Martin boundary of killed random walks on isoradial graphs

Cédric Boutillier (1, 2) , Kilian Raschel (3, 4) , Alin Bostan (5)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 IUF - Institut universitaire de France
3 IDP - Institut Denis Poisson
4 CNRS - Centre National de la Recherche Scientifique
5 SPECFUN - Symbolic Special Functions : Fast and Certified

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.

Mots clés: Martin boundary; Green function; Isoradial graphs; Killed random walk; Discrete exponential function

Domaines: Physique mathématique [math-ph]; Probabilités [math.PR]

Cédric Boutillier, Kilian Raschel, Alin Bostan. Martin boundary of killed random walks on isoradial graphs. 2019. ⟨hal-02422417v1⟩