论文标题
Markov链链过渡概率通过准平台的单调性 - $ C_K \ times z $上的伯努利渗透的申请
Monotonicity of Markov chain transition probabilities via quasi-stationarity -- an application to Bernoulli percolation on $C_k \times Z$
论文作者
论文摘要
令$ x_n，n \ ge 0 $为有限状态空间$ m $的马尔可夫链。如果$ x，y \ in m $使$ x $是暂时的，我们有$ p^y（x_n = x）\ to $ n \ to $ n \ to \ to \ infty $，在轻度的上差条件下，这种收敛性是单调的，因为对于某些$ n $，我们有$ n $ $ \ forall n \ ge n \ ge n：p^y（x__n = x） x）$。我们使用马尔可夫链与其准平台分布的收敛速度的界限，以获得$ n $的明确界限。然后，我们将此结果应用于圆柱图上的参数$ p $ $ c_k \ times z $上的bernoulli渗透。因此，利用马尔可夫链描述每一层的感染模式，因此我们显示了连接概率的单调性的以下均匀结果：$ \ forall k \ ge 3 \，\ forall n \ ge 500k^6 2^k \，\ forall p \ forall p \ in（0,1） \ leftrightArrow（m，n））\ ge p_p（（（0,0）\ leftrightArrow（m，n+1））$。通常，连接概率的这种单调性特性很难建立，并且结果只有很少的结果。
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.
