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

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.