Let $\Lambda$ be a lattice in $\mathbb{R}^d$, $d \ge 2$, thought of as an infinite graph. In percolation theory we consider properties random subgraphs of $\Lambda$. In site percolation $\Lambda^s_p$ every node of $\Lambda$ is included with probability $p$ i.i.d., and we take the induced subgraph on those random nodes. In bond percolation $\Lambda^b_p$ we […]

I’m considering edge percolation on $\mathbb{Z}^2$ with parameter $p$, so that edges are present with probability $p$. Is it known how to express the probability $P(p)$ that $(0,0)$ is in the same connected component as $(1,0)$ as an explicit function of $p$? A very crude bound is $$ p \leqslant P(p) \leqslant 1-(1-p)^4, $$ but […]

Consider a grid $G$ in the $\mathbb{R}^2$ plane formed by the points $(x,y)$ with integer coordinates i.e. $G=\{(x,y)\in\mathbb{R}^2: x\in\mathbb{Z},\;y\in\mathbb{Z} \}$. For $n>0$ let $B_n$ square centered at $(0,0)$ whose sides have length $2^{n+1} + 1$. Note that $B_n=\{-2^n,\ldots,0,\ldots,+2^n \}\times \{-2^n,\ldots,0,\ldots,+2^n \}$. Consider all the circuits as a finite set of points $p_{0},p_1,\ldots, p_{k-1},p_k,p_{k+1}\ldots, p_n$ such […]

Represent the set $R_{n\times n}=\{1,2,\ldots, n\}\times\{1,2,\ldots, n\} $ as a rectangle of $n$ by $n$ points as in the figures below for example. How to calculate the number of circuits that visit a chosen point in this rectangle? What is the most visited point on this rectangle? Making the question more precise we fix the […]

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