Moments of global first passage time and first return time on tree-like fractals

The global first passage time (GFPT) is the first-passage time for a random walker from a randomly selected site to a given site. Here, we find the exact relation between the moments of GFPT and those of first return time (FRT) on general finite networks firstly. The exact relation is meaningful for understanding the dynamic taken place on the networks. It is also helpful to simplify the simulation of random walk on the networks. Then we derive the exact results for the first and second moments, together with asymptotic results for the higher moments, of the GFPT and FRT to a boundary node on the treelike fractal. We find that nth (n ≥ 1) moments of the GFPT and the FRT scale with the network size N as: « GFPTⁿ» ∼ (N_ds/²)ⁿ and «FRTⁿ» ∼ N_ds/^1+2(n-1), where «GFPTⁿ», «FRTⁿ» denote the nth moments of the GFPT and the FRT respectively, d _s is the spectral dimension of the network.