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

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 $n\to \mathrm{\infty }$, and under mild aperiodicity conditions this convergence is monotone in that for some $N$ we have $\mathrm{\forall }n\ge N:P^y(X_n=x)$ $\ge P^y(X_{n+1}=x)$. 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: $\mathrm{\forall }k\ge 3\mathrm{\forall }n\ge 500k^6{2}^k\mathrm{\forall }p\in (0,1)\mathrm{\forall }m\in C_k\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}:$ $P_p((0,0)\leftrightarrow (m,n))\ge P_p((0,0)\leftrightarrow (m,n+1))$. In general these kind of monotonicity properties of connection probabilities are difficult to establish and there are only few pertaining results.