text:
\begin{document}

	\begin{frontmatter}
		\title{On the monotonicity of the critical time in the Constrained-degree percolation model  \\}

		\author[label1]{Charles S. do Amaral\corref{cor1}}
		\address[label1]{Departamento de Matem\'atica - Centro Federal de Educa\c c\~ao Tecnol\'ogica de Minas Gerais, CEFET--MG, Av. Amazonas 7675, Belo Horizonte, MG, Brasil. }
		\ead{charlesmat@cefetmg.br}
		\cortext[cor1]{Corresponding author.}

		\author[label7]{A. P. F. Atman}
		\address[label7]{Departamento de F\'isica - Centro Federal de Educação Tecnológica de Minas Gerais, CEFET-MG and Instituto Nacional de Ci\^encia e Tecnologia - Sistemas Complexos.}

		\author[label6]{Bernardo N. B. de Lima}
		\address[label6]{Departamento de Matem\'atica, Universidade Federal de Minas Gerais, UFMG, Av. Ant\^onio Carlos, 6627, Belo Horizonte, MG, Brasil.}

		\begin{abstract}
			The Constrained-degree percolation model was introduced in [B.N.B. de Lima, R. Sanchis, D.C. dos Santos, V. Sidoravicius, and R. Teodoro, Stoch. Process. Appl. (2020)], where it was proven that this model has a non-trivial phase transition on a square lattice. We study the Constrained-degree percolation model on the $d$-dimensional hypercubic lattice ($\mathbb {Z}^d$) and, via numerical simulations, found evidence that the critical time $ t_{c}^{d}(k) $ is monotonous not increasing in the constrained $k$ if $d=3,4$, like it is when $d=2$. We verify that the lowest constrained value $k$ such that the system exhibits a phase transition is $k=3$ and that the correlation critical exponent $\nu$ for the Constrained-degree percolation model and ordinary Bernoulli percolation are the same.
		\end{abstract}

	\end{frontmatter}

	\section{Introduction}

	The classical percolation model was proposed by Broadbent and Hammersley in 1957~\cite{broadbent} with the idea of modeling the flow of a fluid (deterministic) through a porous medium (random environment). In the ordinary Bernoulli bond percolation model, with parameter $p$, on a graph, each bond is  \textit{open} or \textit{closed}, independently, with probabilities $p$ and $1-p$, respectively. For values of $ p $ greater than one constant $p_c$, called \textit{critical point} (percolation threshold), an infinite sequence of connected open bonds (sites) appears. In this case, we say that \textit{percolation} has occurred. There are several variations of these models in the literature with applications in different areas of science, in [2-17] there are some examples.

	The present study aims to investigate properties of one of these variations, the so called Constrained-degree bond percolation model (CDPM) introduced in \cite{de_lima}. The CDPM is defined in a infinite graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ connected and with bounded degree. The number of bonds that has some end on the vertex $ v $ is denoted for $\mbox{deg}(v)$ (degree of $v$). We fix a sequence of parameters (capacities or constrains) $(k_v)_{v \in \mathcal{V}}$, with $k_v \leq \mbox{deg}(v)$, and consider a time continuous percolation processes defined as follows: let $(U_e)_{e\in\mathcal{E}}$ be a sequence of independent and identically distributed random variables with uniform distribution on $[0,1]$. At the time $t=0$ all bonds are closed. Each bond $ e=<u,v> $ will try to open at time $U_e$, it will succeed if and only if $\overline{\mbox{deg}}(u,U_e) < k_{u}$ and $\overline{\mbox{deg}}(v, U_e) < k_{v}$, where $\overline{\mbox{deg}}(v, t)$ denotes the degree of $v$ considering only the open bonds at time $t$. We say that a site $ v $ percolates in time $ t $ if there is an infinite path of open bonds starting from the vertex $v$. Observe that when $k_v=\mbox{deg}(v),\ \forall v\in\mathcal{V}$, the CDPM reduces to ordinary Bernoulli bond percolation; otherwise, it is a dependent model whose dependence has infinite range, in particular the Fortuin-Kasteleyn-Ginibre (FKG) inequality (see \cite{grimmett}) does not work.

	In 1979, Gaunt proposed \cite{gaunt} a model called \textit{Percolation with restricted-valence} which is analogous to the CDPM, but considering site percolation instead of bond percolation. Kertész, Chakrabarti and Duarte \cite{kertesz} estimated the values of critical points and the \textit{maximum random concentration} for this model on the hypercubic lattice $ \mathbb{Z}^d $ for $ d=2,3 $. The CDPM is related to the study of dimers \cite{furlan} and polymers \cite{soteros, wilkinson1}.  Other types of constrained percolation models, where only specific configurations are allowed on the vertices, were studied in \cite{holroyd, garet}.

	From now on, we consider the CDPM on the hypercubic lattice $\mathbb{Z}^d$ and $k_v=k,\ \forall v\in\mathbb{Z}^d$. In Figure \ref{model_example} we show a part of $ \mathbb{Z}^2 $ considering the same sequence $ (U_e)_{e \in \mathcal{E}} $ for the cases $ k = 2,3 $.

	Let us denote by $\theta^d(t;k)$ the probability of the origin of $\mathbb{Z}^d$ percolates at time $t$ with constrained $k$. Since the function $\theta^d(t;k)$ is monotone in $t$, we can define the critical time as:
	\begin{equation}
	t_{c}^d(k) = \sup\{ t \in [0,1]; \theta^d(t;k)=0 \}.
	\label{def_tc}
	\end{equation}

	\noindent If $\theta^d(1;k)=0$, there is no percolation for any value of $ t $ and we define $ t_{c}^d(k) = \infty $. We denote by $\psi^{d}(t;k)$ the probability of there is at least one vertex that percolates. It is standard to show that $\theta^{d}(t;k)>0$ if and only if $\psi^{d}(t;k)=1$, since the underlying probability measure is translation invariant, that is $\psi^{d}(t;k)\in\{0,1\}$. So we can replace the function $\theta^{d}(t;k)$ in (1) by $\psi^{d}(t;k)$.

	As already mention, when $ k=2d $ this model is the ordinary Bernoulli bond percolation model with parameter $t$, therefore $ t_c^d(2d) = p_c (\mathbb{Z}^d)$ (the ordinary percolation threshold). For the case $k=1 $ there will be only isolated bonds, thus percolation does not occur for all $ t \in [0,1]$ and $d \geq 1$. It was proven in \cite{de_lima} that if $k=2$, $t_{c}^{d}(2)=\infty$ for all $d \geq 2$, that is, percolation does not occur for all times, included $ t = 1 $. The main result in \cite{de_lima} states that for the square lattice ($d=2$) $t_c^2(3)\in(\frac{1}{2},1)$, characterizing a phase transition for $k=3$. Apart these results, there are many questions and few answers.

	One of these intriguing facts is the following: fixed the dimension $d$, is the critical time a non-increasing function in $k$? that is, $t_c^d(k+1)\leq t_c^d(k)$? There is no trivial coupling proving this inequality (in Figure \ref{example2} we present an example of this situation). If the sequence $(t_c^d(k))_k$ is indeed monotone in $k$, we could define, for each dimension $d>2$, the critical restriction $k_c^d=\inf\{k;t_c^d(k)\leq 1\}$. The main purpose of this work is to perform simulations of $t_c^d(k)$ for several values of $d$ and $k$ that support the  monotonicity of the sequence $(t_c^d(k))_k$ as well as $k_c^d=3,\ \forall d\geq 2$. Observe that this is true for the square lattice, as shown in \cite{de_lima}, it holds that $t_c^2(1)=t_c^2(2)=\infty,\ \frac{1}{2}<t_c^2(3)<1$ and $t_c^2(4)=\frac{1}{2}$ (the ordinary percolation threshold for the square lattice proved by Harris \cite{harris} and Kesten \cite{kesten}).

	Using an algorithm based on the Newmann-Ziff algorithm~\cite{ziff01}, we determine $t_{c}^{d}(k)$ for $k~\in~\{2,3..., 2d\}$ ($d=2,3,4$) and the maximum random concentration ($ x_k $), which is defined as the expected value of open bonds in the lattice when $ t = 1 $. In \cite{gaunt}, $x_k$ is the expected value of open sites instead of bonds.

	As mentioned before, in the CDPM, when $k<2d $, the probability of a bond being open or closed depends on the status of the other bonds (long-range dependence), differently from what occurs in the ordinary Bernoulli percolation model. Therefore, it is not possible to use algorithms that assume that each bond is open (closed) independently, for example, the Leath Algorithm \cite{leath} or Invasion Percolation Algorithm \cite{lenormand, chandler, wilkinson2}. Such algorithms generate only a single cluster, rather than an entire lattice configuration and are often used to estimate the critical point of percolation models in higher dimensions, as they have low time and space computational complexities.

	\begin{figure}[t]
		\begin{center}
			\includegraphics[height=7.0cm]{exemplo.jpg}
			\caption{Part of the square lattice $\mathbb{Z}^2$ in which the order of the twelve first bonds that we will try to open is indicated. Bonds represented with dotted lines are closed and continuous lines are open. The configuration shown occurs when the twelfth bond is opened.}
			\label{model_example}
		\end{center}
	\end{figure}

	\begin{figure}
		\begin{center}
			\includegraphics[height=8.0cm]{config.jpg}
			\caption{A piece of $\mathbb{Z}^3$ where the numbers represent the order in which the bonds will be try to open. When $t$ is equal to the opening time of the fourteenth bond, we have that it will open if the constraint is $ k = 4 $, but the same bond will be closed if $ k = 5 $.}
			\label{example2}
		\end{center}
	\end{figure}

	We find numerical evidence that when $ d = 3,4 $, $ t_{c}^d (k) $ is monotonous in $ k $ and we determine that $ k_{c}^d=3$ for these cases. In addition, we also simulated the case $d=2$, verifying the results of \cite{de_lima} and obtaining numerically the value of $t_c^2(3)$ (that was also estimated in \cite{furlan}). Based on this results, we conjecture that $k_{c}^{d}=3$ and $t_{c}^{d}(k)$ is monotonous in $k$ to every $ d \geq 2 $. Furthermore, through the analysis of \textit{correlation length exponent} $\nu$, we found evidence that probably the CDPM (for $k \geq 3$) and the classic percolation model are in the same universality class. In \cite{furlan} it is shown that this occurs, when $ d = 2 $, through a detailed analysis of the critical exponents.

	The remainder of this paper is organized as follows. Section 2 describes our simulation procedure and Section 3 discusses the results obtained. Conclusions are summarized in Section 4.

	\section{Numerical Procedure}

	We consider the regular box $L \times ... \times L$ on $\mathbb{Z}^d$ with periodic boundary. The percolation criterion assumed is that the infinite cluster emerges in the box when there exists a cluster that wraps around one of the $d$ directions. From now on, we will omit $d$ and $k$ in the $t_{c}^{d}(k)$ notation if they are clear from the context.

	We denote by $\psi_{L}(t;k)$ the probability of percolation in the box of length $L$. For each configuration $ (U_e)_{e \in \mathcal{E}} $, we define $\mathcal{O}_t$ as the number of bonds that we will try to open until time $t$, formally we have:
	\begin{equation}
	\mathcal{O}_t = \# \{e \in \mathcal{E}; U_e \leq t \}
	\end{equation}

	\noindent where the $\#$ symbol denotes the number of elements in the set. Note that the probability of a bond belongs to $\mathcal{O}_t$ does not depend on the state of any other bond (the same does not occur when we consider the set formed only by bonds opened up to time $t$). This independence allows us to write the equation (\ref{psi_L}) and estimate $t_{c}$ as follows.

	Using the \textit{Partition Theorem} \cite{grimmett_prob}, the probability of percolation in the box of length $L$ can be write as:
	\begin{equation}
	\psi_{L}(t; k)= \sum_{i=0}^{N} \mathbb{P} (\mathcal{O}_t=i) \cdot \overline{Q_i} = \sum_{i=0}^{N}
	\left(\begin{array}{c}
	N \\
	i
	\end{array}\right) \cdot t^{i} \cdot (1-t)^{(N-i)} \cdot \overline{Q_i}
	\label{psi_L}
	\end{equation}

	\noindent where $\overline{Q_i}$ is the probability of percolation, at time $t$, conditioned that $\mathcal{O}_t=i$. Differentiating (\ref{psi_L}) \cite{choi}, it holds that
	\begin{equation}
	\dfrac{d\psi_{L}(t; k)}{dt}= \sum_{i=0}^{N}
	\left(\begin{array}{c}
	N \\
	i
	\end{array}\right) \cdot (i-Nt)\cdot t^{i-1} \cdot (1-t)^{(N-i-1)} \cdot \overline{Q_i}.
	\label{dpsi_L}
	\end{equation}

	To estimate $\overline{Q}_{i}$, we perform an analysis similar to that used in the Newmann-Ziff algorithm~\cite{ziff01}. A single run of the algorithm is obtained as follows:

	\begin{figure}[t]
		\begin{center}
			\includegraphics[height=6.8cm, width=16.5cm]{psi.jpg}
			\caption{Graphs of the $\psi_{L}(t; k)$ functions when $k=3$ and $d=2,3,4$. The dashed line marks the critical time obtained using the FSS (\ref{tc}). The graph's slope near the critical time grows as the $ L $ increases. }
			\label{psi}
		\end{center}
	\end{figure}

	\begin{itemize}
		\item[(i)]  Create a list with a permutation of all the $ N = dL^d $ bonds ($ e_1, e_2, ..., e_N $).
		\item[(ii)] Try to open the bonds following the order $ e_1 $, ..., $ e_N $ and check if percolation has occurred. If a bond does not satisfy the $ k $ constraint, then it remains closed.
		\item[(iii)] If percolation occurred when the bond $ e_i $ was opened, then we define $Q_j = 1$ for $j \geq i$. Otherwise $ Q_i = 0 $.
	\end{itemize}

	\noindent The estimate of $\overline{Q}_{i}$ is obtained as the mean of a large number of samples of $Q_i$.

	We study system with sizes $32 \leq L \leq 1024$ ($d=2$), $22 \leq L \leq 176$ ($d=3$) and $18\leq L \leq 46$ ($d=4$), with 6 values of $L$ for each $d$. The number of samples varies between $50000$ ($L=48$, $d=4$) until $10^6$ ($L=32$, $d=2$).

	For each $k$ and $d$, the critical time ($t_c$) is obtained by the finite-size scaling (FSS)
	\begin{equation}
	|\overline{t}_L-t_c| \sim L^{a},
	\label{tc}
	\end{equation}

	\noindent where $\overline{t}_L=\int_{0}^{1} t \cdot \frac{d\psi_{L}(t; k)}{dt} dt$ is the average concentration when percolation occurs for the first time and $a$ is a fitting parameter \cite{stauffer94}. The exponent $\nu$ was estimated from the scaling relation
	\begin{equation}
	\mbox{Max}\left[\dfrac{d\psi_{L}(t; k)}{dt} \right] \sim L^{-\frac{1}{\nu}},
	\label{nu}
	\end{equation}

	\begin{table*}[t]
		\centering
		\begin{small}
			\caption*{\ \textbf{Table 1:} Values obtained for $\frac{1}{\nu}$ and $t_c$. When $d=4$, the central values obtained for $t_c(7)$ and $t_c(8)$, marked with $^{*}$, were $ 0.16011996$ and $0.16011964$, respectively.}
			\label{tabela1}
			\begin{tabular}{|c|cc||c|cc||c|cc|}
				\hline

				\multicolumn{3}{|c||}{$d=2$} & \multicolumn{3}{c||}{$d=3$} & \multicolumn{3}{c|}{$d=4$} \\
				\hline
				$k$    &      $t_c$ &   $\frac{1}{\nu}$  &  $k$  &    $t_c$ &  $\frac{1}{\nu}$ &  $k$  &   $t_c$ &  $\frac{1}{\nu}$  \\  \hline
				\multirow{4}{*}{3}      &        \multirow{4}{*}{0.532393(15)}   &        \multirow{4}{*}{0.749(4)}    &  \multirow{1.5}{*}{3}    & \multirow{1.2}{*}{0.277691(10)}   &   \multirow{1.2}{*}{1.15(2)}    &  \multirow{1.2}{*}{3}  & \multirow{1.2}{*}{0.183205(30)} & \multirow{1.2}{*}{1.47(5)} \\
				\multirow{7}{*}{4}  &  \multirow{7}{*}{0.499994(16)}   &     \multirow{7}{*}{0.751(3)}      &   \multirow{3}{*}{4}   & \multirow{3}{*}{0.251319(6)} & \multirow{3}{*}{1.15(2)}     & \multirow{1.4}{*}{4}  &    \multirow{1.4}{*}{0.162716(22)}    &     \multirow{1.4}{*}{1.46(5)} \\
				&    &         &  \multirow{4.7}{*}{5}  & \multirow{4.7}{*}{0.248922(8)} & \multirow{4.7}{*}{1.14(2)}    & \multirow{1.8}{*}{5}    &  \multirow{1.8}{*}{0.160348(15)}    &  \multirow{1.8}{*}{1.47(3)}     \\
				&   &       &      &  &     & \multirow{2.3}{*}{6}     &  \multirow{2.3}{*}{0.160131(14)}  &  \multirow{2.3}{*}{1.47(4)}  \\
				&   &       &  \multirow{4.8}{*}{6}      & \multirow{4.8}{*}{0.248810(7)} &  \multirow{4.8}{*}{1.14(2)}   & \multirow{2.8}{*}{7}    &   \multirow{2.8}{*}{\ 0.160120$(15)^{*}$}    &  \multirow{2.8}{*}{1.46(4)}  \\
				&   &      &        & &     & \multirow{3.2}{*}{8}    &  \multirow{3.2}{*}{\ 0.160120$(15)^{*}$}    &   \multirow{3.2}{*}{1.47(4)}   \\
				&   &      &       &  &  &       &  & \\
				\hline
			\end{tabular}
		\end{small}
	\end{table*}

	\noindent where the left-hand side represents the maximum of $\frac{d\psi_{L}(t; k)}{dt}$ for size $L$.

	We obtained five independent estimates for each parameter studied; the central value is the average of them. The uncertainties were obtained using the standard deviation of the mean.

	\section{Results}

	Although analytical or numerical results already exist, we also simulate the cases $k=2$ and $k=2d$ for each $d$. When $k=2$ we found that $t_c=\infty$ for all $d$, as obtained analytically in \cite{de_lima}. When $d=2$ and $k=4$ , we get $t_c=0.499994(16)$ that is consistent with the exact value $t_c=\frac{1}{2}$ (see \cite{grimmett} for a rigorous proof). For $d=3$ and $k=6$, we estimated $t_c=0.248810(7)$; for $d=4$ and $k=8$, we estimated $t_c=0.160120(15)$. These values are consistent with the numerical results for the ordinary Bernoulli percolation threshold obtained for $d=3$: $0.2488126(5)$ \cite{lorenz_d3_1}, $0.2488125(25)$ \cite{dammer_d3_2} and $0.24881182(10)$ \cite{wang_d3_3}; and for $d=4$: $0.1601310(10)$ \cite{dammer_d3_2}, $0.16013122(6)$ \cite{mertens_d4_1} and $0.1601312(2)$ \cite{xun_d4_2}.

	\begin{figure}[t!]
		\begin{center}
			\hspace{0.11cm}\includegraphics[height=5cm, width=8.5cm]{pcZ2k3.jpg} \quad
			\includegraphics[height=5cm, width=7.3cm]{vZ2k3.jpg} \\[10pt]
			\includegraphics[height=5cm, width=8.5cm]{pcZ3k3.jpg} \quad
			\includegraphics[height=5cm, width=7.3cm]{vZ3k3.jpg} \\[10pt]
			\includegraphics[height=5cm, width=8.5cm]{pcZ4k3.jpg} \quad
			\includegraphics[height=5cm, width=7.3cm]{vZ4k3.jpg}
			\caption{Graphs obtained for $k=3$, considering $d=2,3,4$, to determine $t_c$ and $\frac{1}{\nu}$. The left plots were obtained using the FSS (\ref{tc}), and the right plots through log-log plots using (\ref{nu}). The error bars are smaller than the symbols in the graphics where they are not shown.}
			\label{k3}
		\end{center}
	\end{figure}

	For the case $k=3$ when $d=2,3,4$, Figure \ref{psi} presents graphs of the function $\psi_{L}(t,k)$, and Figure \ref{k3} the graphs obtained through the relations (\ref{tc}) and (\ref{nu}) to estimate $\frac{1}{\nu}$ and $t_c$, respectively. Table 1 shows the results obtained for $\frac{1}{\nu}$ and $t_c$ for all $k$ and $d$ values analyzed. For $d=3$ it becomes clear that $t_c$ is monotonous in $k\geq 3$. If $ d = 4 $, the central values also suggest such behavior, however, considering the uncertainties obtained, we can only say that such behavior will occur for $k \leq 6$.

	In order to obtain more evidence of the possible monotonicity from $t_c(k)$ when $k \geq 6 $, we simulated the model, for $ L = 46 $, considering $M=2 \times 10^5$ configurations $U^{i}=\{U^{i}_{e}\}_{e \in \mathcal{E}}$, $1 \leq i \leq M$. Once a configuration $U^{i}$ is fixed, we denote by $t(k;i)$ as the time that the system percolated for constraint $k$. We found that $t(8;i) \leq t(7;i) \leq t(6; i)$ for all $1\leq i\leq M$. Furthermore, $t(8;i)<t(7;i)$, $t(7;i)<t(6;i)$ and $t(8;i) < t(6;i)$ for $0.096\%$, $3.429\%$ and $3.52\%$ of the analyzed configurations, respectively. Such results support the statement that the critical time will also be monotonous, in $3\leq k \leq 8$, when $d=4$.

	The values obtained for the critical exponent $\nu$, for all $k$, are compatible with the values of ordinary Bernoulli percolation model in $d=2$ ($\frac{4}{3}$ \cite{stauffer94}), $d=3$ ($0.87619(12)$ \cite{xu}, $0.8774(13)$ \cite{koza}) and $d=4$ ($0.6845(23)$ \cite{zhang}). This fact suggests that probably the CDPM is in the same universality class of the ordinary Bernoulli percolation model for all $d$ and $ k \geq 3$.

	The estimates obtained for $x_k(L)$, for $k=1,2,3$, when $d=2$ show that its value varies less than $0.009\%$ when we vary $L$ from $32$ to $1024$. In addition, the uncertainty in the estimate decreases from $\approx 3 \times 10^ {-3}$ to $\approx 8 \times 10^{-5}$. For this reason, we estimate $x_k(L)$ only for the highest value of $L$. Table 2 lists the values obtained. Through a plot of $x_{k}$ versus $k$, we see that for all $d$ its values grow linearly (Figure \ref{lin}).

	\begin{figure}[t]
		\begin{center}
			\includegraphics[height=7.5cm]{max_ocup.jpg}
			\caption{Estimates for maximum random concentration. Dimension $d=2$, triangles; $d=3$, circles; $d=4$, squares. The slopes obtained were $0.2419(16)$, $0.1626(6)$, and $0.1225(3)$ for $d=2,3,4$, respectively. The error bars are smaller than
				the symbols.}
			\label{lin}
		\end{center}
	\end{figure}

	\begin{table*}[]
		\centering
		\begin{small}
			\caption*{\ \textbf{Table 2:} Estimated random maximum concentration ($x_k$) for all $ k $ and $ d $ values.}
			\begin{tabular}{c|ccccccc|}
				\cline{2-8}
				& $x_1$      & $x_2$      & $x_3$      & $x_4$      & $x_5$      & $x_6$      & $x_7$      \\[2pt]  \hline
				\multicolumn{1}{|c|}{$d=2$} & 0.22670(6) & 0.46578(8) & 0.71094(5) & -          & -          & -          & -          \\[3pt]
				\multicolumn{1}{|c|}{$d=3$} & 0.15307(2) & 0.31346(2) & 0.47511(3) & 0.63817(3) & 0.80474(3) & -          & -          \\[3pt]
				\multicolumn{1}{|c|}{$d=4$} & 0.11611(2) & 0.23710(2) & 0.35869(2) & 0.48070(2) & 0.60324(3) & 0.72672(2) & 0.85274(2) \\ \hline
			\end{tabular}
		\end{small}
	\end{table*}

	\section{Conclusion}

	We study proprieties of the critical time for the Constrained-degree percolation model on the hypercubic lattice $\mathbb {Z}^d $ when $d=2,3,4$. It has a long-range dependence and applications in the study of dimers and polymers. One interesting problems related to this model is to check if $t_c(k)$ is a monotone function in the constrained parameter $k$; if the answer is affirmative, it is natural also to ask what is the smallest value of $k$ such that the model has a non-trivial phase transition ($t_c \leq 1$). When $k=1$ there will be only isolated bonds (there is no percolation) and when $k = 2d$ this model reduces to ordinary Bernoulli bond percolation, therefore the non-trivial cases occur when $k \in \{2, ..., 2d-1\}$.

	In \cite{de_lima}, it was proven that: {\it (i)} for all $d\geq 2$ and $k=2$ there is no percolation or an time $t$ and {\it (ii)} for the squares lattice $d=2$, when $k=3$ it holds that $\frac{1}{2}<t_c<1$, that is, there is a non-trivial phase transition (the exact value of $t_c$ is not known when $ k=3$).

	In this work we determined, via numerical simulations, the value of $t_c$ for all non-trivial cases and observed there is a phase transition for constrained $k=3$, for $d=2,3,4$. In addition, we find numerical evidence that $t_c$ is monotonous in $k$,  and that the expected value of open bonds in the lattice when $t=1$ (maximum random concentration, $x_k$) grows linearly in $k$. We also estimated the correlation length exponent $\nu$ and obtained values compatible with those of the ordinary Bernoulli percolation model, suggesting that both models probably are in the same universality class.

	This present study can contribute to the analysis of applied models related to the Constrained-degree percolation model on $\mathbb{Z}^{d}$, besides helping researchers in the analytical and numerical study of this model.

	\section*{Acknowledgments}
	We thank the authors of \cite{de_lima} for the example of Figure 2. The research of A.P.F.A.\ is supported in part by CNPq grant 308792/2018-1 and FAPEMIG. The research of B.N.B.L.\ is supported in part by CNPq grant 305811/2018-5 and FAPERJ (Pronex E-26/010.001269/2016).


TITLE: Proving a graph doesn't have a perfect matching
QUESTION [1 upvotes]: Consider the following graph:

Find a perfect matching or prove one doesn't exist.

I don't think a perfect matching exists here, as the vertices $a_2, a_3$ and $a_4$ are problematic to us, but I'm having some trouble proving this. Using Hall's theorem, we can prove that a matching of a certain cardinality doesn't exist, but how am I supposed to know the cardinality of the perfect matching in order to prove my claim? Can someone give me a hint how to apply the theorem here?
EDIT: Can I assume that the cardinality of the perfect matching $|M| = 2$, as the smallest vertex cover is {$a_5, a_4$}, and then find two vertices that break Hall's condition?

REPLY [2 votes]: As Daniel Mathias gave the hint;
The graph $G$ is disconnected. Subgraph generated by $\{a_2,b_2,b_3,a_5,a_6,b_5,b_6\}$ is one component and subgraph generated by $\{a_1,a_3,a_4,b_1,b_4\}$ is another component.
Now if $G$ has a perfect matching then both components also have perfect matching. But none of the components have perfect matching because each has odd number of vertices (Reason: A graph has perfect matching with $M$ edges then it necessarily has $2M$ vertices).


TITLE: Linearisation of a Quasilinear Elliptic PDE
QUESTION [1 upvotes]: I have noticed that the word 'linearisation' can have different meanings in different places in the literature.
For example, if one has a second-order quasilinear elliptic PDE, what would be the linearisation of this PDE be and what would a solution be?  I assume it's not just discarding non-linear terms.
The equation (for $f$) is this one, where $g$ is a $3$-metric, $k$ denotes the corresponding second fundamental form.  $\phi$ is determined using another equation but for now, the form of $\phi$ could be assumed to obtain solutions.
$ \Bigg( g^{ij} - \frac{\phi^2 f^i f^j}{1 + \phi^2 |\nabla f|^2} \Bigg) \Bigg( \frac{\phi \nabla_{ij}f + \phi_i f_j + \phi_j f_i}{\sqrt{1 + \phi^2 |\nabla f|^2}} - k_{ij} \Bigg) =0,  $
$\phi_j$ denotes $\partial \phi/ \partial x_j$.  So essentially, what would one mean by a linearisation of this equation?

REPLY [1 votes]: A solution is a function which satisfies the PDE.
Given a solution $u_0$, the linearisation of aPDE about $u_0$ is obtained by substituting $u(x) = u_0(x)+\varepsilon v(x)$ then only keeping terms of order $\varepsilon$. This gives a linear PDE where the unknown function is now $v$. Note that "the linearisation of a PDE" doesn't make sense without specifying the solution about which you are linearising -  this is analogous to a Taylor series expansion which is always taken about a given point. Often, however, $u_0$ is obvious from context (it is usually an equilibrium solution for example) in which case authors will frequently just refer to the "linearisation about $u_0$" as simply "the linearisation".
Here is an example. Consider the equation $$(1+ \vert \nabla u \vert^2)\Delta u - \sum_{i,j=1}^n \frac{\partial u}{\partial x_i} \frac{\partial u}{\partial x_j} \frac{\partial^2u}{\partial x_ix_j}=0\qquad \text{in } B_1 \tag{$\ast$} $$ with $u=0$ on $\partial B_1$. (If you're curious this is the equation for prescribed zero mean curvature. I chose this because, to me, your equation looks of mean curvature type). An obvious solution to $(\ast)$ is $u_0(x) =0$ for all $x\in B_1$.
Now let us substitute $u(x) = u_0(x) + \varepsilon v(x)=\varepsilon v(x)$ into $(\ast)$. This gives $$0= \varepsilon(1+ \varepsilon^2\vert \nabla v \vert^2)\Delta v - \varepsilon^3\sum_{i,j=1}^n \frac{\partial v}{\partial x_i} \frac{\partial v}{\partial x_j} \frac{\partial^2v}{\partial x_ix_j}= \varepsilon\Delta v+ \varepsilon^3 \bigg (\vert \nabla v \vert^2\Delta v - \sum_{i,j=1}^n \frac{\partial v}{\partial x_i} \frac{\partial v}{\partial x_j} \frac{\partial^2v}{\partial x_ix_j} \bigg ). $$ From inspection, the order $\varepsilon$ equation is $$\Delta v=0 \qquad \text{in } B_1$$ with $v=0$ on $\partial B_1$. This is the linearisation of $(\ast)$  about the solution $u_0=0$. Tip: If it is not clear from inspection what the order $\varepsilon$ terms are then an effective strategy is to calculate $\frac{d}{d\varepsilon}\big \vert_{\varepsilon=0}$ for both sides.


text:
\begin{document}
	\title{Documentation of quantumarticle.cls v\quantumarticleversion}
	\maketitle

	\tableofcontents

	\clearpage

	\section{Introduction}
	The quantumarticle document class is the preferred document class for papers that will be submitted to \href{https://quantum-journal.org/}{Quantum -- the open journal of quantum science}. It is based on the widely used \texttt{article} document class and designed to allow a seamless transition from documents typeset with \texttt{article}, \texttt{revtex4-1} and the \texttt{elsarticle} document classes.

	As Quantum is an arXiv overlay journal, all papers have to be submitted to the arXiv. To make this submission process as user-friendly as possible, the quantumarticle document class implements a number of arXiv-specific checks, which however can be deactivated via the option \texttt{noarxiv}.

        An example for this is a check that is meant to make sure that the arXiv produces a PDF file and that hyperlinks are correctly broken across multiple lines.
        To ensure this, the arXiv \href{https://arxiv.org/help/submit_tex}{recommends} putting
	\begin{verbatim}
	\pdfoutput=1
	\end{verbatim}
	within the first 5 lines of your main LaTeX file. By default, the quantumarticle document class will throw an error if this line is missing. Just like other similar checks, this can be deactivated by means of the \texttt{noarxiv} option, or only this specific check can be deactivated via the option \texttt{nopdfoutputerror}.

        Giving the \texttt{noarxiv} option also disables a number of other features of quantum article and removes any Quantum related branding from the document. Authors who would like to keep the checks active but still use this document class for manuscripts not intended for submission to Quantum and therefore without Quantum branding can use the \texttt{unpublished} option.

        One feature deactivated by both \texttt{unpublished} and \texttt{noarxiv} for example, is the ``title click feature'' of quantumarticle. As this document class can be used for arbitrary documents, Quantum implements a feature where readers can, by clicking on the title of a manuscript, verify whether this manuscript was actually published by Quantum. Obviously this is an unwanted feature in any manuscript not accepted in Quantum and it can thus be deactivated with the \texttt{unpublished} option.

	\section{Typesetting documents}

        The following are guidelines for the usage of the quantumarticle document class for manuscripts to be submitted to or accepted in Quantum. A detailed description of the functionality and options of the document class follow in Section~\ref{sec:options}.

	\subsection{Recommendations on structure}
	In the \texttt{twocolumn} layout and without the \texttt{titlepage} option a paragraph without a previous section title may directly follow the abstract.
	In \texttt{onecolumn} format or with a dedicated \texttt{titlepage}, this should be avoided.

	Longer articles should include a section that, early on, explains the main results, their limitations, and assumptions.
	This section can be used to, for example, present the main theorem, or provide a summary of the results for a wider audience.

	\subsection{Title information}
	The title of the document is given via the common \texttt{title} command. Note that clicking the title performs a search for that title on \href{http://quantum-journal.org}{quantum-journal.org}.
	In this way readers can easily verify whether a work using the \texttt{quantumarticle} class was actually published in Quantum. By giving the \texttt{accepted=YYYY-MM-DD} option, with \texttt{YYYY-MM-DD} the acceptance date, the note ``Accepted in Quantum YYYY-MM-DD, click title to verify'' can be added to the bottom of each page to clearly mark works that have been accepted in Quantum.

	You should call \texttt{\textbackslash{}maketitle} before your running text starts.

	\subsubsection{Authors and affiliations}
	You can provide information on authors and affiliations in the common format also used by \texttt{revtex}:
	\begin{verbatim}
	\author{Author 1}
	\author{Author 2}
	\affiliation{Affiliation 1}
	\author{Author 3}
	\affiliation{Affiliation 2}
	\author{Author 4}
	\affiliation{Affiliation 1}
	\affiliation{Affiliation 3}
	\end{verbatim}
	In this example affiliation 1 will be associated with authors 1, 2, and 4, affiliation 2 with author 3 and affiliation 3 with author 4.
	Repeated affiliations are automatically recognized and typeset in \texttt{superscriptaddress} style.
	Alternatively you can use a format similar to that of the \texttt{authblk} package and the \texttt{elsarticle} document class to specify the same affiliation relations as follows:
	\begin{verbatim}
	\author[1]{Author 1}
	\author[1]{Author 2}
	\author[2]{Author 3}
	\author[1,3]{Author 4}
	\affil[1]{Affiliation 1}
	\affil[2]{Affiliation 1}
	\affil[3]{Affiliation 1}
	\end{verbatim}

	\subsubsection{Other author related information}
	The quantumarticle document class supports further commands that are author specific:
	\begin{commands}
		\command{email}{
			E-Mail address of the author, displayed in the bottom of the page.
		}

		\command{homepage}{
			Homepage of the author, displayed in the bottom of the page.
		}

		\command{thanks}{
			Additional text that is displayed in the bottom of the page.
		}

		\command{orcid}{
			If the ORCiD of the author is given, his name will become a link to his ORCiD profile.
		}
	\end{commands}

	\subsection{Abstract}
	The abstract is typeset using the common \texttt{abstract} environment. In the standard, \texttt{twocolumn}, layout the abstract is typeset as a bold face first paragraph.
	In \texttt{onecolumn} layout the abstract is placed above the text.
	Both can be combined with the \texttt{titlepage} option to obtain a format with dedicated title and abstract pages that are not included in the page count.
	This format can be more suitable for long articles.
	The \texttt{abstract} environment can appear both before and after the \texttt{\textbackslash{}maketitle} command and calling \texttt{\textbackslash{}maketitle} is optional, as long as there is an \texttt{abstract}.
	Both \texttt{abstract} and \texttt{\textbackslash{}maketitle} however must be placed after all other \texttt{\textbackslash{}author}, \texttt{\textbackslash{}affiliation}, etc.\ commands.

	\subsection{Sectioning}
	Sections, subsections, subsubsections, and paragraphs should be typeset with the standard LaTeX commands. The paragraph is the smallest unit of sectioning. Feel free to end the paragraph title with a full stop if you find this appropriate.

	\subsection{Equations}
	You can use the standard commands for equations. For multi-line equations \texttt{align} is preferable over \texttt{eqnarray}, please refrain from using the latter.	For complex equations you may want to consider using the \texttt{IEEEeqnarray} environment from the \texttt{IEEEtrantools} package.

	How you refer to equations is up to you, but please be consistent and use the \texttt{\textbackslash{}eqref\{\dots\}} command instead of writing \texttt{(\textbackslash{}ref\{\dots\})}.

	As a courtesy for your readers and referees, please suppress equation numbers only if there is a specific reason to do so, to not make it unnecessarily difficult to refer to individual results and steps in derivations.

	Very wide equations can be shown expanding over both columns using the \texttt{widetext} environment.
	In \texttt{onecolumn} mode, the \texttt{widetext} environment has no effect.
	To enable this feature in \texttt{twocolumn} mode, \texttt{quantumarticle} relies on the package \texttt{ltxgrid}.
	Unfortunately this package has a bug that leads to a sub-optimal placement of extremely long footnotes.

	\subsection{Floats}
	Every floating element must have an informative caption and a number. The caption can be placed above, below, or to the side of the figure, as you see fit.

	Feel free to place them at the top or bottom of the page, or in the middle of a paragraph as you see fit. Try to place them on the same page as the text referring to them. A figure on the first page can help readers remember and recognize your work more easily.

	\subsubsection{Figures}
	Figures are typeset using the standard \texttt{figure} environment for single-column figures and \texttt{figure*} for multi-column figures.

	\subsubsection{Tables}
	Tables are typeset using the standard \texttt{table} environment for single-column tables and \texttt{table*} for multi-column tables.

	\subsection{Plots}
	Quantum provides a Jupyter notebook to create plots that integrate seamlessly with \texttt{quantumarticle}.

	\subsection{Footnotes}
	Footnotes are typeset using the \texttt{footnote} command. They will appear in the bottom of the page. Please do only use footnotes when appropriate and do not mix footnotes with references.

	\subsection{References}
	Citations to other works should appear in the References section at the end of the work.

	\paragraph{Important:} As Quantum is a member of Crossref, all references to works that have a DOI \emph{must} be hyperlinked according to the DOI. Those links must start with \texttt{https://doi.org/} (preferred), or \texttt{http://dx.doi.org/}. Direct links to the website of the publisher are not sufficient. This can be achieved in several ways, depending on how you are formatting your bibliography.

	\subsubsection{Manual bibliography}
	Suppose the DOI of an article that you want to cite is \texttt{10.22331/idonotexist}. If you are formatting your bibliography manually, you can cite this work using the following in your \texttt{thebibliography} environment:
	\begin{verbatim}
	\bibitem{examplecitation}
	Name Surname,
	\href{https://doi.org/10.22331/
	idonotexist}{Quantum
	\textbf{123}, 123456 (1916).}
	\end{verbatim}

	\paragraph{Important:} If you are formatting your bibliography manually, please do not group multiple citations into one \texttt{\textbackslash{}bibitem}.
	Having to search through multiple references to find the cited result makes your work less accessible for authors and grouping references can screw up our automatic extraction of citations.

	\subsubsection{BibTeX bibliography}
	We encourage the use of BibTeX to generate your bibliography from the BibTeX meta-data provided by publishers.
	For DOI linking to work, the BibTeX file must contain the \texttt{doi} field as for example in:
	\begin{verbatim}
	@article{examplecitation,
	author = {Surname, Name},
	title = {Title},
	journal = {Quantum},
	volume = {123},
	page = {123456},
	year = {1916},
	doi = {10.22331/idonotexist},
	}
	\end{verbatim}
	Several authors had problems because of Unicode characters in their BibTeX files.
	Be advised that \href{http://wiki.lyx.org/BibTeX/Tips}{BibTeX does not support Unicode characters}.
	All special characters must be input via their respective LaTeX commands.

	\paragraph{natbib}
	If you are using BibTeX, you can load the \texttt{natbib} package by putting
	\begin{verbatim}
	\usepackage[numbers,sort&compress]{natbib}
	\end{verbatim}
	in the preamble of your document and then use the \texttt{plainnat} citation style by including your BibTeX bibliography \texttt{mybibliography.bib} where you want the bibliography to appear as follows:
	\begin{verbatim}
	\bibliographystyle{plainnat}
	\bibliography{mybibliography}
	\end{verbatim}
	The quantumarticle class automatically detects that the \texttt{natbib} package was loaded and redefines the \texttt{\textbackslash{}doi} command to create hyperlinks.
	This is likely the easiest option if you are converting from another document class.

	\paragraph{BibLaTeX}
	If you want to use BibLaTeX, you can instead add
	\begin{verbatim}
	\usepackage[backend=bibtex]{biblatex}
	\addbibresource{mybibliography.bib}
	\end{verbatim}
	to the preamble of your document and then output the bibliography with
	\begin{verbatim}
	\printbibliography
	\end{verbatim}
	where appropriate.
	You then have to upload the .bbl file along with the other source files when submitting to the arXiv.
	Due to incompatibilities between different BibLaTeX versions we unfortunately cannot recommend this option.

	The quantumarticle class automatically detects that the \texttt{biblatex} package was loaded, sets the default option \texttt{doi=true} to include the DOI in the bibliography, and declares a suitable field format to make it a hyperlink.
	Due to issues with \texttt{biber} we recommend to use the \texttt{bibtex} backend of \texttt{biblatex}.

	Feel free to change the appearance of citations in any way you like by using a different \texttt{bibliographystyle} or via the advanced mechanisms provided by BibLaTeX.
	The only two requirements are that citations must uniquely identify the cited work and that they must contain a DOI hyperlink whenever possible.

	\subsection{Appendix}
	Quantum allows the usage of appendices. Ideally, the command \texttt{\textbackslash{}appendix} should be put before the appendices to get appropriate section numbering. The appendices are then numbered alphabetically, with numeric (sub)subsection numbering.
	Equations continue to be numbered sequentially.
	You are free to change this in case it is more appropriate for your article, but a consistent and unambiguous numbering of sections and equations must be ensured.

	If you want your appendices to appear in \texttt{onecolumn} mode but the rest of the document in \texttt{twocolumn} mode, you can insert the command \texttt{\textbackslash{}onecolumn\textbackslash{}newpage} before \texttt{\textbackslash{}appendix}.

	\subsection{Recommended packages}
	Quantum encourages you to load the following extra packages:
	\begin{verbatim}
	\usepackage[utf8]{inputenc}
	\usepackage[english]{babel}
	\usepackage[T1]{fontenc}
	\usepackage{amsmath}
	\usepackage{hyperref}
	\end{verbatim}
	If you do not load the \texttt{hyperref} package, quantumarticle automatically loads it for you.
	Packages that change font settings, such as \texttt{times} or \texttt{helvet} should be avoided.

	\section{Package options}
        \label{sec:options}
	The quantumarticle document class provides a multitude of package options that change the appearance of the final document to the liking of the user. Default options that need not be given are marked with \texttt{(default)}. Options that are only provided for compatibility reasons and that will not alter the typesetting of the document are marked with \texttt{(C)}.

	\subsection{Document setup}

	\subsubsection{Document format}
	The document class does not specify a default paper size and an error is raised if none is specified. The reason for this is that different LaTeX compilers and localizations can cause paper sizes to vary between setups and this can lead to unexpected results when multiple authors collaborate on a document and the final typesetting is performed by a third party, such as the arXiv.

	\begin{options}
		\option{a4paper}{
			Sets the paper size to ISO A4.
		}
		\option{a5paper}{
			Sets the paper size to ISO A5.
		}
		\option{b5paper}{
			Sets the paper size to ISO B5.
		}
		\option{letterpaper}{
			Sets the paper size to ANSI Letter.
		}
		\option{legalpaper}{
			Sets the paper size to ANSI Legal.
		}
		\option{executivepaper}{
			Sets the paper size to ANSI Executive.
		}
	\end{options}

	\subsubsection{Orientation}
	By default, the orientation is portrait.

	\begin{options}
		\option{landscape}{
			Sets the orientation of the document to landscape.
		}
	\end{options}

	\subsubsection{Fontsize}
	By default, the fontsize is 10pt.

	\begin{options}
		\defaultoption{10pt}{
			Sets the normal fontsize to 10pt.
		}
		\option{11pt}{
			Sets the normal fontsize to 11pt.
		}
		\option{12pt}{
			Sets the normal fontsize to 12pt.
		}
	\end{options}

	\subsubsection{Sides}
	By default, the setting is oneside.

	\begin{options}
		\defaultoption{oneside}{
			Prepares the document onesided.
		}
		\option{twoside}{
			Prepares the document twosided.
		}
	\end{options}

	\subsubsection{Titlepage}
	By default, the document class is typeset with no extra titlepage.

	\begin{options}
		\option{titlepage}{
			Displays the title on a separate titlepage.
		}
		\compatibilityoption{notitlepage}{
			Displays the title on the same page as the text.
		}
	\end{options}

	\subsubsection{Draft}
	By default, the document class is typeset as final.

	\begin{options}
	\option{draft}{
		Marks the document as a draft.
	}
	\defaultoption{final}{
		Does not mark the document as a draft.
	}
	\end{options}

	\subsection{Style options}

	\subsubsection{Equations}
	By default, equations are typeset centered, with equation numbers on the right.

	\begin{options}
		\option{fleqn}{
			Display equations left aligned.
		}
		\option{leqno}{
			Display equation numbers to the left of the equation.
		}
	\end{options}

	\subsubsection{Bibliography}

	\begin{options}
		\option{openbib}{
			Display elements of bibliography entries on separate lines, \href{https://tex.stackexchange.com/questions/56628/custom-references-page-with-additional-line-breaks}{see here}.
		}
	\end{options}

	\subsection{Pre-loaded packages}
	The document class only loads packages necessary for its operation, such as \texttt{geometry} or \texttt{hyperref}. For compatibility reasons, there are commands that load certain widely used packages for the user.

	\begin{options}
		\option{amsfonts}{
			Load the package \texttt{amsfonts}.
		}
		\compatibilityoption{noamsfonts}{
			Don't load the package \texttt{amsfonts}.
		}
		\option{amssymb}{
			Load the package \texttt{amssymb}.
		}
		\option{amssymbol}{
			\ditto
		}
		\compatibilityoption{noamssymb}{
			Don't load the package \texttt{amssymb}.
		}
		\option{amsmath}{
			Load the package \texttt{amsmath}.
		}
		\compatibilityoption{noamsmath}{
			Don't load the package \texttt{amsmath}.
		}
	\end{options}

	\subsection{Publication}
	By default, all documents that are created with the \texttt{quantumarticle} document class are treated as possible submissions to Quantum.

	\begin{options}
		\option{accepted=YYYY-MM-DD}{
			Adds the note 'Accepted in Quantum YYYY-MM-DD' to the document.
		}
		\option{unpublished}{
			Intended for works that are not published in Quantum. Disables all Quantum-related branding.
		}
		\option{noarxiv}{
			Intended for works that are not meant to be uploaded to the arXiv. Disables all arXiv-related sanity checks in the document. This option also sets \texttt{unpublished}.
		}
	\end{options}

	\subsection{Compatibility}
	The \texttt{quantumarticle} class aims to be maximally compatible with documents that were previously typeset with other document classes. For this reason, a number of options are present for the sole purpose of compatibility but do not have any effect on how your document will be typeset These include:

	\begin{options}
		\compatibilityoption{checkin}{No effect.}
		\compatibilityoption{preprint}{\ditto}
		\compatibilityoption{reprint}{\ditto}
		\compatibilityoption{manuscript}{\ditto}
		\compatibilityoption{noshowpacs}{\ditto}
		\compatibilityoption{showpacs}{\ditto}
		\compatibilityoption{showkeys}{\ditto}
		\compatibilityoption{noshowkeys}{\ditto}
		\compatibilityoption{balancelastpage}{\ditto}
		\compatibilityoption{nobalancelastpage}{\ditto}
		\compatibilityoption{nopreprintnumbers}{\ditto}
		\compatibilityoption{preprintnumbers}{\ditto}
		\compatibilityoption{hyperref}{\ditto}
		\compatibilityoption{bibnotes}{\ditto}
		\compatibilityoption{nobibnotes}{\ditto}
		\compatibilityoption{footinbib}{\ditto}
		\compatibilityoption{nofootinbib}{\ditto}
		\compatibilityoption{altaffilletter}{\ditto}
		\compatibilityoption{altaffilsymbol}{\ditto}
		\compatibilityoption{superbib}{\ditto}
		\compatibilityoption{citeautoscript}{\ditto}
		\compatibilityoption{longbibliography}{\ditto}
		\compatibilityoption{nolongbibliography}{\ditto}
		\compatibilityoption{eprint}{\ditto}
		\compatibilityoption{noeprint}{\ditto}
		\compatibilityoption{author-year}{\ditto}
		\compatibilityoption{numerical}{\ditto}
		\compatibilityoption{galley}{\ditto}
		\compatibilityoption{raggedbottom}{\ditto}
		\compatibilityoption{tightenlines}{\ditto}
		\compatibilityoption{lengthcheck}{\ditto}
		\compatibilityoption{reprint}{\ditto}
		\compatibilityoption{eqsecnum}{\ditto}
		\compatibilityoption{secnumarabic}{\ditto}
		\compatibilityoption{floats}{\ditto}
		\compatibilityoption{stfloats}{\ditto}
		\compatibilityoption{endfloats}{\ditto}
		\compatibilityoption{endfloats*}{\ditto}
		\compatibilityoption{osa}{\ditto}
		\compatibilityoption{osameet}{\ditto}
		\compatibilityoption{opex}{\ditto}
		\compatibilityoption{tops}{\ditto}
		\compatibilityoption{josa}{\ditto}
		\compatibilityoption{byrevtex}{\ditto}
		\compatibilityoption{floatfix}{\ditto}
		\compatibilityoption{nofloatfix}{\ditto}
		\compatibilityoption{ltxgridinfo}{\ditto}
		\compatibilityoption{outputdebug}{\ditto}
		\compatibilityoption{raggedfooter}{\ditto}
		\compatibilityoption{noraggedfooter}{\ditto}
		\compatibilityoption{frontmatterverbose}{\ditto}
		\compatibilityoption{linenumbers}{\ditto}
		\compatibilityoption{nomerge}{\ditto}
		\compatibilityoption{hypertext}{\ditto}
		\compatibilityoption{frontmatterverbose}{\ditto}
		\compatibilityoption{inactive}{\ditto}
		\compatibilityoption{groupedaddress}{\ditto}
		\compatibilityoption{unsortedaddress}{\ditto}
		\compatibilityoption{runinaddress}{\ditto}
		\compatibilityoption{superscriptaddress}{\ditto}
		\compatibilityoption{aps}{Triggers revtex compatibility mode}
		\compatibilityoption{pra}{\ditto}
		\compatibilityoption{prb}{\ditto}
		\compatibilityoption{pre}{\ditto}
		\compatibilityoption{prl}{\ditto}
		\compatibilityoption{prx}{\ditto}
		\compatibilityoption{aip}{\ditto}
	\end{options}

	\section{Contributing}
	In case you encounter problems using the article class please consider opening a bug report in our \href{https://github.com/quantum-journal/quantum-journal/issues}{bug-tracker on GitHub}. You can also contact us via email under \texttt{latex@quantum-journal.org}, but it may take significantly longer to get a response. In any case we need the full source of a document that produces the problem and the log file showing the error to help you.

	Improvements submitted as pull requests against the \href{https://github.com/quantum-journal/quantum-journal/tree/develop}{develop} branch are very much appreciated!

	\section{Copyright}
	Copyright 2019 Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften (\url{http://quantum-journal.org/about/}).

	The document class \texttt{quantumarticle} is derived from \texttt{article} available from \url{https://www.ctan.org/pkg/article}.

	It may be distributed and/or modified under the conditions of the LaTeX Project Public License, either version 1.3c of this license or (at your option) any later version. The latest version of this license is in \url{http://www.latex-project.org/lppl.txt} and version 1.3c or later is part of all distributions of \LaTeX version 2005/12/01 or later.
\end{document}

config: arxiv