{"year":"2022","abstract":[{"text":"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 :\r\nP_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. ","lang":"eng"}],"citation":{"ama":"Richthammer T, König P. Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z. Published online 2022.","chicago":"Richthammer, Thomas, and Philipp König. “Monotonicity of Markov Chain Transition Probabilities via Quasi-Stationarity - an Application to Bernoulli Percolation on C_k × Z,” 2022.","ieee":"T. Richthammer and P. König, “Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z.” 2022.","short":"T. Richthammer, P. König, (2022).","bibtex":"@article{Richthammer_König_2022, title={Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z}, author={Richthammer, Thomas and König, Philipp}, year={2022} }","mla":"Richthammer, Thomas, and Philipp König. <i>Monotonicity of Markov Chain Transition Probabilities via Quasi-Stationarity - an Application to Bernoulli Percolation on C_k × Z</i>. 2022.","apa":"Richthammer, T., &#38; König, P. (2022). <i>Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z</i>."},"status":"public","type":"preprint","title":"Monotonicity of Markov chain transition probabilities via quasi-stationarity - an application to Bernoulli percolation on C_k × Z","language":[{"iso":"eng"}],"date_updated":"2026-02-18T12:27:38Z","_id":"64216","date_created":"2026-02-18T12:27:28Z","author":[{"first_name":"Thomas","full_name":"Richthammer, Thomas","id":"62054","last_name":"Richthammer"},{"first_name":"Philipp","full_name":"König, Philipp","last_name":"König"}],"user_id":"62054"}