Monotonicity of Markov chain transition probabilities via quasi-stationarity -- an application to Bernoulli percolation on $C_k \times Z$

TL;DR

This work investigates monotonicity of Markov chain transition probabilities through the lens of quasi-stationary distributions, and applies the framework to Bernoulli percolation on the layered cylinder graph $C_k^\times$. It builds a Markov chain of layer-by-layer infection patterns and proves uniform onset bounds for monotonicity of connection probabilities, leveraging explicit quasi-stationary convergence rates from Champagnat–Villemonais. The authors derive a concrete, although large, bound $N(k) = 500 k^6 1.95^k$ such that for all $n \ge N(k)$, $p \in (0,1)$ and $m \in C_k$, $P_p((0,0) \leftrightarrow (m,n)) \ge P_p((0,0) \leftrightarrow (m,n+1))$, with complementary results on the expected number of infected vertices per layer for small or large $p$. The approach, combining a general monotonicity criterion with a combinatorial percolation analysis on $C_k^\times$, provides a concrete pathway to proving monotonicity phenomena in layered graphs, and complements related bunkbed-type conjectures in percolation theory.

Abstract

Let $X_n, n \ge 0$ be a Markov chain with finite state space $M$. If $x,y \in M$ such that $x$ is transient we have $P^y(X_n = x) \to 0$ for $n \to \infty$, and under mild aperiodicity conditions this convergence is monotone in that for some $N$ we have $\forall n \ge N: P^y(X_n = x)$ $\ge P^y(X_{n+1} = x)$. We use bounds on the rate of convergence of the Markov chain to its quasi-stationary distribution to obtain explicit bounds on $N$. We then apply this result to Bernoulli percolation with parameter $p$ on the cylinder graph $C_k \times Z$. Utilizing a Markov chain describing infection patterns layer per layer, we thus show the following uniform result on the monotonicity of connection probabilities: $\forall k \ge 3\, \forall n \ge 500k^6 2^k \,\forall p \in (0,1) \, \forall m \in C_k\!\!:$ $P_p((0,0) \leftrightarrow (m,n)) \ge P_p((0,0) \leftrightarrow (m,n+1))$. In general these kind of monotonicity properties of connection probabilities are difficult to establish and there are only few pertaining results.

Figures (11)

• Figure 1: The picture shows a realization of Bernoulli percolation on $C_5^\times$ restricted to $C_5 \times \{-1,0,1,2,3\}$. The vertices on the right side should be identified with the corresponding vertices on the left side, the origin $(0,0)$ is marked with a thick dot, open edges are drawn, closed edges are dotted. The corresponding values of the chain $\mathcal{X}_n$ are given on the right.
• Figure 2: For $C_k$ with $k = 8$, this shows a bond configuration in $E_{[1,3]}$ contributing to $\{\mathcal{X}_3^{y} = x\}$, where $x = \{\{*,0,2,6\}, \{1\}, \{3,5\}, \{4\}, \{7\}\}$ and $y \in M^*$ such that $0 \sim_y *$. The top layer is $V_3$, drawn bonds are open, dotted bonds are closed, and the right side should be thought of identified with the left side. We note that in the given construction $C_1^1 = \{0,2,6\}$, $C_1^2 = \{7\}$, $B_1^1 = \{0,...,6\}$, $B_1^2 = \{7\}$, $A_1 = \{1,3,4,5\}$, $x_1 = \{\{*,0,1,2,3,4,5,6\},\{7\}\}$, $C_2^1 = \{1\}$, $C_2^2 = \{3,5\}$, $B_2^1 = \{1\}$, $B_2^2 = \{3,4,5\}$, $A_2 = \{4\}$, $x_2 = \{\{*,0,2,6\}, \{1\}, \{3,4,5\},\{7\}\}$, $C_3^1 = B_3^1 = \{4\}$, $A_3 = \emptyset$, $x_3 = x$. Exactly $24 - 4$ of the 48 bonds are open. The partition $x$ consists of $l= 5$ components and has a nesting number of $m = 3$.
• Figure 3: The picture shows a realization of Bernoulli percolation on $C_{12}^\times$ restricted to $E_{[1,3]}$, where the dot marks the vertex $(0,0)$, drawn bonds are open, dotted bonds are closed, and the right side should be identified with the left side. If $0$ is infected with the respect to the pattern $y$, then the shown bond configuration contributes to $B \cap B' \subset \{\mathcal{X}_3^y= x_*\}$.
• Figure 4: The picture shows a realization of Bernoulli percolation on $C_{12}^\times$ restricted to $E_{[1,9]}$, where the dot marks the vertex $(0,0)$, drawn bonds are open, dotted bonds are closed, and the right side should be identified with the left side. If $0$ is infected with the respect to the pattern $y$, $m' = 5$ and $x = \{ \{0\}, \{1,2\}, \{3\}, \{4\}, \{*,5,6\}, \{7,8\}, \{9,10,11\}\}$, then the shown bond configuration contributes to $B_1 \cap ... \cap B_4 \subset \{\mathcal{X}_{m'+4}^y= x\}$.
• Figure 5: Connecting horizontally adjacent vertices in $\mathbbm{Z}^2$ with shortest open paths of length 1,3,5,7. Open bonds are drawn, closed bonds are dotted. All paths go at most two layers below (and do not go above) the starting layer.

Theorems & Definitions (35)

• Theorem 1
• Remark 1
• Definition 1
• Theorem 2
• Remark 2
• Theorem 3
• Remark 3
• Definition 2
• Lemma 1
• Lemma 2