Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 204 / 2 / d / Solution

Write ${\mathcal{W}}_n$ for the deterministic set of $n$-step self-avoiding walks from $0$. Each uses $n$ distinct edges, so independence givesFor $n\ge 1$, the function $t\mapsto t^{1/n}$ on $[0,\infty )$ is a concave function. The Jensen inequality gives the root-moment bound for open self-avoiding walksEquivalently, the Lyapunov moment inequality compares exponents $1/n$ and $1$, both strictly positive. Taking the limit superior and the connective constant limit provesThis includes $p=0$ and $n=1$. No independence between the different self-avoiding walk indicators is asserted or needed.