Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces

TL;DR

This paper proves that joint eigenfunctions on certain locally symmetric spaces become delocalized on average as these spaces converge to a symmetric space, under specific spectral and geometric conditions.

Contribution

It establishes quantum ergodicity results for sequences of locally symmetric spaces converging in the Benjamini--Schramm sense, extending understanding of eigenfunction behavior.

Findings

Eigenfunctions delocalize on average in the Benjamini--Schramm limit

Results hold for spaces with uniform spectral gap and discreteness

Applicable to a broad class of symmetric spaces

Abstract

We prove that for almost all symmetric spaces $X$ and for any sequence of compact locally symmetric spaces $Y_n$ which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to $X$, the joint eigenfunctions of all invariant differential operators on $Y_n$ delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.