Local limit theorems for sequences of simple random walks on graphs

TL;DR

This paper establishes local limit theorems for simple random walks on various graphs, including percolation clusters, fractals, and trees, providing insights into their asymptotic behaviors.

Contribution

It introduces new local limit theorems applicable to diverse graph models, extending understanding of random walk behaviors on complex structures.

Findings

Local limit theorems for random walks on percolation clusters

Results for random walks on fractal graphs like Sierpinski carpets

Insights into convergence of random walks on graph sequences

Abstract

In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.