Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 214 / 2

Because $p<p_c={\tilde{p}}_c$, choose a finite set $S\ni 0$ with ${\varphi }_p(S)=\rho <1$. Let $L$ exceed the ${\ell }^{\infty }$-distance from $0$ to every endpoint of an edge in $\partial S$, and putOn the one-arm event to distance $n>L$, take the first oriented boundary edge $(x,y)\in \partial S$ used by an open self-avoiding path. The connection $0\stackrel{S}{\leftarrow }x$, the open edge $(x,y)$, and the remaining connection from $y$ to $\partial B(n)$ occur disjointly. The van den Berg-Kesten inequality and translation invariance therefore giveIteration yields $q_n\le {\rho }^{\lfloor n/L\rfloor }$ up to an inessential finite-scale adjustment. Since $p<1$, each of the finitely many remaining $q_n$ is strictly below one, so reducing the exponent if necessary produces a constant $c>0$ valid for every $n\ge 1$:This is the finite-size proof of exponential decay of subcritical percolation.

By Tonelli theorem,If $p<p_c$, part (a) bounds the summand by $e^{-c\Vert x{\Vert }_{\infty }}$. There are only polynomially many vertices at each radius, so the series converges and $\chi (p)<\infty$.

Conversely, suppose $\chi (p)<\infty$. Then $p<1$. Choose $q>p$ so close to $p$ thatUse the standard sprinkling coupling for Bernoulli percolation: first expose the $p$-open clusters, then independently open each remaining edge with probability $\alpha =(q-p)/(1-p)$. Explore the $q$-cluster of the origin cluster by following sprinkled edges. Each discovered $p$-cluster has at most $2d$ times its number of vertices as many incident edges, so the exploration is dominated by a Galton-Watson process of mean at most $2d\alpha \chi (p)<1$. This process dies out almost surely, and hence there is no infinite $q$-open cluster. Thus $q\le p_c$, and $p<q$ implies $p<p_c$. Therefore