Explicit bounds for the return probability of simple random walks
Consider the simple random walk on the \(d\)-dimesional Euclidean lattice \(\mathbb{Z}^d\) and let \(P^d_n(i,j)\) be the probability of going from \(i\in \mathbb{Z}\) to \(j\in\mathbb{Z}^d\) in \(n\) steps. A classical result of Polya asserts that \(P_{2n}^d(0,0)\sim 2\left(\frac{d}{4\pi n}\right)^{d/2}\). In the paper under review, the authors give the following bound: \[ P_{2n}^d(0,0)\leq 2\left(\frac{d}{4\pi n}\right)^{d/2}\left[1-\frac{d}{8n}+\frac{7}{2}d(d+2)\left(\frac{d}{4n}\right)^2\right]+ \left(1+\frac{1}{\sqrt{n}}\right)e^{-2n/d} \] for all \(n\geq 3\) and \(d\leq n\). From this they deduce that \[ P_{2n}^d(0,0)\leq 2\left(\frac{d}{4\pi n}\right)^{d/2}\tag{1} \] when \(n\geq \frac{7}{4}d^2(d+1)+1\), giving a partial answer to a conjecture of Russ Lyons, who asked to prove (1) for all \(d\geq 1\) and \(n\geq 1\) (for \(d=1,2\) the conjecture is easy to prove).
scientific article; zbMATH DE number 2175941