Sampling Sphere Packings with Continuum Glauber Dynamics

TL;DR

This paper proves spectral gap results for Continuum Glauber dynamics in Gibbs point processes, enabling efficient sampling of sphere packings with arbitrary-range repulsive potentials and improving previous sampling thresholds.

Contribution

It extends spectral independence techniques to continuous settings, providing new efficient sampling algorithms for Gibbs point processes with repulsive interactions.

Findings

Established spectral gap for continuum Glauber dynamics with repulsive potentials.

Developed continuous analogs of spectral independence and localization.

Improved sampling thresholds for fixed-size sphere packings.

Abstract

Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.