Complete spectrum of stochastic master equation for random walks on treelike fractals

TL;DR

This paper derives the complete eigenvalue spectrum of the stochastic master equation for random walks on treelike fractals, linking the smallest eigenvalue to the mean trapping time, and offers a recursive method for eigenvalue calculation.

Contribution

It provides a full spectral analysis of the stochastic master equation for random walks on treelike fractals, including explicit recursive relations for eigenvalues.

Findings

Eigenvalues are obtained through recursive relations.

The reciprocal of the smallest eigenvalue approximates the mean trapping time.

Method can be adapted to other treelike fractal structures.

Abstract

We study random walks on a family of treelike regular fractals with a trap fixed on a central node. We obtain all the eigenvalues and their corresponding multiplicities for the associated stochastic master equation, with the eigenvalues being provided through an explicit recursive relation. We also evaluate the smallest eigenvalue and show that its reciprocal is approximately equal to the mean trapping time. We expect that our technique can also be adapted to other regular fractals with treelike structures.