|Y_n| > 0]} \\
	&= \frac{\E_\TT[|Y_n|^k}{n^{k-1}} \cdot\frac{1}{n\cdot\P_\TT
		[|Y_n| > 0]} \\
	&\to k! W\lambda^{-(k-1)}\cdot\frac{1}{\lambda W} \\
	&= k! \lambda^{-k}\,.\qedhere
	\end{align*}

\end{proof}

More can be said about the structure of the open percolation cluster of
the root conditioned on $\rtt\conn\TT_n$, but we require two
general, more or less standard lemmas first.

\begin{lemma} \label{lem:condition}
	For any events $A$ and $B$ with $\P[B] \neq0$,

	\[
	|\P[A \| B] - \P[A]| \leq\P[B^c]\,.
	\]

\end{lemma}

\begin{proof}
	Expand

	\[
	\P[A] = \P[A \| B](1 - \P[B^c]) + \P[A \| B^c] \P[B^c]
	\]

	and solve

	\[
	\P[A] - \P[A\| B] = (\P[A \| B^c] - \P[A \| B]) \P[B^c]\,.
	\]

	Taking absolute values and bounding $|\P[A \| B^c] - \P[A \| B]| \leq
	1$ completes the proof.
\end{proof}

\begin{lemma} \label{lem:sum-tails}
	Let $X_k$ be i.i.d. centered random variables with $\E[|X_1|^p] <
	\infty$ for some $p \in[2,\infty)$. Then there exists a constant
	$C_p$ so that

	\[
	\P\left[ \left|\sum_{k = 1}^n \frac{X_k}{n}\right| > t \right]
	\leq C_p t^{-p}n^{-p/2} + 2 \exp\left(-\frac{n t^2}{\Var
		[X_1]}\right)
	\]

	for all $t > 0$.
\end{lemma}

\begin{proof}
	This is a straightforward application of \cite[Theorem
	$2.1$]{chesneau} which states that for independent random variables
	$M_i$ with $\E[M_i] = 0$ and $\E[|M_i|^p] < \infty$ for some $p > 2$
	we have

	\[
	\P\left[\left|\sum_{i=1}^n M_i\right| \geq t \right] \leq C_p
	t^{-p}\max\left(r_{n,p}(t), (r_{n,2}(t))^{p/2}\right) + \exp\left
	(- \frac{t^2}{16 b_n}\right)
	\]

	where $r_{n,u}(t) = \sum_{i = 1}^n \E(|M_i|^u \one_{|M_i| \geq
		3b_n/t}),$ $b_n = \sum_{i = 1}^n \E[M_i^2]$ and $C_p$ is a positive
	constant. Setting $M_i = X_i/n$ completes the proof.
\end{proof}

For a fixed tree and $m < n$, define $B_m(n)$ to be the event that
$\rtt\conn\TT_n$ through precisely one vertex at depth $m$.

\begin{proposition} \label{pr:spread}
	Suppose $\E[Z^p] < \infty$ for all $p \geq1$. There exists an $N =
	N(\TT)$ with $N < \infty$ almost surely so that for all $n \geq N$,
	we have

	\begin{enumerate}

		\item[$(a)$] $\P_\TT[B_m(n)^c \| \rtt\conn\TT_n] < Cn^{-1/4}$ for
		$m = m(n) := \lceil\frac{\log n}{4 \log\mu}\rceil$

		\item[$(b)$] $\max_{v \in\TT_n} \P_\TT[v \in Y_n \| \rtt\conn
		\TT_n] = O(n^{-1/8})$
	\end{enumerate}

	for some constant $C > 0$.
\end{proposition}

\emph{Proof.} Note first that for the choice of $m$ as in part $(a)$,
we have $ \frac{1}{2\mu} W n^{1/4} \leq Z_m \leq2 \mu W n^{1/4}$ for
sufficiently large $n$.

$(a)$ Using Theorem \ref{thm:surv-prob} and Lemma \ref
{lem:prop-sandwich}, we bound

\begin{align}
\P_\TT[B_m(n)^c \| \rtt\conn\TT_n] &\leq\frac{\left(\sum_{v
		\in T_m} \P_\TT[v \conn\TT_n] \right)^2}{\P_\TT[\rtt\conn\TT
	_n]} \nonumber\\
&\leq\left(\frac{2}{1 - p_c}\right)^2 \left(\frac{\sum_{v \in
		\TT_m} \overline{W}(v)}{Z_m} \right)^2 \frac{Z_m^2}{(n - m)^2 \P
	_\TT[\rtt\conn\TT_n]} \nonumber\\
&\leq C \left(\frac{\sum_{v \in\TT_m} \overline{W}(v)}{Z_m}
\right)^2 W n^{-1/2} \label{eq:B-n-complement-bound}
\end{align}

for $n$ sufficiently large, and some choice of $C > 0$ depending on the
distribution of $Z$. Applying Lemma \ref{lem:sum-tails} for $p = 9$
gives

\[
\P\left[ \left|\frac{\sum_{v \in\TT_m} \overline{W}(v)}{Z_m} -
\E[\overline{W}] \right| > n^{1/8} \right] \leq C_9 n^{-9/8} + 2
\exp\left(- n^{1/4} / \Var[\overline{W}] \right)
\]

where we use the trivial bound of $1 \leq Z_m.$ Since this is summable
in $n$, the Borel-Cantelli Lemma implies that this event only occurs
finitely often. In particular, this means that for sufficiently large
$n$

\begin{equation} \label{eq:B_m-bound}
\P_\TT[B_m(n)^c \, | \, \rtt\conn\TT_n] \leq C W n^{-1/4}
\end{equation}

for some constant $C > 0$ depending only on the distribution of $Z$.

$(b)$ Applying Lemma \ref{lem:condition} to the measure $\P_\TT[
\cdot\| \rtt\conn\TT_n]$ and recalling $B_m(n) \subseteq\rtt\conn
\TT_n$,

\begin{align*}
\Big|\P_\TT[v \in Y_n \, | \,\rtt\conn\TT_n] - \P_\TT[v \in Y_n
\| B_m(n)] \Big| &\leq\P_\TT\left[B_m(n)^c \| \rtt\conn\TT_n
\right]
\end{align*}

which is $O(n^{-1/4})$ by part $(a)$. It is thus sufficient to bound
$\P_\TT[v \in Y_n \| B_m(n)]$.
For a vertex $v \in\TT_n$ and $m < n$, let $P_m(v)$ be the ancestor
of $v$ in $\TT_m$. We then have

\[
\P_\TT[v \in Y_n \| B_m(n)] \leq\P_\TT[\rtt\conn P_m(v) \conn\TT
_n \| B_m(n)]\,.
\]

Conditioned on $B_m(n)$, there exists a unique vertex $w \in\TT_m$ so
that $\rtt\conn w \conn\TT_n$; this vertex $w$ is chosen with
probability bounded above by

\begin{align}
\P_\TT[\rtt&\conn w \conn\TT_n \| B_m(n)] \nonumber\\
&\leq\frac{\P_\TT[\rtt\conn w \conn\TT_n]}{\sum_{u \in\TT_m}
	\P_\TT[\rtt\conn u \conn\TT_n] - \sum_{(u_1,u_2) \in\binom{\TT
			_m}{2}} \P_\TT[\rtt\conn(u_1,u_2) \conn\TT_n] } \nonumber\\
&\leq\frac{\P_\TT[w \conn\TT_n]}{ \sum_{u \in\TT_m} \P_\TT[u
	\conn\TT_n] - \left(\sum_{u \in\TT_m}\P_\TT[u \conn\TT_n]
	\right)^2 } \nonumber\\
&\leq\frac{c (n - m)^{-1}\overline{W}(w)}{(1 + o(1))\sum_{u \in\TT
		_m} \P_\TT[u \conn\TT_n]} \label{eq:point-B_m-bound}
\end{align}

where the latter inequality is by applying the bound of Lemma \ref
{lem:prop-sandwich} to the numerator and arguing as in \eqref
{eq:B-n-complement-bound} to almost-surely bound the denominator. In
particular, the $o(1)$ term is uniform in $w$.

We want to take the maximum over all possible $w \in\TT_m$, and note
that for any $\alpha> 0$,

\begin{align*}
\P\left[\max_{w \in\TT_m} \overline{W}(w) > n^{\alpha}\right]
&=\E\left[\P\left[\max_{w \in\TT_m} \overline{W}(w) > n^{\alpha
}\, \big|\, \T_m \right]\right] \\
&\leq \E[Z_m] \P[\overline{W}> n^\alpha] \\
&\leq\mu^m \cdot\frac{\E[\overline{W}^{2/\alpha}]}{n^{2}} \\
&= O(n^{-7/4})
\end{align*}

which is summable, implying that for any fixed $\alpha\!>\!0$, we
eventually have $\max_{w \in\TT_m} \overline{W}(w)\!\leq\!n^{\alpha
}.$ It merely remains to bound the denominator of \eqref{eq:point-B_m-bound}.

Note that by Proposition \ref{pr:factorial-moments}, the lower bound
given in Lemma \ref{lem:prop-sandwich} converges almost surely to
$\frac{W \lambda}{2}$ as $n \to\infty$. In particular, this means
that if we set

\[
p_n := \P\left[\frac{W \lambda}{4} \leq n\P_\TT[|Y_n| > 0] \right],
\]

then $p_n \to1$. By Hoeffding's inequality together with
Borel-Cantelli, the number of vertices $u \in\TT_m$ for which we have

\[
\frac{W(u) \lambda}{4} \leq(n - m) \P_\TT[u \conn\TT_n]
\]

is almost surely at least $1/2$ of $\TT_m$ for $n$ sufficiently large.
This gives

\[
(n-m)\sum_{u \in\TT_m} \P_\TT[u \conn\TT_n] \geq\frac{\lambda
}{4}\sum_{u \in\TT_m} W(u) \one_{W(u)\lambda/4 \leq(n - m)\P_\TT
	[u \conn\TT_n] } = \Omega(Z_m) \,.
\]

Recalling that $Z_m = \Theta(Wn^{-1/4})$ and plugging the above into
\eqref{eq:point-B_m-bound} completes the proof. $\Cox$

\subsection{Incipient infinite cluster}

As in \cite{kesten-IIC}, we sketch a proof of the construction of the
IIC. For an infinite tree $T$, define $T{[n]}$ to be the finite subtree
of $T$ obtained by restricting to vertices of depth at most $n$.

\begin{lemma}\label{lem:constr}
	Suppose $\E[Z^4] < \infty$; for a subtree $t$ of $\TT{[n]}$, we have

	\[
	\lim_{M \to\infty} \P_\TT[\mathcal{C}_{p_c}[n] = t \| \rtt\conn
	\TT_M ] = \frac{\sum_{v \in t_n}W(v)}{W}\P_\TT[\mathcal
	{C}_{p_c}[n] = t ]
	\]

	almost surely for each tree $t$.

	The random measure $\mu_\TT$ on subtrees of $\TT$ with marginals

	\[
	\mu_\TT\big|_{\T_n}[t] := \frac{\sum_{v \in t_n}W(v)}{W}\P_\TT
	[\mathcal{C}_{p_c}[n] = t ]
	\]

	has a unique extension to a probability measure on rooted infinite
	trees $\GW$ almost surely. The IIC is thus the random subtree of $\TT
	$ with law $\mu_\TT$.
\end{lemma}

\begin{proof}
	Since each $\TT$ has countably many vertices, Theorem \ref
	{thm:surv-prob} assures that $n\P_\TT[v \conn\TT_{n + |v|}] =
	\lambda W(v)$ for each vertex $v$ of $\TT$ a.s. When all of these
	limits hold, we then have

	\begin{align*}
	\P_\TT[\mathcal{C}_{p_c}[n] = t \| \rtt\conn\TT_M] &= \frac{ \P
		_\TT[\mathcal{C}_{p_c}[n] = t, \rtt\conn\TT_M]}{\P_\TT[\rtt
		\conn\TT_M]} \\
	&= \P_\TT[\mathcal{C}_{p_c}[n] = t]\left(\frac{\sum_{v \in t_n}
		\P_\TT[v \conn\TT_M] + O(|t_n|^2 M^{-2})}{\P_\TT[\rtt\conn\TT
		_M]} \right) \\
	&\xrightarrow{M\to\infty} \P_\TT[\mathcal{C}_{p_c}[n] = t]\frac
	{\sum_{v \in t_n} W(v)}{W}
	\end{align*}

	for each $t$. To show that the measure $\mu_\TT$ can be extended, we
	note that its marginals are consistent, as can be seen via the
	recurrence $W(v) = p_c \sum_{w} W(w)$ where the sum is over all
	children of $v$. Applying the Kolmogorov extension theorem \cite
	[Theorem $2.1.14$]{durrett4} completes the proof.
\end{proof}

It is easy to show that the law of the IIC can in fact be generated by
conditioning on $p > p_c$ percolation to survive and then taking $p \to
p_c^+$:

\begin{corollary} For a subtree $t$ of $\TT[n]$, we have

	\[
	\lim_{p \to p_c^+} \P_\TT[\mathcal{C}_p[n] = t \| |\mathcal{C}_p|
	= \infty] = \frac{\sum_{v \in t_n}W(v) }{W} \P_\TT[\mathcal
	{C}_{p_c}[n] = t]
	\]

	almost surely.
\end{corollary}

\begin{proof}
	As shown in \cite{mpr-quenched}, we have

	\[
	\lim_{p \to p_c} \frac{\P_\TT[|\mathcal{C}_p| = \infty]}{p - p_c}
	= K W
	\]

	almost-surely for some constant $K$ depending only on the offspring
	distribution. The Corollary follows from Bayes' theorem in the same
	manner as Lemma \ref{lem:constr}.
\end{proof}

In light of Lemma \ref{lem:constr}, it is natural to guess that the
number of vertices in the IIC at depth $n$ will asymptotically be the
size-biased version of $(|Y_n| \| \rtt\conn\TT_n )$: the sum $\sum
_{v \in t_n} W(v)$ will be relatively close to $|t_n| W$, therefore
biasing each choice of $t$ by a factor of $|t_n|$. In order to make
this argument rigorous, we will invoke Proposition \ref{pr:spread}
which shows that no single vertex has high probably of surviving
conditionally. Throughout, we use the notation $n(a,b) = (na,nb)$ for
$a < b$ and $\mathbf{C}$ to denote the IIC.

\begin{thm} \label{thm:IIC}
	Suppose $\E[Z^p] < \infty$ for each $1 \leq p < \infty$. Then for
	each $0 \leq a < b$,

	\[
	\lim_{n \to\infty} \P_\TT[\mathbf{C}_n \in n(a,b)] = \int_{a}^b
	\lambda^2 x e^{-\lambda x} \,dx
	\]

	almost surely. In fact, $\mathbf{C}_n / n$ converges in distribution
	to the random variable with density $\lambda^2 x e^{-\lambda x}$ for
	$\GW$-almost every $\TT$.
\end{thm}

\begin{proof}
	To see that convergence in distribution follows from the almost sure
	limit, apply the almost sure limit to each interval $(a,b)$ with $a,b
	\in\mathbb{Q}$; since there are only countably many such intervals,
	there exists a set of full $\GW$ measure on which these limits
	simultaneously exist for each rational interval, thereby implying
	convergence in distribution \cite[Theorem 3.2.5]{durrett4}.

	We have

	\[
	\P_\TT[\mathbf{C}_n \in n(a,b)] = \lim_{M \to\infty} \P_\TT[Y_n
	\in n(a,b)\| \rtt\conn\TT_{n + M}]\,.
	\]

	For a fixed $n$, write

	\begin{align}
	\P_\TT[&|Y_n| \in n(a,b) \| \rtt\conn\TT_{n+M}] \nonumber\\ &=
	\frac{\P_\TT[\rtt\conn\TT_{n + M} \| |Y_n| \in n(a,b)]\cdot\P
		_\TT[|Y_n| \in n(a,b) \| \rtt\conn\TT_n] \cdot\P_\TT[\rtt\conn
		\TT_n]}{\P_\TT[\rtt\conn\TT_{n + M}]}\,. \label{eq:Y_n-size-cond}
	\end{align}

	We then calculate

	\begin{align*}
	\P_\TT&[\rtt\conn\TT_{n + M} \| |Y_n| \in n(a,b)]\\
	&= \sum_{S} \P_\TT[Y_n = S \| |Y_n| \in n(a,b) ] \P_\TT[S \conn\TT_{n + M}] \\
	&= \sum_{S} \P_\TT[Y_n = S \| |Y_n| \in n(a,b) ] \sum_{v \in S} \P
	_\TT[v \conn\TT_{n + M}] + O(M^{-2}) \\
	&= \sum_{v \in\TT_n} \P_\TT[v \in Y_n \| |Y_n| \in n(a,b)] \P_\TT
	[v \conn\TT_{n + M}] + O(M^{-2})\,.
	\end{align*}

	For a fixed $n$, we take $M \to\infty$ and utilize Theorem \ref
	{thm:surv-prob} to get

	\begin{equation} \label{eq:p-IIC}
	\lim_{M\to\infty} \frac{\P_\TT[\rtt\conn\TT_{n + M} \| |Y_n|
		\in n(a,b)]}{\P_\TT[\rtt\conn\TT_{n + M}]} = \frac{1}{W} \sum_{v
		\in\TT_n} \P_\TT[v \in Y_n \| |Y_n| \in n(a,b)]\cdot W(v)\,.
	\end{equation}

	We plug this into \eqref{eq:Y_n-size-cond} to get the limit

	\begin{align*}
	\lim_{M \to\infty} & \P_\TT[|Y_n| \in n(a,b) \| \rtt\conn\TT
	_{n+M}] \nonumber\\
	&= \left(\sum_{v \in\TT_n} \frac{ \P_\TT[v \in Y_n \| |Y_n| \in
		n(a,b)] }{n} \cdot W(v)\right)\nonumber\\
	&\quad\times \left(\P_\TT[|Y_n| \in n(a,b) \|
	\rtt\conn\TT_n]\right) \left( \frac{n\cdot\P_\TT[\rtt\conn
		\TT_n]}{W} \right)\,.
	\end{align*}

	Theorems \ref{thm:surv-prob} and \ref{thm:cond-surv} show that the
	latter two factors above have almost sure limits $\int_a^b \lambda
	e^{-\lambda x} \,dx$ and $\lambda$ as $n \to\infty$, leaving only
	the first term. We note that

	\begin{align*}
	\E\left[\sum_{v \in\TT_n} \frac{ \P_\TT[v \in Y_n \| |Y_n| \in
		n(a,b)] }{n} \cdot W(v) \,\Bigg|\, \T_n \right] &= \sum_{v \in\TT
		_n}\frac{ \P_\TT[v \in Y_n \| |Y_n| \in n(a,b)] }{n} \\
	&= \E_\TT\left[\frac{|Y_n|}{n} \, \bigg| \, |Y_n| \in n(a,b)
	\right] \\
	&= \frac{\E_\TT\left[\frac{|Y_n|}{n}\cdot\one_{|Y_n|/n \in
			(a,b)} \| \rtt\conn\TT_n \right]}{\P_\TT\left[\frac{|Y_n|}{n}
		\in(a,b) \| \rtt\conn\TT_n \right]}\\
	&\to\frac{\int_{a}^{b} \lambda x e^{-\lambda x} \,dx}{\int_{a}^{b}
		\lambda e^{-\lambda x}\,dx}
	\end{align*}

	where the limit is by the continuous mapping theorem \cite[Theorem
	3.2.4]{durrett4} and Theorem \ref{thm:cond-surv}. It's thus sufficient
	to show that

	\begin{equation}\label{eq:need} \left|\sum_{v \in\TT_n} \frac{ \P
		_\TT[v \in Y_n \| |Y_n| \in n(a,b)] }{n} \cdot(W(v) - 1) \right|
	\xrightarrow{n \to\infty} 0
	\end{equation}

	almost surely.

	Our strategy is to use a conditional version of the Borel-Cantelli
	Lemma together with Chebyshev's inequality. We bound the conditional
	variance

	\begin{align}
	\Var\Bigg[\sum_{v \in\TT_n}&\frac{ \P_\TT[v \in Y_n \| |Y_n|
		\in n(a,b)] }{n} \cdot(W(v) - 1) \, \bigg| \, \T_n \Bigg] \nonumber
	\\
	&= \Var(W)\sum_{v \in\TT_n} \frac{\P_\TT[v \in Y_n \| |Y_n| \in
		n(a,b)]^2 }{n^2} \nonumber\\
	&\leq\Var(W) \max_{v \in\TT_n} \P_\TT[v \in Y_n \| |Y_n| \in
	n(a,b)] \sum_{v \in\TT_n} \frac{\P_\TT[v \in Y_n \| |Y_n| \in
		n(a,b)] }{n^2} \nonumber\\
	&\leq\Var(W) \max_{v \in\TT_n} \P_\TT[v \in Y_n \| |Y_n| \in
	n(a,b)] \cdot\frac{\E[Y_n \| |Y_n| \in n(a,b)] }{n^2} \nonumber\\
	& \leq\Var(W) \cdot\frac{b}{n} \cdot\max_{v \in\TT_n} \P_\TT
	[v \in Y_n \| |Y_n| \in n(a,b)] \label{eq:var-bound} \,.
	\end{align}

	We want to show that this is summable, and thus look to bound the $\max
	$ term. Applying Lemma \ref{lem:condition} to the measure $\P_\TT
	[\cdot\| |Y_n| \in n(a,b)]$ gives

	\begin{align}
	&\left|\P_\TT[v \in Y_n \| |Y_n| \in n(a,b)] - \P_\TT[v \in Y_n \|
	|Y_n| \in n(a,b), B_m(n)] \right|\nonumber\\
	&\qquad\qquad\qquad\qquad\qquad\qquad\quad\leq\P_\TT[B_m(n)^c \| |Y_m| \in n(a,b)] \nonumber\\
	&\qquad\qquad\qquad\qquad\qquad\qquad\quad\leq\frac{\P_\TT[B_m(n)^c \| \rtt\conn\TT_n]}{\P_\TT[|Y_n|
		\in n(a,b) \| \rtt\conn\TT_n]} \nonumber \\
	&\qquad\qquad\qquad\qquad\qquad\qquad\quad= O(n^{-1/4}) \label{eq:point-difference}
	\end{align}

	by Proposition \ref{pr:spread} and Theorem \ref{thm:cond-surv}.
	Similarly,

	\begin{align}
	\P_\TT[v \in Y_n \| |Y_n| \in n(a,b), B_m(n)] &= \frac{\P_\TT[v
		\in Y_n, |Y_n| \in n(a,b), B_m(n)]}{\P_\TT[|Y_n| \in n(a,b), B_m(n)]}
	\nonumber\\
	&\leq\frac{ \P_\TT[v \in Y_n,B_m(n)]}{\P_\TT[|Y_n| \in n(a,b),
		B_m(n)]} \nonumber\\
	&= \frac{\P_\TT[v \in Y_n \| B_m(n)]}{\P_\TT[|Y_n| \in n(a,b) \|
		B_m(n)]} \label{eq:point-up}\,.
	\end{align}

	Using Lemma \ref{lem:condition} once again expands the denominator

	\[
	\Big| \P_\TT[|Y_n| \in n(a,b) \| B_m(n)] - \P_\TT[|Y_n| \in n(a,b)
	\| \rtt\conn\TT_n]\Big| \leq\P_\TT[B_m(n)^c \| \rtt\conn\TT
	_n] \leq C n^{-1/4}
	\]

	by Proposition \ref{pr:spread}. Plugging into \eqref{eq:point-up}
	gives the upper bound

	\begin{equation}
	\P_\TT[v \in Y_n \| |Y_n| \in n(a,b), B_m(n)] \leq\frac{\P_\TT[v
		\in Y_n \| B_m(n)]}{\P_\TT[|Y_n| \in n(a,b) \| \rtt\conn\TT_n] - C
		n^{-1/4}} \label{eq:point-final}\,.
	\end{equation}

	Combining \eqref{eq:point-difference}, \eqref{eq:point-final} and
	Proposition \ref{pr:spread} bounds

	\[
	\max_{v \in\TT_n} \P_\TT[v \in Y_n \| |Y_n| \in n(a,b)] =
	O(n^{-1/8})\,.
	\]

	Thus, by \eqref{eq:var-bound}, the conditional variance is almost
	surely summable. For any fixed $\delta> 0$, Chebyshev's inequality
	then implies

	\[
	\P\left[\left|\sum_{v \in\TT_n} \frac{\P_\TT[v \in Y_n \|
		|Y_n| \in(a,b)]}{n}\cdot(W(v) - 1) \right| > \delta\,\Bigg|\, \T
	_n\right]
	\]

	is summable almost surely. Applying a conditional Borel-Cantelli Lemma
	(e.g. \cite{chen-bc}) shows that \eqref{eq:need} holds almost surely.
\end{proof}

\section*{Acknowledgements}
The author would like to thank Josh Rosenberg for helpful
conversations.

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