The range of random walk on trees and related trapping problem (Q1363005)

Let \(T_N\) \((N\geq 2)\) be the infinite tree with \(N+1\) branches emanating from each vertex. Let also \(\{X_n\}_{n\geq 0}\) be the simple random walk on \(T_N\) with the probability law \(\{P_x\}_{x\in T_N}\). The range of \(\{X_n\}_{n\geq 0}\) up to time \(n\) is denoted by \(R_n= \#\{X_0, X_t,\dots, X_n\}\). Let also \(\zeta\) denote a standard normal variable and let \(E_x\) be the expectation with respect to \(P_x\). The main result of the paper is as follows: (i) \(\lim_{n\to\infty} R_n/n= (N- 1)/N\), \(P_0\)-a.s., (ii) \(\lim_{n\to\infty}\text{ var}(R_n)/n= (N^2+ 1)/[N^2(n- 1)]\), (iii) \((R_n- E_0R_n)/n^{1/2} @>d>>\zeta(N^2+ 1)/[N^2(N- 1)]\), \(n\to\infty\). Some results concerning asymptotic behavior for the mean trapping time and survival probability are presented, too.