Exact mean first-passage time on the T-graph

TL;DR

This paper derives an exact formula for the average first-passage time to the center of a T-fractal for a random walk, confirming known asymptotic behaviors and diffusion laws on fractal structures.

Contribution

It provides an explicit analytical expression for the mean first-passage time on the T-fractal, advancing understanding of diffusion on low-dimensional fractals.

Findings

Exact mean first-passage time formula derived

Results agree with known asymptotic diffusion laws

Analytic techniques based on decimation procedures used

Abstract

We consider a simple random walk on the T-fractal and we calculate the exact mean time $\tau^g$ to first reach the central node $i_0$. The mean is performed over the set of possible walks from a given origin and over the set of starting points uniformly distributed throughout the sites of the graph, except $i_0$. By means of analytic techniques based on decimation procedures, we find the explicit expression for $\tau^g$ as a function of the generation $g$ and of the volume $V$ of the underlying fractal. Our results agree with the asymptotic ones already known for diffusion on the T-fractal and, more generally, they are consistent with the standard laws describing diffusion on low-dimensional structures.