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

TL;DR

This paper proves the monotonicity of certain Markov chain transition probabilities using quasi-stationarity and applies this to show a uniform monotonicity result for connection probabilities in Bernoulli percolation on cylinder graphs.

Contribution

It introduces explicit bounds on convergence to quasi-stationary distributions and applies them to establish monotonicity of connection probabilities in percolation models.

Findings

Monotonicity of transition probabilities for Markov chains with transient states.

Explicit bounds on the convergence rate to quasi-stationary distributions.

A uniform monotonicity result for connection probabilities in Bernoulli percolation on cylinders.

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.