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Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces

Kai Hippi

TL;DR

We address the problem of establishing large-scale quantum ergodicity and quantum mixing on sequences of compact hyperbolic surfaces by developing a spectral-geometric framework that leverages the hyperbolic wave propagator $P_t$ with $h_t(x)=\frac{\sin(tx)}{x}$ and a modified propagator $P_{t,\tau}$. The method replaces Nevo's ergodic theorem with exponential mixing of the geodesic flow, and proves both a large-scale analogue for physical-space observables and a probabilistic analogue for Weil–Petersson random surfaces, under Benjamini-Schramm convergence, uniform injectivity radius, and expander-type spectral gaps. The analysis decomposes into a spectral component, controlled via a sharp bound on $|\frac{1}{T}\int h_{t,\tau}(a)h_t(b)\,dt|^{-1}$, and a geometric component encoded in the weight function $F_{t,t',\rho}$, with the Hilbert-Schmidt norm playing a central role. These results extend Zelditch-type quantum mixing to a large-scale regime and provide a robust bridge between quantum ergodicity on manifolds and large-scale randomness models such as Weil–Petersson surfaces, with potential consequences for arithmetic cases and random-geometry quantum chaos.

Abstract

We study compact hyperbolic surfaces and multiplicative observables, establishing a large-scale analogue of Zelditch's quantum mixing theorem with hypotheses that hold for both arithmetic and Weil-Petersson random surfaces of large genus. This complements the large-scale quantum ergodicity theorems of Le Masson and Sahlsten, which themselves serve as large-scale analogues of the quantum ergodicity theorem of Shnirelman, Zelditch, and de Verdiere, thereby providing a more complete picture of the asymptotic behaviour of observables in the large-scale limit. Our approach does not rely on ball averaging operators or Nevo's ergodic theorem. Instead, we introduce a new method based on the hyperbolic wave equation and the quantitative exponential mixing of the geodesic flow established by Ratner and Matheus.

Quantum Mixing and Benjamini-Schramm Convergence of Hyperbolic Surfaces

TL;DR

We address the problem of establishing large-scale quantum ergodicity and quantum mixing on sequences of compact hyperbolic surfaces by developing a spectral-geometric framework that leverages the hyperbolic wave propagator with and a modified propagator . The method replaces Nevo's ergodic theorem with exponential mixing of the geodesic flow, and proves both a large-scale analogue for physical-space observables and a probabilistic analogue for Weil–Petersson random surfaces, under Benjamini-Schramm convergence, uniform injectivity radius, and expander-type spectral gaps. The analysis decomposes into a spectral component, controlled via a sharp bound on , and a geometric component encoded in the weight function , with the Hilbert-Schmidt norm playing a central role. These results extend Zelditch-type quantum mixing to a large-scale regime and provide a robust bridge between quantum ergodicity on manifolds and large-scale randomness models such as Weil–Petersson surfaces, with potential consequences for arithmetic cases and random-geometry quantum chaos.

Abstract

We study compact hyperbolic surfaces and multiplicative observables, establishing a large-scale analogue of Zelditch's quantum mixing theorem with hypotheses that hold for both arithmetic and Weil-Petersson random surfaces of large genus. This complements the large-scale quantum ergodicity theorems of Le Masson and Sahlsten, which themselves serve as large-scale analogues of the quantum ergodicity theorem of Shnirelman, Zelditch, and de Verdiere, thereby providing a more complete picture of the asymptotic behaviour of observables in the large-scale limit. Our approach does not rely on ball averaging operators or Nevo's ergodic theorem. Instead, we introduce a new method based on the hyperbolic wave equation and the quantitative exponential mixing of the geodesic flow established by Ratner and Matheus.

Paper Structure

This paper contains 13 sections, 26 theorems, 361 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Main theorems
  4. Further discussions
  5. Preliminaries
  6. The hyperbolic plane and hyperbolic surfaces
  7. Hyperbolic surfaces
  8. Strategy of proofs
  9. Spectral data
  10. Propagators
  11. Geometric data
  12. Weight function
  13. Proof of main theorems

Key Result

Theorem 1.1

Let $(M,g)$ be a compact Riemannian manifold (possibly with a boundary) with quantum Hamiltonian $\sqrt{-\Delta}$ with eigenvalues $\{ \mu_{j} = \sqrt{\lambda_{j}} \}$ and eigenvalues $\{ \psi_{j} \}$, where $\{ \lambda_{j} \}$ and $\{ \psi_{j} \}$ are the eigenvalues and eigenfunctions of the Lapla Then the geodesic flow $\varphi_t$ is ergodic on $(S^*M, d\mu)$ if and only if properties (i) and (

Figures (5)

  • Figure 1: Contour plot of $G_{\tau,T}(a,b)$. Two pronounced "ridges" are visible along the lines $b = a + \tau$ and $b = a - \tau$. The ridges get pronounced as $T$ increases. For a fixed $\delta$, $G_{\tau,t}(a,b)$ has a positive lower bound when $a-b-\tau \in ( -\delta, \delta )$. Notice also how the ridges decay as $a$ and $b$ grow.
  • Figure 2: The case of $t > t' + \rho$. The integration domain $B( z,t ) \cap B( z',t' )$ is just the smaller ball $B( z', t' )$ for it is enveloped by the bigger ball $B( z,t )$. The bound in this case is established using the polar coordinates around $z'$.
  • Figure 3: The case of $\rho< t < t' + \rho$, where $a + b = \rho$. The integration domain $B( z,t ) \cap B( z',t' )$ is the intersection of two balls. The middle point $z'$ of the smaller ball is contained in the bigger ball. The integration domain is split into two parts, $D_{1}$ and $D_{2}$ by two geodesics of length $t'$ going from $z'$ to points $w$ and $w'$ which are the points where the two balls intersect. The bound in this case is established separately for $D_{1}$ and $D_{2}$. For $D_{1}$, we use the polar coordinates around $z'$ and for $D_{2}$ the polar coordinates around $z$.
  • Figure 4: The case of $t < \rho$. The integration domain $B( z,t ) \cap B( z',t' )$ is the intersection of two balls. The middle point $z'$ of the smaller ball is not contained in the bigger ball. The bound in this case is established using the polar coordinates around $m$ which is the intersection of geodesics $[z,z']$ and $[w,w']$, where $w$ and $w'$ are the intersections of the two balls.
  • Figure 5: The case is split into two case which are bounded separately.

Theorems & Definitions (45)

  • Theorem 1.1: Semi-classical quantum mixing, Sun97Zel05
  • Theorem 1.2: Position space exponential mixing
  • Remark 1.3
  • Theorem 1.4: Large-scale quantum ergodicity and quantum mixing
  • Theorem 1.5: Quantitative version
  • Remark 1.6
  • Theorem 1.7: Probabilistic version
  • Proposition 4.1: Spectral data
  • proof : Proof of Proposition \ref{['Prp: Spectral data']}
  • Proposition 5.1: Propagators
  • ...and 35 more