Quantum ergodicity for quantum graphs without back-scattering

TL;DR

This paper establishes quantum ergodicity for certain quantum graphs without back-scattering, using estimates of quantum variance and properties of expander graphs, including Ramanujan and random regular graphs.

Contribution

It provides the first quantum ergodicity results for quantum graphs with boundary scattering matrices that prohibit back-scattering, extending understanding of quantum chaos in these systems.

Findings

Quantum variance estimates for specific quantum graphs.

Quantum ergodicity proven for Ramanujan and random regular graphs.

Conditions identified under which quantum ergodicity holds.

Abstract

We give an estimate of the quantum variance for $d$-regular graphs quantised with boundary scattering matrices that prohibit back-scattering. For families of graphs that are expanders, with few short cycles, our estimate leads to quantum ergodicity for these families of graphs. Our proof is based on a uniform control of an associated random walk on the bonds of the graph. We show that recent constructions of Ramanujan graphs, and asymptotically almost surely, random $d$-regular graphs, satisfy the necessary conditions to conclude that quantum ergodicity holds.