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Sampling Sphere Packings with Continuum Glauber Dynamics

Aiya Kuchukova, Santosh Vempala, Daniel J. Zhang

TL;DR

This work develops a continuum-general framework for sampling Gibbs point processes, focusing on hard-sphere packings, by proving a spectral gap for Continuum Glauber dynamics under Strong Spatial Mixing. It extends discrete notions of Spectral Independence and Negative Fields Localization to continuous space, introducing an Influence Operator and a continuous localization scheme to transfer low-activity gaps to higher activity. The main results give a domain-size–independent spectral gap for $\lambda< e/\Delta_{\varphi}$ and, in the hard-sphere case, a concrete bound $\lambda< e^{-\delta} e/2^d$, yielding explicit mixing times and improved fixed-size-sample (canonical) sampling algorithms. The methods combine operator-theoretic SI analysis, variance-conservation-type localization, and burn-in techniques to produce practical, polynomial-time sampling guarantees for canonical and grand-canonical ensembles. These insights advance efficient simulation of dense repulsive point processes with broad implications for statistical physics and high-dimensional packing problems.

Abstract

We establish a spectral gap for Continuum Glauber dynamics on the hard sphere model assuming strong spatial mixing, thereby extending the range of parameters in which Continuum Glauber is provably rapidly mixing. To do this, we introduce continuous extensions of spectral independence and negative fields localization. Our techniques apply to general Gibbs point processes with finite-range repulsive pair potentials. As a corollary, we improve the threshold up to which packings of a fixed number of spheres can be sampled from a bounded domain.

Sampling Sphere Packings with Continuum Glauber Dynamics

TL;DR

This work develops a continuum-general framework for sampling Gibbs point processes, focusing on hard-sphere packings, by proving a spectral gap for Continuum Glauber dynamics under Strong Spatial Mixing. It extends discrete notions of Spectral Independence and Negative Fields Localization to continuous space, introducing an Influence Operator and a continuous localization scheme to transfer low-activity gaps to higher activity. The main results give a domain-size–independent spectral gap for and, in the hard-sphere case, a concrete bound , yielding explicit mixing times and improved fixed-size-sample (canonical) sampling algorithms. The methods combine operator-theoretic SI analysis, variance-conservation-type localization, and burn-in techniques to produce practical, polynomial-time sampling guarantees for canonical and grand-canonical ensembles. These insights advance efficient simulation of dense repulsive point processes with broad implications for statistical physics and high-dimensional packing problems.

Abstract

We establish a spectral gap for Continuum Glauber dynamics on the hard sphere model assuming strong spatial mixing, thereby extending the range of parameters in which Continuum Glauber is provably rapidly mixing. To do this, we introduce continuous extensions of spectral independence and negative fields localization. Our techniques apply to general Gibbs point processes with finite-range repulsive pair potentials. As a corollary, we improve the threshold up to which packings of a fixed number of spheres can be sampled from a bounded domain.
Paper Structure (48 sections, 108 theorems, 408 equations, 1 figure, 3 algorithms)

This paper contains 48 sections, 108 theorems, 408 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Sampling Sphere Packings.
  3. Continuum Glauber Dynamics and Gibbs point processes.
  4. Main results
  5. Preliminaries
  6. Technical overview
  7. The discrete setting.
  8. Spectral Independence in Continuous Space.
  9. Strong Spatial Mixing.
  10. Localization Schemes.
  11. Mixing and Runtime.
  12. Sampling from the canonical model.
  13. Technical challenges.
  14. Related work
  15. Acknowledgments.
  16. ...and 33 more sections

Key Result

Theorem 1.1

For $\lambda < \frac{e}{\Delta_{\varphi}}$, the spectral gap of Continuum Glauber for any Gibbs point process with finite-range repulsive pair potentials is lower bounded by a positive constant independent of $|\Lambda|$.

Figures (1)

  • Figure 1: Proof outline

Theorems & Definitions (253)

  • Theorem 1.1: Spectral Gap of CG
  • Theorem 1.2: Spectral Gap of CG for Hard Spheres
  • Theorem 1.3: Mixing
  • Theorem 1.4: Runtime of CG
  • Theorem 1.5: Canonical Model
  • Definition 1.6: Poisson point process
  • Definition 1.7: Hard Sphere Model
  • Definition 1.8: Gibbs point process
  • Definition 1.9
  • Remark 1.10
  • ...and 243 more