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Spectral gap for Pollicott-Ruelle resonances on random coverings of Anosov surfaces

Julien Moy

TL;DR

The paper proves a near-optimal spectral gap for Pollicott–Ruelle resonances of the geodesic flow on random degree‑n coverings of a closed Anosov surface, in the large‑n limit. It combines strong convergence of permutation representations (via Magee–Puder–van Handel) with a detailed low‑frequency analysis using the spherical mean operator on the universal cover, all within Faure–Sjöstrand anisotropic Sobolev spaces to control resolvents. The main result shows that, for any compact set K in the half-plane Re z > δ0 (with δ0 = (1/2)Pr(−2ψ^u)), asymptotically almost surely there are no new Pollicott–Ruelle resonances in K as the degree grows, establishing a probabilistic spectral gap and highlighting the optimal regime in the hyperbolic case where a Laplacian–PR resonance correspondence is available. The work connects dynamical decay rates with random-cover spectral theory, offering methods that extend beyond constant-curvature settings to variable negative curvature and other random-surface models. The results have potential implications for quantitative decay of correlations on large random surfaces and deepen the link between geometric group representations, resonances, and stochastic models of geometry.

Abstract

Let $(M,g)$ be a closed Riemannian surface with Anosov geodesic flow. We prove the existence of a spectral gap for Pollicott--Ruelle resonances on random finite coverings of $M$ in the limit of large degree, which is expected to be optimal. The proof combines the recent strong convergence results of Magee, Puder and van Handel for permutation representations of surface groups with an analysis of the spherical mean operator on the universal cover of $M$.

Spectral gap for Pollicott-Ruelle resonances on random coverings of Anosov surfaces

TL;DR

The paper proves a near-optimal spectral gap for Pollicott–Ruelle resonances of the geodesic flow on random degree‑n coverings of a closed Anosov surface, in the large‑n limit. It combines strong convergence of permutation representations (via Magee–Puder–van Handel) with a detailed low‑frequency analysis using the spherical mean operator on the universal cover, all within Faure–Sjöstrand anisotropic Sobolev spaces to control resolvents. The main result shows that, for any compact set K in the half-plane Re z > δ0 (with δ0 = (1/2)Pr(−2ψ^u)), asymptotically almost surely there are no new Pollicott–Ruelle resonances in K as the degree grows, establishing a probabilistic spectral gap and highlighting the optimal regime in the hyperbolic case where a Laplacian–PR resonance correspondence is available. The work connects dynamical decay rates with random-cover spectral theory, offering methods that extend beyond constant-curvature settings to variable negative curvature and other random-surface models. The results have potential implications for quantitative decay of correlations on large random surfaces and deepen the link between geometric group representations, resonances, and stochastic models of geometry.

Abstract

Let be a closed Riemannian surface with Anosov geodesic flow. We prove the existence of a spectral gap for Pollicott--Ruelle resonances on random finite coverings of in the limit of large degree, which is expected to be optimal. The proof combines the recent strong convergence results of Magee, Puder and van Handel for permutation representations of surface groups with an analysis of the spherical mean operator on the universal cover of .
Paper Structure (54 sections, 56 theorems, 369 equations)

This paper contains 54 sections, 56 theorems, 369 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Spectral properties of large random surfaces
  4. Main results
  5. Almost sure spectral gap for uniformly random Anosov surface covers
  6. Strong convergence of representations and spectral gaps
  7. High frequency spectral gap
  8. The case of hyperbolic surfaces
  9. Related works
  10. Eigenvalues of random lifts of graphs
  11. Eigenvalues of the Laplacian on random negatively curved covers
  12. Other models of random surfaces
  13. Overview of the proof
  14. Geometric preliminaries
  15. Geometry of $SM$
  16. ...and 39 more sections

Key Result

Theorem 1.1

The family of operators $(X+z)^{-1}:C^\infty(\mathcal{M})\to \mathcal{D}'(\mathcal{M})$, initially defined and holomorphic in $\{\operatorname{Re}(z)>0\}$, extends meromorphically to $z\in \mathbb{C}$. Its poles have finite rank and are called Pollicott--Ruelle resonances.

Theorems & Definitions (124)

  • Theorem 1.1: Butterley--Liverani, Faure--Sjöstrand
  • Theorem 1.2: Tsujii, Nonnenmacher--Zworski
  • Corollary 1.3: Exponential decay of correlations
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10: Magee--Puder--van Handel
  • ...and 114 more