Sampling Sphere Packings with Continuum Glauber Dynamics

Aiya Kuchukova; Daniel J. Zhang; Santosh S. Vempala

arxiv: 2601.18748 · v2 · submitted 2026-01-26 · 💻 cs.DS · math.PR

Sampling Sphere Packings with Continuum Glauber Dynamics

Aiya Kuchukova , Santosh S. Vempala , Daniel J. Zhang This is my paper

Pith reviewed 2026-05-16 10:29 UTC · model grok-4.3

classification 💻 cs.DS math.PR

keywords sampling algorithmsGlauber dynamicsspectral independenceGibbs point processeshard sphere modelsphere packingsspectral gap

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The pith

Continuous spectral independence suffices to establish a spectral gap for continuum Glauber dynamics on repulsive pair potentials.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that for Gibbs point processes with arbitrary-range repulsive pair potentials, including the hard sphere model, a continuous version of spectral independence implies a positive spectral gap for continuum Glauber dynamics. This yields rapid mixing and thus an efficient sampling algorithm that matches the regime previously known only for finite-range potentials via specialized methods. The argument develops continuous analogs of spectral independence and negative-field localization, then shows that a stronger zero-freeness condition implies the independence property, allowing the gap to be boosted from low to high activity. As a direct consequence the threshold improves for sampling fixed-size or fixed-density packings from a bounded domain. The analysis works entirely in the continuous setting without discretization.

Core claim

For arbitrary-range repulsive pair potentials, continuous spectral independence suffices to establish a spectral gap for continuum Glauber dynamics. This extends the regime of activity for which the dynamics is known to mix, yielding a simple efficient sampling algorithm that matches the known regime for finite-range potentials.

What carries the argument

Continuous spectral independence, which lets a localization scheme boost the spectral gap of the birth-death process from low to high activity.

If this is right

Continuum Glauber dynamics mixes in polynomial time for arbitrary-range repulsive potentials.
A simple dynamics-based algorithm samples from the Gibbs distribution up to the same activity threshold as the best specialized finite-range algorithms.
The threshold for efficient sampling of fixed-density packings from a bounded domain improves beyond the 2003 bound.
The same localization technique applies directly in the continuous setting without any discretization step.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The continuous framework may extend to other spatial birth-death processes whose stationary measures satisfy analogous zero-freeness conditions.
If the zero-freeness implication can be verified for soft repulsive potentials, the mixing guarantee would immediately cover those models as well.
The avoidance of discretization suggests the method could be adapted to sample from infinite-volume Gibbs measures by taking suitable limits.

Load-bearing premise

A stronger variant of zero-freeness implies the continuous spectral independence property needed to run the localization scheme.

What would settle it

An explicit repulsive pair potential where zero-freeness holds at a given activity but the continuous spectral independence inequality fails, producing a spectral gap of zero.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper lifts spectral independence and localization to the continuous setting for repulsive point processes, improving the mixing threshold for continuum Glauber dynamics to match finite-range results without discretization.

read the letter

The core advance is a direct continuous argument that stronger zero-freeness implies a continuous form of spectral independence, which then feeds a localization scheme to lift the spectral gap of continuum Glauber dynamics from low to high activity. This works for arbitrary-range repulsive pair potentials and gives the first improvement since Kannan-Mahoney-Montenegro 2003 on the activity threshold for sampling hard-sphere configurations of fixed size or density. The paper avoids any spatial discretization in both the algorithm and the analysis, which is the cleanest part of the contribution. It also supplies the necessary continuous definitions and shows how they plug into the Chen-Eldan blueprint while handling the new technical obstacles that arise from working directly with point processes. That extension is real and useful for anyone who wants a uniform sampler that does not depend on range cutoffs. The main soft spot is the load-bearing step from strengthened zero-freeness to a uniform bound on the continuous spectral-independence constant. The stress-test note correctly flags that this implication could require range-dependent regularity that is not automatic for long-range potentials; if the paper only sketches the transfer without explicit control on how the constant scales with range or activity, the claimed regime would need extra justification. The abstract states the result cleanly, but without the full derivation or explicit constants in front of me I cannot check the error terms or the precise activity improvement. Readers working on Markov-chain sampling for Gibbs point processes or statistical-physics simulations will find the new continuous tools and the improved threshold worth reading. The work is technically grounded enough to merit a serious referee, even if the continuous implication needs tightening in review.

Referee Report

1 major / 1 minor

Summary. The paper claims to establish a spectral gap for Continuum Glauber dynamics on Gibbs point processes with arbitrary-range repulsive pair potentials (including hard spheres) by developing continuous analogs of spectral independence and negative-fields localization. It shows that a stronger variant of zero-freeness implies continuous spectral independence, which in turn permits a localization scheme to boost the spectral gap from small to large activity; this extends the known mixing regime for the dynamics and yields an efficient sampling algorithm matching the finite-range case, while also improving the activity threshold for sampling fixed-size/density packings in a bounded domain beyond the 2003 Kannan-Mahoney-Montenegro result. The argument follows the Chen-Eldan blueprint but works directly in the continuous setting without discretization.

Significance. If the central implication from strengthened zero-freeness to a range-independent continuous spectral independence constant holds, the work supplies a unified, discretization-free route to spectral gaps for a broad class of repulsive potentials, closing the gap between finite-range and arbitrary-range results and furnishing the first improvement on the Kannan et al. packing-sampling threshold in two decades. The explicit construction of continuous spectral independence and localization constitutes a reusable technical contribution for other spatial birth-death processes.

major comments (1)

[continuous spectral independence development] The load-bearing step (stronger zero-freeness implying a continuous spectral independence constant bounded solely in terms of activity, uniformly in potential range) is stated in the abstract and developed in the continuous-analogs section, but the provided manuscript sketch does not exhibit an explicit bound or regularity condition that rules out range-dependent growth of the SI constant; without such control the localization boost cannot be guaranteed to reach the claimed activity regime for arbitrary-range potentials.

minor comments (1)

[preliminaries] Notation for the continuous Glauber generator and the precise definition of the strengthened zero-freeness region should be introduced with a dedicated display equation early in the preliminaries to aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading, positive assessment of the significance, and constructive feedback on the manuscript. We address the single major comment below and will revise the presentation accordingly.

read point-by-point responses

Referee: The load-bearing step (stronger zero-freeness implying a continuous spectral independence constant bounded solely in terms of activity, uniformly in potential range) is stated in the abstract and developed in the continuous-analogs section, but the provided manuscript sketch does not exhibit an explicit bound or regularity condition that rules out range-dependent growth of the SI constant; without such control the localization boost cannot be guaranteed to reach the claimed activity regime for arbitrary-range potentials.

Authors: We appreciate this observation on the presentation. The continuous-analogs section (specifically the development from strengthened zero-freeness to continuous spectral independence) does establish that the SI constant is bounded by a function of activity alone. The argument proceeds by controlling the total variation distance between perturbed and unperturbed measures via the zero-freeness radius, which is independent of potential range by the repulsive-pair assumption; the resulting bound appears in the estimates leading to Theorem 3.4 and is used directly in the localization analysis of Section 5. The uniformity follows because the pair-potential regularity conditions (monotonicity and integrability) enter only through the zero-freeness hypothesis, which itself carries no range dependence. That said, we agree the sketch does not foreground an explicit corollary isolating the range-independent bound. We will add such a corollary (with the precise functional dependence on activity) immediately after the main SI theorem to make the control fully transparent and to streamline the subsequent localization argument. revision: partial

Circularity Check

0 steps flagged

No circularity; continuous extension relies on new definitions and external discrete blueprint

full rationale

The paper develops continuous analogs of spectral independence and negative-field localization as new objects, then proves that a strengthened zero-freeness condition implies the continuous spectral-independence property. This implication is used to run a localization scheme that boosts the spectral gap. The high-level structure follows Chen-Eldan 2022 (external) but the continuous definitions, the zero-freeness strengthening, and the direct continuous analysis are introduced here without reducing any target quantity to a parameter fitted inside the same paper or to a self-citation chain. No equation is shown to equal its own input by construction, and the cited priors (Chen-Eldan 2022, Kannan et al. 2003) are independent of the present authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard probability axioms plus the new continuous definitions of spectral independence and zero-freeness. No free parameters are fitted inside the proof; the activity threshold is the quantity being bounded rather than chosen by hand. No new particles or forces are postulated.

axioms (2)

standard math Standard axioms of probability and measure theory on point processes
Invoked throughout to define the Gibbs distribution and the birth-death process.
domain assumption Existence of a continuous spectral independence property for repulsive potentials
The paper states that a stronger zero-freeness variant implies this property; it is the key new assumption transferred from the discrete setting.

pith-pipeline@v0.9.0 · 5553 in / 1371 out tokens · 58891 ms · 2026-05-16T10:29:32.886350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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...
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection (coupling combiner forces bilinear branch) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.23. Suppose ... bounded complex density up to activity λ0>0. Then ... ρ(Ψλ−Id)≤C/ε e^λCϕ.

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