Riemann moduli spaces are quantum ergodic

TL;DR

This paper demonstrates that Riemann moduli spaces with the Weil--Petersson metric exhibit quantum ergodicity when the genus and number of punctures satisfy certain conditions, extending to other singular spaces with ergodic geodesic flows.

Contribution

It establishes quantum ergodicity for Riemann moduli spaces with the Weil--Petersson metric and identifies additional singular spaces with similar properties.

Findings

Riemann moduli spaces are quantum ergodic for 3g+n ≥ 4

Quantum ergodicity holds for certain singular spaces with ergodic geodesic flow

Provides examples of spaces with ergodic geodesic flow where quantum ergodicity applies

Abstract

In this note we show that the Riemann moduli spaces $M_{g, n}$ equipped with the Weil--Petersson metric are quantum ergodic for $3g+n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.