Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 26 / 3 / c

For every site, $P(R(x)\ge r)={\beta }_r$. This event depends only on edges with both endpoints in $x+{\Lambda }_r$: any connecting path can be stopped on its first hit of that boundary. Since $\lambda >0$, ${\beta }_r\to 0$, so all clusters have finite radius almost surely, by a countable union over sites. Set the radius of an isolated site to zero.

Write $a_n=(d/\lambda )\log n$. It suffices to take $0<ϵ<1$, since larger error intervals contain one such interval. For any small $\delta >0$, the decay-rate limit gives, for all large $r$,For the upper tail use $r_{+}=\lceil (1+ϵ)a_n\rceil$ and a union bound:provided $\delta <\lambda ϵ/(1+ϵ)$.

For the lower tail let $r_{-}=\lfloor (1-ϵ)a_n\rfloor +1$. Choose sites in ${\Lambda }_{n-r_{-}}$ spaced by $2r_{-}+1$ in each coordinate. Their radius-$r_{-}$ boxes are vertex-disjoint, so their local connection events are independent. There are $N_n\ge c(n/r_{-}{)}^d$ such sites for large $n$. If $M_n\le (1-ϵ)a_n$, none of these events occurs. Thuswhereif $\delta <\lambda ϵ/(1-ϵ)$. Choose one $\delta$ satisfying both restrictions. The integer choices give the required strict inequalities. Thus the maximum cluster radius under exponential one-arm decay has convergence in probability:The logarithmic box-packing loss does not change the leading constant $d/\lambda$.