Martin boundary of killed random walks on isoradial graphs

TL;DR

This paper analyzes the Martin boundary of killed random walks on isoradial graphs, revealing asymptotic behaviors and boundary properties despite the graphs' non-homogeneity and lack of translation invariance.

Contribution

It extends Martin boundary analysis to isoradial graphs, a non-homogeneous class, providing new asymptotic results and boundary minimality insights.

Findings

Computed asymptotics of the Martin kernel on isoradial graphs

Established the Martin boundary as minimal for these graphs

Extended classical results to non-translation-invariant structures

Abstract

We consider killed planar random walks on isoradial graphs. Contrary to the lattice case, isoradial graphs are not translation invariant, do not admit any group structure and are spatially non-homogeneous. Despite these crucial differences, we compute the asymptotics of the Martin kernel, deduce the Martin boundary and show that it is minimal. Similar results on the grid $\mathbb Z^d$ are derived in a celebrated work of Ney and Spitzer.