Large Deviation Analysis for Canonical Gibbs Measures

Christian Hirsch; Martina Petr\'akov\'a

Large Deviation Analysis for Canonical Gibbs Measures

T0 review · 0 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Fixed-particle Gibbs point processes satisfy a large-deviation principle, with the rate function given by the grand-canonical entropy-plus-interaction formula re-centered to have minimum zero.

desk verdict Genuinely new LDPs for canonical Gibbs processes; the unbounded-interaction theorem is conditional in a way the abstract doesn't say. read the letter →

arxiv 2411.18483 v2 pith:NSGMJICF submitted 2024-11-27 math.PR

Christian Hirsch , Martina Petráková This is my paper · ORCID

classification math.PR MSC 60K3560F1060G5582B21

keywords Gibbspointprocesscanonicalensemblelargedeviationprincipleempiricalfieldbinomialhard-coreinteractionStraussfreeenergy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes large-deviation principles (LDPs) for Gibbs point processes in which the number of points is fixed, the canonical ensemble, as the sampling window grows. The central result shows that the random individual empirical field of a binomial Gibbs process satisfies an LDP with speed equal to window volume and a rate function obtained from the grand-canonical rate, specific entropy plus mean interaction energy, re-centered by its minimum. The same framework yields LDPs for unbounded increasing interactions, provided the finite-volume partition function has a finite free-energy limit, and for the hard-core Strauss process when the intensity is below the packing threshold. This matters because many physically central models, including Coulomb and Riesz gases, are inherently fixed-particle systems yet lacked a systematic large-deviation theory.

What carries the argument

The bridge is the elementary identity that the binomial point process is the Poisson point process conditioned on the point count, $\mathrm{d}B_n=\Pi_n(|\omega|=n)^{-1}\mathbf 1_{[|\omega|=n]}\,\mathrm{d}\Pi_n$, combined with the grand-canonical LDP in the $\tau_{L_0}$-topology. Since the conditioning event is closed, upper bounds are immediate; the technical work is a family of couplings for the lower bounds. In the unbounded case the deletion/sprinkling operations of the Poisson setting are replaced by a move operation that relocates so-called $b$-dense points into sparse cubes, keeping the particle number fixed, while the hard-core case couples the binomial and Poisson processes and thins away inadmissible close pairs.

What would settle it

For the unbounded theorem, check the free-energy condition: if a concrete $V$ (say, a superlinearly growing increasing local interaction) has $\lim_n |W_n|^{-1}\log Z_n=-\infty$, the claimed LDP cannot hold, and Remark 2.12 already flags such cases. For the bounded theorem, a numerical test of $\mathbb P(R^o_{n,\rho_n}\in U)$ against $\exp(-|W_n|\inf_U J)$ for a small interaction radius would quickly expose any failure of the rate function.

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Extended reading notes

Core claim

The sequence of individual empirical fields $R^o_{n,\rho_n}$ driven by the binomial Gibbs process satisfies the LDP with speed $|W_n|$ and rate $J(m)=\widetilde J(m)-\inf_{m'}\widetilde J(m')$, where $\widetilde J(m)=I(P)+P^o(V)$ when $m=P^o$ for some stationary probability $P$ satisfying $P^o(1)=\lambda$, and $+\infty$ otherwise. Here $I(P)$ is the specific entropy and $P^o(V)$ the Palm expectation of the interaction. Thus the canonical rate differs from the grand-canonical one only by the normalization that makes its minimum zero. This is proved first for bounded local interactions, then for unbounded increasing interactions under the additional condition that $\lim |W_n|^{-1}\log Z_n>-\infty$, and for the hard-core Strauss process under $\lambda R^d v_d<1$; boundary-condition variants are also treated.

Load-bearing premise

For the unbounded case, the load-bearing premise is that the canonical partition function $Z_n$ does not decay faster than exponentially, i.e. $\lim |W_n|^{-1}\log Z_n>-\infty$; the stated assumptions on $V$ do not force this, and the paper notes it can fail.

Editorial extensions

If this is right

For any bounded local observable, the empirical average satisfies the same LDP, so deviations of such statistics are exponentially controlled by the canonical rate.
For unbounded observables such as edge counts in random geometric graphs, the results give lower-tail upper bounds and two-sided lower bounds, reflecting localization effects that can make full LDPs fail.
The canonical partition function has the free-energy limit $-A$ in the bounded case, and in the unbounded case when the finite-limit condition holds.
The hard-core Strauss process obeys the LDP for local bounded functionals whenever $\lambda R^d v_d<1$, including a formula for its free energy.
Similar LDPs hold for Gibbs measures with alternative boundary-condition Hamiltonians in the bounded setting, so the results are not tied to periodicity alone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the finite-free-energy condition in the unbounded theorem can be verified for Riesz-type interactions, the move coupling may supply the first canonical LDP for such fixed-charge systems; the paper already draws on the move idea from number-rigidity work.
The normalization $J(m)=\widetilde J(m)-\inf\widetilde J$ means the minimizers of $\widetilde J$ are exactly the typical empirical fields; testing this against simulations at moderate $n$ could measure the practical radius of the LDP.
The hard-core constraint $\lambda R^d v_d<1$ is exactly the regime in which the partition function is positive; an analogous obstruction should appear for more general infinite interactions, so the finite free-energy condition is likely necessary rather than technical.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a serious paper and deserves a real referee. It delivers the first systematic LDPs for canonical (fixed-number) Gibbs processes outside the Coulomb/Riesz setting, and the proofs are detailed enough to check. The bounded-interaction Theorem 2.7 is the clean centerpiece; the move-coupling for unbounded interactions is genuinely new and looks sound. Credit where due: the paper states assumptions clearly, uses the grand-canonical LDP as an external benchmark rather than reproving it, and handles the hard-core Strauss case with a thinning/sprinkling argument that respects the fixed-number constraint.

The main soft spot is exactly what the stress-test note flags. The abstract and introduction advertise Theorem 2.11 as an LDP for 'non-negative possibly unbounded increasing local interaction' without carrying the finiteness condition. Read carefully, the theorem's second paragraph is explicitly conditional on lim (1/|W_n|) log Z_n > -∞, and Remark 2.12 concedes the limit can be -∞. So the theorem is correctly stated, but the framing overstates scope. Anyone citing it as an unconditional LDP would be wrong. This is a presentation problem, not a mathematical hole.

Minor points: Lemma 4.3 is only sketched, though the sketch is reasonably detailed. The hard-core result requires λ R^d v_d < 1 and covers only the pure hard-core Strauss interaction; the authors say so up front. Theorem 2.16 gives an LDP for local bounded functionals rather than the full empirical-field LDP, which is a slightly weaker target but consistent with the hard-core section's stated scope.

I don't see a load-bearing flaw. The variational rate function is the expected canonical analogue, the upper bound via conditioning on the closed event {|ω|=n} is clean, and the lower-bound couplings are the real work. Citation pattern looks honest; self-citations are proof tools, not padding. For probabilists in continuum statistical mechanics or stochastic geometry, this is worth a read. Recommendation: send to peer review. An editor should not desk-reject this. A referee should ask for a revised abstract/introduction that states the finiteness condition in Theorem 2.11 explicitly, and possibly a bit more detail around Lemma 4.3, but the core mathematics is in good shape.

Referee Report

0 major / 3 minor

Summary. This paper develops large-deviation principles for canonical (binomial) Gibbs point processes in growing windows. The central object is the individual empirical field R^o_{n,rho_n} under the binomial Gibbs measure. Theorem 2.7 gives a full LDP for bounded r-local interactions with rate function J(m)=tilde J(m)-inf tilde J(m), where tilde J(m)=I(P)+P^o(V). Theorem 2.11 extends this to non-negative increasing r-local cardinality-bounded interactions, but only under the explicit additional hypothesis that lim |W_n|^{-1} log Z_n > -infinity; Remark 2.12 notes that this limit can be -infinity. Theorem 2.16 treats the hard-core Strauss process under lambda v_d R^d < 1 for bounded local score functions, and Theorem 2.18 covers two boundary-condition Hamiltonians. The proofs condition the Poisson process on {|omega|=n}, use the Georgii-Zessin LDP as an external benchmark, and construct couplings via move operations for unbounded interactions and thinning/sprinkling for the hard-core case.

Significance. If the results hold, the paper fills a real gap by giving systematic canonical analogues of grand-canonical variational LDPs for Gibbs point processes, with an explicit rate function in the spirit of Georgii-Zessin. The move-operation coupling is a substantive technical contribution, particularly because deletion/addition is not available in the canonical setting. The hard-core result and the boundary-condition extensions broaden the applicability, and the examples (Strauss process, k-wise interactions) are useful illustrations. The proofs are detailed and all lemmas are proved, with the exception of Lemma 4.3, which is presented as a sketch. The paper is also honest about the conditional nature of the unbounded-interaction result: Remark 2.12 states that the relevant free-energy limit can be -infinity. The explicit hypotheses and the careful statements are strengths of the paper.

minor comments (3)

[Abstract and Section 1, bullet (1)] The abstract and the introduction describe Theorem 2.11 as establishing an LDP for a possibly unbounded non-negative increasing local interaction without mentioning the additional condition lim |W_n|^{-1} log Z_n > -infinity. Since Remark 2.12 explicitly notes that this limit can be -infinity, in which case the LDP is not established, the advertised scope should carry this caveat. Theorem 2.11 itself is correctly conditional; the issue is purely one of presentation.
[Section 4, Lemma 4.3] The proof of Lemma 4.3 is presented only as a sketch. Because Corollaries 2.10 and 2.14 depend on this lemma, it would be helpful to write out the reduction of (45) to (30) explicitly, or at least to state precisely how the parameters b, delta, and epsilon are chosen for vector-valued thresholds a.
[Section 2.2.1, Theorem 2.16] Theorem 2.16 is stated at the level of bounded local score functions, i.e., finite-dimensional projections, rather than for the individual empirical field itself. The abstract says the large-deviation principle is formulated for distributions of individual empirical fields, so the hard-core section should explicitly clarify that the stated result is at the functional level, or state the corresponding empirical-field version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the canonical LDP is derived from the external grand-canonical LDP of Georgii-Zessin plus new coupling arguments; the conditional finite-free-energy hypothesis is an explicit assumption, not a disguised input.

full rationale

The paper's central derivation is self-contained against the external benchmark [14]. The rate function J(m)=tilde J(m)-inf tilde J is the canonical analogue of the Georgii-Zessin variational formula, but the whole content of Theorems 2.7, 2.11, and 2.16 is to show that the fixed-number binomial Gibbs measures satisfy an LDP with that rate function. The upper bounds are direct applications of Corollary 3.2 in [14] to the closed event {|omega|=n}, which is legitimate because the binomial process is the Poisson process conditioned on n points. The lower bounds require new coupling constructions (moving points from dense to sparse regions) and are proved in detail in Sections 3.1, 5, and 6; they do not reduce to the statement being proved. The paper cites the authors' earlier work [15] for the b-dense-point technique, but it re-proves the needed rareness lemma (Lemma 5.2) and constructs its own move coupling, so the citation is a proof tool rather than a load-bearing self-citation. The finite-free-energy condition in Theorem 2.11 ((lim 1/|W_n|) log Z_n > -infinity) is an explicit hypothesis, and Remark 2.12 acknowledges it can fail; this is a scoping limitation, not a circular reduction. No fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation. The derivation chain is legitimate and the central claims have independent content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. It relies on standard large-deviation theory, concentration inequalities, and the externally cited grand-canonical LDP. The only extra assumptions are well-posedness conditions (lambda R^d v_d < 1 for hard-core) and finiteness of the free-energy limit in the unbounded case, both stated explicitly.

assumptions (5)

standard math Georgii-Zessin LDP for tau_Lo-continuous local bounded functionals of Poisson point processes (Corollary 3.2 in [14])
Used as the fundamental external benchmark in Lemmas 3.1, 3.4, 4.1, 6.1, 6.6.
standard math Tail bound for binomial distributions (Lemma 1.1 in [19])
Invoked in Lemmas 3.3, 6.5 and Theorem 2.18 to control rare events.
standard math Stirling's formula gives lim (1/|W_n|) log Pi_n(|omega|=n) = 0 (Lemma 3.7)
Essential for transferring Poisson-based estimates to the canonical conditioning.
domain assumption Intensity constraint lambda R^d v_d < 1 for hard-core interactions
Ensures the hard-core partition function is positive (Section 2.2.1) and used in Lemma 6.7.
domain assumption Finiteness of lim (1/|W_n|) log Z_n in the unbounded case
Extra assumption in Theorem 2.11; can fail as noted in Remark 2.12.

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Pith. "Pith review of Large Deviation Analysis for Canonical Gibbs Measures." pith.science (2026). https://pith.science/paper/NSGMJICF

@misc{pith2026241118483,
  author       = {Pith},
  title        = {Pith review of: Large Deviation Analysis for Canonical Gibbs Measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSGMJICF}},
  note         = {Machine review of arXiv:2411.18483}
}

read the original abstract

In this paper, we present a large-deviation theory developed for functionals of canonical Gibbs processes, i.e., Gibbs processes with respect to the binomial point process. We study the regime of a fixed intensity in a sequence of increasing windows. Our method relies on the traditional large-deviation result for local bounded functionals of Poisson point processes noting that the binomial point process is obtained from the Poisson point process by conditioning on the point number. Our main methodological contribution is the development of coupling constructions allowing us to handle delicate and unlikely pathological events. The presented results cover three types of Gibbs models - a model given by a bounded local interaction, a model given by a non-negative possibly unbounded increasing local interaction and the hard-core interaction model. The derived large deviation principle is formulated for the distributions of individual empirical fields driven by canonical Gibbs processes, with its special case being a large deviation principle for local bounded observables of the canonical Gibbs processes. We also consider unbounded non-negative increasing local observables, but the price for treating this more general case is that we only get large-deviation bounds for the tails of such observables. Our primary setting is the one with periodic boundary condition, however, we also discuss generalizations for different choices of the boundary condition.

Figures

Figures reproduced from arXiv: 2411.18483 by the authors.

**Figure 1.** Graphic representation of the partition of the window Wn from the proof of Lemma 5.2 in dimension d = 2. Left: The partition corresponding to the choice wn L = 2, i.e. cn = 32 wn L 2 = 36. Different shades of gray correspond to the partition of the indices {1, . . . , 36} into subsets A1, . . . , A9 of size Kn = wn L 2 = 4. Right: A point configuration ω (round points) in the partitioned window Wn into cu… view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Adams, A

S. Adams, A. Collevecchio, and W. K¨ onig. A variational f ormula for the free energy of an interacting many-particle s ystem. Ann. Probab., 39(2):683–728, 2011

work page 2011

[2] [2]

Adams and Q

S. Adams and Q. Vogel. Space-time random walk loop measur es. Stochastic Process. Appl. , 130(4):2086–2126, 2020

work page 2020

[3] [3]

Chatterjee and M

S. Chatterjee and M. Harel. Localization in random geome tric graphs with too many edges. Ann. Probab., 48(1):574–621, 2020

work page 2020

[4] [4]

Dereudre, A

D. Dereudre, A. Hardy, T. Lebl´ e, and M. Ma ¨ ıda. DLR equations and rigidity for the sine-beta process. Comm. Pure Appl. Math., 74(1):172–222, 2021

work page 2021

[5] [5]

Dereudre and P

D. Dereudre and P. Houdebert. Phase transition for conti nuum Widom-Rowlinson model with random radii. J. Stat. Phys. , 174(1):56–76, 2019

work page 2019

[6] [6]

Dereudre and C

D. Dereudre and C. Renaud-Chan. Liquid-gas phase transi tion for Gibbs point process with Quermass interaction. arXiv preprint arXiv:2309.08338, 2023

work page arXiv 2023

[7] [7]

Dereudre and T

D. Dereudre and T. Vasseur. Number-rigidity and β -circular Riesz gas. Ann. Probab., 51(3):1025–1065, 2023

work page 2023

[8] [8]

Dickson and Q

M. Dickson and Q. Vogel. Formation of inﬁnite loops for an interacting bosonic loop soup. Electron. J. Probab. , 29:Paper No. 24, 39, 2024

work page 2024