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

Fix $k\ge 1$ and write $n=qk+r$ with $1\le r\le k$ and $q\ge 0$. Repeatedly use the given inequality with its second index equal to $k$. This yieldsThere are only finitely many possible remainders for fixed $k$, soTaking $k=1$ rules out positive infinity for this upper limit. Let $L=\mathrm{liminf}{x}_n/n$, which may be negative infinity, and choose $k_j\to \infty$ with $x_{k_j}/k_j\to L$. The assumption ${\alpha }_{k_j}/k_j\to 0$ then gives $\mathrm{limsup}{x}_n/n\le L$. This also works when $L=-\infty$, by taking an arbitrarily negative upper bound. Therefore the asymmetrically almost-subadditive sequence hasThe last infimum identity follows from the fixed-$k$ bound and from the convergence of the corrected ratios to $\gamma$. No positivity assumption on ${\alpha }_n$ is required.

A connection to a box boundary can be witnessed by a finite self-avoiding open path stopped at its first boundary hit. On the event of reaching radius $m+n$, split such a path at its first graph vertex $z\in \partial {\Lambda }_n$. Its prefix witnesses connection from zero to $z$ within ${\Lambda }_n$. Its remaining segment reaches a point $y$ with $\Vert y{\Vert }_{\infty }=m+n$, so $\Vert y-z{\Vert }_{\infty }\ge m$; truncate it at its first hit of $z+\partial {\Lambda }_m$. The two witnessing edge sets are disjoint, giving disjoint occurrence of increasing events.

Use the BK inequality: for increasing finite-coordinate events in a Bernoulli product measure, $P(A\square B)\le P(A)P(B)$, where the square means disjoint open-edge witnesses. For each fixed $z$, the first event has probability at most ${\beta }_n$, and the second has probability exactly ${\beta }_m$ by translation invariance. A union bound over $z$ givesThe two boxes can overlap, so replacing BK by an independence claim would be incorrect. Restricting each event to paths stopped at its local boundary makes all the relevant coordinate sets finite, as required by the stated inequality.

Put $x_n=\log {\beta }_n$ and ${\alpha }_n=\log |\partial {\Lambda }_n|$. Here $|\partial {\Lambda }_n|=(2n+1{)}^d-(2n-1{)}^d$, so ${\alpha }_n=O(\log n)=o(n)$. A fixed straight open path of length $n$ gives $p^n\le {\beta }_n\le 1$. Part (a) therefore gives the almost-subadditive percolation decay rateIn particular the limit is finite for the prescribed $0<p<1$.

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 $d/\lambda$.