Patterson-Sullivan distributions of finite regular graphs

TL;DR

This paper introduces Patterson-Sullivan distributions for eigenfunctions on finite regular graphs, linking them to Wigner and Ruelle distributions, thus extending concepts from quantum chaos and hyperbolic geometry to discrete graph settings.

Contribution

It constructs Patterson-Sullivan distributions on finite regular graphs and establishes their connections to Wigner and Ruelle distributions, providing discrete analogues of hyperbolic surface results.

Findings

Patterson-Sullivan distributions are constructed for eigenfunctions on finite regular graphs.

Distributions are related to Wigner distributions via pseudo-differential calculus.

Distributions are also connected to invariant Ruelle distributions from the geodesic flow.

Abstract

On finite regular graphs, we construct Patterson-Sullivan distributions associated with eigenfunctions of the discrete Laplace operator via their boundary values on the phase space. These distributions are closely related to Wigner distributions defined via a pseudo-differential calculus on graphs, which appear naturally in the study of quantum chaos. Using a pairing formula, we prove that Patterson-Sullivan distributions are also related to invariant Ruelle distributions arising from the transfer operator of the geodesic flow on the shift space. Both relationships provide discrete analogues of results for compact hyperbolic surfaces obtained by Anantharaman-Zelditch and by Guillarmou-Hilgert-Weich.