On the norms of the random walks on planar graphs (Q1374015)

Let us consider a connected planar graph \(X\) such that the degree of each vertex, i.e. the number of edges adjacent to the vertex, is finite and such that there are a finite number of vertices in any compact subset of the plane. Let us also suppose that there are no loops or multiple edges. On this graph we consider a random walk that goes from a vertex to one of its neighbors picked uniformly at random. We associate with this random walk a random walk operator \(M\), \(Mf(q)= {1\over N(q)} \sum_{p\sim q} f(p)\) for \(f\in \ell^2 (X,N)\), where \(N(q)\) is the degree of vertex \(q\) and where \(p\sim q\) means that \(\{q,p\}\) is an edge, i.e. \(p\) and \(q\) are neighbors. The operator \(M\) is self-adjoint on the space \(\ell^2 (X,N)\). We establish some upper bounds for \(\| M\|\), the norm of this operator, acting on \(\ell^2 (X,N)\). If \(P^n (q,p)\) is the probability of going from \(q\) to \(p\) in \(n\) steps, then we know [see \textit{W. Woess}, Bull. Lond. Math. Soc. 26, No. 1, 1-60 (1994; Zbl 0830.60061)] that \(\| M\|= \lim_{n\to \infty} (P^{2n} (q,q))^{1/2n}\).