  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.
