In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(...In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ < m or λ > m, respectively;(2) when λ = m, 0 < α < 1,(Xn)n≥ 0 is transient/recurrent according to whether A < 0 or A > 0, respectively.In particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.11531001 and 11626245)
文摘In a Galton-Watson tree generated by a supercritical branching process with offspring N and EN =:m > 1, the conductance assigned to the edge between the vertex x and its parent x* is denoted by C(x) and given by C(x) =(λ +A/|x|α)-|x|, where |x| is the generation of the vertex x. For(Xn)n≥0, a C(x)-biased random walk on the tree, we show that (1) when λ≠ m, α > 0,(Xn)n≥0 is transient/recurrent according to whether λ < m or λ > m, respectively;(2) when λ = m, 0 < α < 1,(Xn)n≥ 0 is transient/recurrent according to whether A < 0 or A > 0, respectively.In particular, if P(N = 1) = 1, the C(x)-biased random walk is Lamperti’s random walk on the nonnegative integers(see Lamperti(1960)).
