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arxiv: 2606.28009 · v1 · pith:MHTOIA2Mnew · submitted 2026-06-26 · 🧮 math-ph · math.MP· math.PR

Uniqueness, analyticity and mixing for Gibbs point processes via spectral gaps

Andreas G\"obel , Matthew Jenssen , Marcus Michelen , Marcus Pappik , Will Perkins , Leon Schiller This is my paper

Pith reviewed 2026-06-29 02:27 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.PR
keywords Gibbs point processeshard-sphere modelspectral gapuniquenessanalyticity of pressurespatial mixingphase transitionsbirth-death dynamics
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The pith

Spectral gaps for continuum birth-death dynamics imply uniqueness of infinite-volume Gibbs measures and analytic pressure up to a new activity threshold.

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

The paper shows that whenever a spectral gap holds for the associated Glauber-like birth-death process, the Gibbs point process has a unique infinite-volume measure, the pressure is analytic in activity, and both spatial and temporal mixing occur. This threshold, called λ_spec, improves the known uniqueness and analyticity bounds for the hard-sphere model in every fixed dimension d ≥ 2, with the improvement becoming exponentially large as d grows. The authors also construct repulsive radial potentials for which λ_spec is infinite, so the corresponding Gibbs process remains unique and analytic at every positive activity; two such examples are potentials whose ground states are the E8 and Leech lattices.

Core claim

The existence of a spectral gap for the continuum birth-death dynamics up to activity λ_spec implies uniqueness of the infinite-volume Gibbs measure, analyticity of the pressure, and various mixing properties. For the hard-sphere model this produces strictly better thresholds than classical bounds in each fixed dimension, and the gap to those bounds grows exponentially with dimension; the same argument yields an optimal mixing-time bound for heat-bath dynamics up to expected density Θ(d/2^d). Certain repulsive radial potentials are shown to satisfy λ_spec = +∞, hence to have no phase transition at any finite activity, including potentials whose zero-temperature ground states are the E8 and L

What carries the argument

The spectral threshold λ_spec, defined as the supremum of activities at which the spectral gap for the Glauber-like continuum birth-death dynamics continues to hold.

If this is right

  • The infinite-volume Gibbs measure is unique for every activity up to λ_spec.
  • The pressure is an analytic function of activity up to λ_spec.
  • Spatial and temporal mixing hold for the process up to λ_spec.
  • Heat-bath dynamics for hard spheres mix in optimal time up to expected density Θ(d/2^d).
  • Repulsive radial potentials with infinite λ_spec have no phase transition at any positive activity.

Where Pith is reading between the lines

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

  • The same spectral-gap argument could be applied to other pair potentials once their own λ_spec is located.
  • Because λ_spec can be infinite, the method separates the question of phase transition from the mere existence of a spectral gap.
  • Numerical approximation of the spectral gap would give concrete, computable bounds on the location of any phase transition for a given potential.
  • The exponential improvement in high dimensions suggests the approach captures a larger fraction of the regime where uniqueness is expected to hold.

Load-bearing premise

The spectral gap proved for the birth-death dynamics in the 2013 work remains valid all the way up to λ_spec and is strong enough to imply uniqueness, analyticity, and mixing.

What would settle it

Exhibiting two distinct infinite-volume Gibbs measures for the hard-sphere model at an activity strictly below the value of λ_spec computed from the spectral gap would show that the gap does not imply uniqueness.

Figures

Figures reproduced from arXiv: 2606.28009 by Andreas G\"obel, Leon Schiller, Marcus Michelen, Marcus Pappik, Matthew Jenssen, Will Perkins.

Figure 1
Figure 1. Figure 1: The graph of implications. Dashed lines indicate implications known from the literature. Corollary 1.10. For the hard-sphere potential, ψ is analytic for all α ∈ (0, cd2 −d ) for each fixed c < log(2/ √ 3) and d sufficiently large. We also show that if λspec = ∞ then αspec = ∞, and so for such potentials ψ is analytic for all α > 0. Corollary 1.11. Let ϕ be a repulsive, translation invariant potential that… view at source ↗
Figure 2
Figure 2. Figure 2: Structure of the induction step. Moreover, for Part 3 of the induction step, we use the following elementary lemma. Lemma 5.3. Let t > 0, and let f : [0, t] → C \ {0} be absolutely continuous. It holds that f(t) f(0) = exp Z t 0 f ′ (s) f(s) ds  , where the integral should be understood as a Lebesgue integral and f ′ is almost everywhere a derivative of f. Proof. Since f is continuous and non-zero, we ha… view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the regions considered in step 4. The region ∆1 = Λ ∩ BS(x) is represented by the checkered pattern, the region ∆2 = Λ ∩ B2S(x) is represented by the lined pattern. In the proof, we handle the expected influence of points in the ball ∆1 and the annulus ∆a on the region Λ2 separately. Part 4: ((i) and (iii) for n, and (iv) for < n imply (iv) for n): Fix a region Λ ∈ Bb that can be covered by… view at source ↗
read the original abstract

A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $\lambda$, a radius $r$, and a bounded set $\Lambda \subset \mathbb{R}^d$ one samples a Poisson process of intensity $\lambda$ in $\Lambda$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $\lambda_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $\Theta(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $\lambda_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $\lambda > 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $\lambda_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $\lambda_{spec}$ for interesting potentials.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that a spectral gap for a Glauber-like continuum birth-death dynamics (taken from the 2013 Kondratiev-Kuna-Ohlerich result) up to an implicitly defined activity threshold λ_spec implies uniqueness of the infinite-volume Gibbs measure, analyticity of the pressure, and several forms of spatial and temporal mixing for Gibbs point processes. For the hard-sphere model it derives explicit bounds on λ_spec that improve classical uniqueness radii for each fixed d ≥ 2, with the improvement growing exponentially in d; it also gives an optimal mixing-time bound for heat-bath dynamics up to density Θ(d/2^d). For certain repulsive radial pair potentials (including examples in d=8 and d=24 whose ground states are the E8 and Leech lattices) it shows λ_spec = ∞, hence absence of phase transition at any activity.

Significance. If the implication chain from spectral gap to the listed equilibrium and mixing properties holds, the work supplies a systematic route from dynamical spectral information to absence of phase transition in continuum systems. The exponential improvement for hard spheres in high dimension and the asymptotic matching of the Parisi-Zamponi rapid-mixing threshold are concrete advances; the construction of potentials with λ_spec = ∞, backed by the Cohn-Kumar-Miller-Radchenko-Viazovska ground-state theorems, gives rigorous examples of interaction potentials with no phase transition at any density. The paper is transparent about its reliance on the 2013 black-box result and focuses its own contribution on the implication theorems and the explicit analysis of λ_spec.

major comments (2)
  1. [Definition of λ_spec and implication theorems] The central claim that the 2013 spectral gap continues to hold exactly up to the defined λ_spec and is sufficient for all listed consequences is load-bearing; the manuscript must therefore contain a self-contained statement (in the section introducing λ_spec) of the precise range of activities for which the 2013 theorem applies and of the precise functional inequalities that are deduced from the gap.
  2. [Hard-sphere analysis] The exponential improvement over classical bounds as d → ∞ for hard spheres is a headline quantitative result; the explicit lower bound or asymptotic expression for λ_spec (hard-sphere case) must appear in the main text (not only in the abstract) so that the growth rate can be verified directly.
minor comments (2)
  1. [Notation and preliminaries] Notation for the continuum birth-death generator and for the various mixing notions should be introduced once, with a short table or list of definitions, to avoid repeated cross-references.
  2. [References] The reference list should include the precise citation for the Parisi-Zamponi prediction that is being matched asymptotically.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Definition of λ_spec and implication theorems] The central claim that the 2013 spectral gap continues to hold exactly up to the defined λ_spec and is sufficient for all listed consequences is load-bearing; the manuscript must therefore contain a self-contained statement (in the section introducing λ_spec) of the precise range of activities for which the 2013 theorem applies and of the precise functional inequalities that are deduced from the gap.

    Authors: We agree that a self-contained statement of the precise activity range from the 2013 Kondratiev-Kuna-Ohlerich theorem and the functional inequalities deduced from the spectral gap would improve transparency. In the revised manuscript we will insert this statement in the section introducing λ_spec. revision: yes

  2. Referee: [Hard-sphere analysis] The exponential improvement over classical bounds as d → ∞ for hard spheres is a headline quantitative result; the explicit lower bound or asymptotic expression for λ_spec (hard-sphere case) must appear in the main text (not only in the abstract) so that the growth rate can be verified directly.

    Authors: We agree that the explicit lower bound or asymptotic expression for λ_spec in the hard-sphere case should be stated in the main text. We will move or add this material to the main body in the revision. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation takes the spectral gap result as an external input from the 2013 Kondratiev-Kuna-Ohlerich paper (distinct authors), defines λ_spec implicitly from that gap, proves that any such gap implies uniqueness/analyticity/mixing, and then computes explicit bounds on λ_spec for hard spheres and other potentials. No step reduces by construction to a fitted input, self-citation, or ansatz smuggled from prior work by the same authors. The implication chain from gap to thermodynamic properties is presented as independent content, and the paper is self-contained against the external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the spectral-gap theorem from the 2013 Kondratiev-Kuna-Ohlerich paper and on standard existence results for Gibbs measures in continuum statistical mechanics; no free parameters or new invented entities are introduced.

axioms (2)
  • domain assumption Existence and uniqueness of the continuum birth-death dynamics with the stated spectral gap up to λ_spec (Kondratiev-Kuna-Ohlerich 2013)
    Invoked as the starting point whose consequences are derived in the present work.
  • standard math Standard measurability and integrability conditions for the pair potential and activity parameter
    Required for the definition of the Gibbs point process and the dynamics.

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Reference graph

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