Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 30 / 4

Set $L={\inf }_{k\ge 1}{x}_k/k$. It may be $-\infty$, but cannot be $+\infty$ because $x_1$ is finite. Fix $k$ and write $n=qk+r$, $0\le r<k$. Repeated use of the subadditive sequence inequality giveswhere $C_1=0$. Dividing by $n$ and letting $n\to \infty$ gives $\mathrm{limsup}{x}_n/n\le x_k/k$. Since this holds for every $k$, the upper limit of a sequence is at most $L$, while every ratio is at least $L$. If $L$ is finite these bounds agree. If $L=-\infty$, choose $k$ with $x_k/k$ below any prescribed real number; the same upper-limit bound proves convergence to $-\infty$. This establishes the extended-real Fekete lemma:A finite real limit of a sequence requires a lower linear bound. For example $x_n=-n^2$ satisfies the premise but has normalized limit of a sequence $-\infty$, so that qualification cannot be dropped.

Write $\varphi ={\varphi }_{p,q}^1$ and $C_n=\varphi (0\leftrightarrow e_n)$. We use these standard facts about the infinite-volume wired random-cluster measure for $q\ge 1$: it is translation invariant and invariant under coordinate reflections/permutations; it has positive association of the random-cluster model; and every conditional edge-open probability is at leastThe last bound follows because an edge either leaves its component count unchanged on opening, giving probability $p$, or merges two components, giving $p/(p+q(1-p))$. These are standard infinite-volume properties of the limit obtained from the finite wired laws by weak convergence of probability measures. Positive association of random variables applies to connection events by increasing approximation with finite-path events.

Opening the $n$ edges of the straight segment gives $C_n\ge r^n$, so its logarithm is finite. The events $A=\{0\leftrightarrow e_m\}$ and $B=\{e_m\leftrightarrow e_{m+n}\}$ are increasing, and their intersection implies connection to $e_{m+n}$. ThusThe sequence $a_n=-\log C_n$ is a subadditive sequence, with $0\le a_n\le -n\log r$. Part (a) proves the random-cluster inverse correlation length exists and is finite:Since $a_n/n\ge \alpha$ for every $n$, exponentiating yields the requested bound

Let $B_n=\varphi (0\leftrightarrow \partial {\Lambda }_n)$ and $s_n=|\partial {\Lambda }_n|$. This connection event is the union over $x\in \partial {\Lambda }_n$ of $\{0\leftrightarrow x\}$. The union bound therefore supplies a graph vertex with connection probability at least $B_n/s_n$.

Every boundary graph vertex has some coordinate equal to $n$ or $-n$. By applying a permutation and a graph automorphisms, which preserve both the law and the boundary, choose such a graph vertex in the form $x=(n,x_2,\dots ,x_d)$ while retaining its probability bound. Now $e_{2n}-x=(n,-x_2,\dots ,-x_d)$ is a reflection of $x$. Translation and reflection invariance givePositive association of random variables applied to the two connection events, whose intersection implies $0\leftrightarrow e_{2n}$, proves the reflection comparison of random-cluster connections:Also $C_n\le B_n$, since any path to $e_n$ meets the box boundary. Consequentlyand henceHere $s_n=(2n+1{)}^d-(2n-1{)}^d=O(n^{d-1})$, so $\log s_n/n\to 0$. Both outside logarithmic connection rates converge to $\alpha (p,q)$ by part (b). Squeezing gives the rate for every box size, and therefore for the requested even subsequence: