Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 214 / 1

For bond percolation, the Harris-FKG inequality states that any two increasing events $A,B$ satisfyThe same inequality holds for two decreasing events, either by reversing the coordinate order or by taking complements in the two-event identity. Intersections of decreasing events are decreasing events, so induction givesWrite $a={\mathbb{P}}_p(A_1)=\cdots ={\mathbb{P}}_p(A_n)$ and $u={\mathbb{P}}_p({\bigcup }_j{A}_j)$. Then $1-u\ge (1-a{)}^n$. Taking the nonnegative $n$-th root and rearranging yields the square-root trick for positively associated events:No independence among the events is needed.

Put $I=\{x:|C(x)|=\infty \}$, the union of all infinite percolation clusters. Since $p>p_c$, the percolation probability $\theta (p)$ is positive. The translation ergodicity of Bernoulli percolation makes $\{I\ne \varnothing \}$ a zero-one event; it has positive probability because ${\mathbb{P}}_p(0\in I)=\theta (p)>0$, hence it occurs almost surely.

Let $D_m=\{B(m)\cap I\ne \varnothing \}$. These events increase to $\{I\ne \varnothing \}$, so continuity from below of a measure gives ${\mathbb{P}}_p(D_m)\to 1$. On $D_m$, some vertex of $B(m)$ has an infinite open path in a graph. For every $N\ge m$, the segment up to its first visit to $\partial B(N)$ lies entirely in $B(N)$. Thus even with this restriction on the connecting path in a graph,This proves the requested uniformity and supplies the version needed for crossings later.

For each of the $2d$ coordinate faces $F$ of $B(N)$, define the increasing event $A_F(m,N)$ that an open path in a graph lying in $B(N)$ joins $B(m)$ to $F$. Their union is exactly the event of connection from $B(m)$ to $\partial B(N)$ within $B(N)$. The bond percolation law and the centered boxes are invariant under coordinate permutations and sign changes, so all $2d$ events have the same probability.

Apply the square-root trick for positively associated events from part (a), with $2d$ in place of the number of events. If ${\delta }_m=1-{\mathbb{P}}_p(D_m)$, part (b.i) gives, for every $N=m+n$,In particular this holds for the left and right faces, and it proves the stronger statement in which each connecting path in a graph stays inside the outer box.

We use the uniqueness of the infinite percolation cluster on ${\mathbb{Z}}^d$: for $p>p_c$, there is almost surely exactly one infinite percolation cluster, denoted $I$. Fix $m$ and let $N>m$. Two observations account for the possible distinct endpoints of the face connections.

First, the probability that some finite percolation cluster meeting $B(m)$ reaches $\partial B(N)$ tends to zero as $N\to \infty$. There are only finitely many vertices in $B(m)$, and each of their finite percolation clusters has finite radius; apply the union bound and continuity from above of a measure.

Second, all vertices of $I\cap B(m)$ are joined to one another inside $B(N)$ with probability tending to one. On each configuration, this is a finite set of vertices of one connected percolation cluster. Choose a finite open path in a graph from one such vertex to each of the others; the union of these finitely many paths is contained in some finite box. The assertion is vacuous if the set is empty.

By part (b.ii), each of the two face-connection events within $B(N)$ fails with probability at most ${\delta }_m^{1/(2d)}$. Outside the two exceptional events just described, their endpoints in $B(m)$ belong to $I$ and can be joined inside $B(N)$. Concatenating the left-face path, that joining path, and the right-face path produces a crossing of $B(N)$. ConsequentlyNow let $m\to \infty$. Since ${\delta }_m\to 0$,The order of limits matters: $m$ is fixed while the finitely many relevant percolation clusters are connected inside increasingly large boxes.