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

Let X_n, n ≥ 0 be a Markov chain with finite state space M . If x, y ∈ M such that x is transient we have P_y (X_n = x) → 0 for n → ∞, and under mild aperiodicity conditions this convergence is monotone in that for some N we have ∀n ≥ N : P_y (X_n = x) ≥ Py (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 × Z. Utilizing a Markov chain describing infection patterns layer per layer, we thus show the following uniform result on the monotonicity of connection probabilities: ∀k ≥ 3 ∀n ≥ 500k^62^k ∀p ∈ (0, 1) ∀m ∈ C_k : P_p((0, 0) ↔ (m, n)) ≥ P_p((0, 0) ↔ (m, n + 1)). In general these kind of monotonicity properties of connection probabilities are difficult to establish and there are only few pertaining results.