Equidistribution of random walks on spheres
The authors analyze some properties related to the distribution of the \(n\)-step transition probabilities of random walks. Particularly, they consider irreducible random walks of nearest neighbour type on locally finite graphs, that is random walks \((X,P)\), where \(X\) is the vertex set of a connected graph, \(P=(p(x,y))_{x,y\in X}\) is the stochastic transition matrix which describes the one-step transitions of a Markov chain \(\{Z_n\}_{n\in N}\), with state space \(X\) and where \(P\) has the property that \(p(x,y)>0\) if and only if \(x\) and \(y\) are neighbours. In Section 2 the equidistribution of the \(n\)-step transition probabilities \(p^{(n)}(x,y)\) on the spheres \(S(x,k)= \{y\in X:d(x,y)=k\}\) is considered \((d\) is the natural distance on the graph). Section 3 is devoted to prove a necessary and sufficient condition for the asymptotical isotropy of a recurrent, strongly periodic (or strongly aperiodic) random walk (see Theorem 3.2). In Section 4 the authors study the Green function \(G_{Z^2}((0,0), (a,b)\mid z)\) of the simple random walk on \(Z^2\); while in Section 5 the asymptotic behaviour of the transition probabilities on the free product of the simple random walk on \(Z^2\) and a random walk on a group \(\Gamma\) is studied. It is a good study by which the authors show that the property of isotropy is strongly linked to geometrical properties of the graph, more than the asymptotical isotropy.
scientific article; zbMATH DE number 1334368
scientific article