arxiv: 2605.09180 · v1 · submitted 2026-05-09 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

Fluctuations for the critical free Bose gas

Quirin Vogel

Pith reviewed 2026-05-12 02:16 UTC · model grok-4.3

classification 🧮 math.PR

keywords free Bose gascritical regimecritical exponentsmacroscopic loopsMinakshisundaram-Pleijel expansionFredholm determinantGreen operatorfluctuations

0 comments

The pith

For the critical free Bose gas in three dimensions, the critical exponents and macroscopic loops depend on domain geometry and boundary conditions.

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

The paper examines the probabilistic behavior of the free Bose gas at criticality in bounded domains. It shows that unlike in subcritical or supercritical regimes, the critical exponents here vary with the shape and boundaries of the container. This dependence arises because the second term in the Minakshisundaram-Pleijel expansion governs the appearance of large loops. The work also derives non-Gaussian limiting distributions for the particle number fluctuations, expressed through a regularized Fredholm determinant involving the Green operator.

Core claim

We study the critical free Bose gas from a probabilistic vantage with a focus on the three-dimensional case. We obtain the critical exponents. These exponents and the occurrence of macroscopic loops subtly depend on the geometry of the domain and the boundary conditions, contrary to the subcritical and supercritical case. In particular, the second term of the Minakshisundaram-Pleijel expansion determines the emergence of large loops. We furthermore obtain non-Gaussian limit laws for the fluctuations, governed by the regularized Fredholm determinant of the Green operator.

What carries the argument

the second term of the Minakshisundaram-Pleijel expansion of the heat kernel, which controls the emergence of macroscopic loops, together with the regularized Fredholm determinant of the Green operator, which determines the non-Gaussian fluctuation laws

Load-bearing premise

The Minakshisundaram-Pleijel expansion is valid for the relevant operator in the considered bounded domains and boundary conditions.

What would settle it

A computation or simulation of loop size distributions and fluctuation statistics for the critical free Bose gas in two domains with different second coefficients in the Minakshisundaram-Pleijel expansion, such as a ball and a cube, to check whether the predicted geometric differences appear.

Figures

Figures reproduced from arXiv: 2605.09180 by Quirin Vogel.

**Figure 1.** Figure 1: A simulation of the Bose gas for ΛL = 30T 3 , projected onto the first two coordinates. The density is kept constant while β is varied, to illustrate the nature of phase transition of the permutation. The largest loop is colored in red. The particles are represented by bold dots. 1.2. Past results. In [KVZ25], a probabilistic analysis of P (can) L,β,ρ and ZL,β,ρ was conducted. A phase transition occurs at … view at source ↗

**Figure 2.** Figure 2: Phase diagram of the free Bose gas in dimension d = 3. The present article studies the behavior on the critical line. 1.3. High-level summary. In this article, the case ρ = ρc is studied. The critical regime exhibits substantially richer behavior. In contrast to the subcritical and supercritical phases, where the relevant quantities display exponential or asymptotically constant behavior, criticality is c… view at source ↗

read the original abstract

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 shows geometry and boundary conditions affect critical exponents and macroscopic loops for the 3D free Bose gas in ways they do not sub- or supercritically, with fluctuations governed by a regularized Fredholm determinant.

read the letter

The core result is that at criticality the second coefficient in the Minakshisundaram-Pleijel expansion of the Green operator controls whether large loops appear, and this term picks up the domain shape and boundary data. Away from criticality those effects drop out, so the critical regime really is different. The non-Gaussian fluctuation laws are then expressed through the same regularized determinant. That is the new piece: a clean probabilistic derivation that tracks the expansion precisely where the usual Gaussian approximations break down. The argument stays inside the standard loop soup or random walk representation of the free Bose gas, which keeps the technical overhead reasonable. The stress-test note is right that no obvious circularity or unjustified limit swap shows up from the stated claims. The main soft spot is that the abstract (and the summary we have) leaves the justification for using the Minakshisundaram-Pleijel expansion right at criticality a bit thin; one wants to see that the remainder terms do not interfere with the geometry dependence that is being extracted. If the full proof handles the error estimates uniformly in the domain, the claim holds; if not, the geometry effect could be an artifact of the cutoff. The paper is aimed at people working on Bose-Einstein condensation, loop soups, and finite-size scaling in quantum statistical mechanics. A reader already comfortable with the probabilistic formulation of the ideal Bose gas will get the most out of it. It is worth sending to referees because the question it answers is well-posed and the tools are standard enough that a careful check is feasible. I would bring it to a reading group only if the group already has someone who knows the Minakshisundaram-Pleijel literature.

Referee Report

0 major / 2 minor

Summary. The manuscript develops a probabilistic analysis of the critical free Bose gas in three dimensions. It derives critical exponents whose values and the emergence of macroscopic loops depend on the domain geometry and boundary conditions through the second coefficient of the Minakshisundaram-Pleijel expansion of the Green operator. It further establishes non-Gaussian limiting distributions for the fluctuations, expressed in terms of the regularized Fredholm determinant of the same operator.

Significance. If the derivations hold, the work is significant because it isolates the precise mechanism by which geometry enters at criticality, a feature absent from the subcritical and supercritical regimes. The reliance on the standard probabilistic representation of the Bose gas together with the Minakshisundaram-Pleijel expansion and regularized Fredholm determinants supplies a coherent and falsifiable framework. The explicit identification of the second MP coefficient as the carrier of geometry dependence and the concrete form of the non-Gaussian laws constitute concrete, testable advances.

minor comments (2)

The abstract states that the second term of the Minakshisundaram-Pleijel expansion determines the emergence of large loops; a brief reminder in the introduction of the precise form of this expansion (including the coefficient that carries the geometry dependence) would help readers connect the abstract claim to the later theorems.
Notation for the regularized Fredholm determinant should be introduced with an explicit reference to its definition (e.g., via the zeta-function regularization or the Hadamard finite-part) at the first appearance, to avoid ambiguity with other common regularizations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work on the critical free Bose gas and for recommending minor revision. No specific major comments were raised in the report, so we have no points requiring rebuttal or substantive changes. We will incorporate any minor editorial adjustments in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation chain starts from the standard probabilistic formulation of the free Bose gas together with the Minakshisundaram-Pleijel expansion of the Green operator, both external and independently established. Critical exponents and macroscopic-loop occurrence are read off from the second coefficient of this known expansion, while non-Gaussian fluctuation laws are expressed via the regularized Fredholm determinant of the same operator. No equation reduces to a self-definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation whose validity is internal to the paper. The geometry dependence is carried by an external asymptotic fact rather than by any construction internal to the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available, so ledger is minimal. The work assumes the standard free Bose gas model and heat-kernel expansion without listing fitted parameters or new entities.

axioms (2)

domain assumption Standard probabilistic formulation of the free Bose gas on a bounded domain with given boundary conditions.
Invoked implicitly by the focus on critical free Bose gas and domain geometry.
domain assumption Validity of the Minakshisundaram-Pleijel expansion for the Green operator in the domain.
Cited as determining the emergence of large loops.

pith-pipeline@v0.9.0 · 5365 in / 1384 out tokens · 45278 ms · 2026-05-12T02:16:54.872882+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 unclear
We obtain the critical exponents. These exponents and the occurrence of macroscopic loops subtly depend on the geometry of the domain and the boundary conditions... governed by the regularized Fredholm determinant of the Green operator.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
the second term of the Minakshisundaram–Pleijel expansion determines the emergence of large loops

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

[1]

and Pleijel,

Minakshisundaram, S. and Pleijel,. Some properties of the eigenfunctions of the. Canadian Journal of Mathematics , volume=. 1949 , publisher=

work page 1949
[2]

2012 , publisher=

Brownian Motion and Classical Potential Theory , author=. 2012 , publisher=

work page 2012
[3]

Progress of Theoretical Physics , volume=

Quantum-statistical theory of liquid helium , author=. Progress of Theoretical Physics , volume=. 1951 , publisher=

work page 1951
[4]

Journal of Spectral Theory , volume=

Heat kernel estimates for general boundary problems , author=. Journal of Spectral Theory , volume=

work page
[5]

Journal of mathematical physics , volume=

Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure , author=. Journal of mathematical physics , volume=. 2006 , publisher=

work page 2006
[6]

and Vogel, Q

Dickson, M. and Vogel, Q. , year =. Formation of infinite loops for an interacting bosonic loop soup , volume =. Electronic Journal of Probability , publisher =. doi:10.1214/24-ejp1085 , number =

work page doi:10.1214/24-ejp1085
[7]

, year =

Vogel, Q. , year =. Emergence of interlacements from the finite volume Bose soup , volume =. doi:10.1016/j.spa.2023.104227 , journal =

work page doi:10.1016/j.spa.2023.104227 2023
[8]

Electronic Journal of Probability , volume=

Spatial random permutations and Poisson-Dirichlet law of cycle lengths , author=. Electronic Journal of Probability , volume=. 2011 , publisher=

work page 2011
[9]

Journal of mathematical physics , volume=

Limit theorems for statistics of combinatorial partitions with applications to mean field Bose gas , author=. Journal of mathematical physics , volume=. 2005 , publisher=

work page 2005
[10]

2003 , publisher=

Logarithmic combinatorial structures: a probabilistic approach , author=. 2003 , publisher=

work page 2003
[11]

2005 , publisher=

Trace ideals and their applications , author=. 2005 , publisher=

work page 2005
[12]

2018 , publisher=

Lectures on the Poisson process , author=. 2018 , publisher=

work page 2018
[13]

Some applications of functional integration in statistical mechanics

Ginibre, J. Some applications of functional integration in statistical mechanics. Mécanique statistique et théorie quantique des champs: Proceedings, Ecole d'Eté de Physique Théorique, Les Houches, France, July 5-August 29, 1970. 1971

work page 1970
[14]

Probability theory and related fields , volume=

Spatial random permutations with small cycle weights , author=. Probability theory and related fields , volume=. 2011 , publisher=

work page 2011
[15]

Communications in mathematical physics , volume=

Spatial random permutations and infinite cycles , author=. Communications in mathematical physics , volume=. 2009 , publisher=

work page 2009
[16]

, journal=

Feynman, R. , journal=. Atomic theory of the transition in. 1953 , publisher=

work page 1953
[17]

1981 , publisher=

Operator algebras and quantum statistical mechanics II , author=. 1981 , publisher=

work page 1981
[18]

2005 , publisher=

The mathematics of the Bose gas and its condensation , author=. 2005 , publisher=

work page 2005
[19]

Communications in Mathematical Physics , volume=

New lower bounds for the (near) critical Ising and 4 models’ two-point functions , author=. Communications in Mathematical Physics , volume=. 2025 , publisher=

work page 2025
[20]

Probability Theory and Related Fields , volume=

One-arm exponent of critical level-set for metric graph Gaussian free field in high dimensions , author=. Probability Theory and Related Fields , volume=. 2025 , publisher=

work page 2025
[21]

arXiv preprint arXiv:2406.02397 , year=

One-arm probabilities for metric graph Gaussian free fields below and at the critical dimension , author=. arXiv preprint arXiv:2406.02397 , year=

work page arXiv
[22]

Journal of Physics A: Mathematical and General , volume=

Percolation transition in the Bose gas , author=. Journal of Physics A: Mathematical and General , volume=

work page
[23]

arXiv preprint arXiv:2405.14329 , year=

A confined random walk locally looks like tilted random interlacements , author=. arXiv preprint arXiv:2405.14329 , year=

work page arXiv
[24]

2024 , publisher=

Random walks and physical fields , author=. 2024 , publisher=

work page 2024
[25]

The Annals of Probability , pages=

Markov loops and renormalization , author=. The Annals of Probability , pages=. 2010 , publisher=

work page 2010
[26]

Stochastic Processes and their Applications , volume=

Space--time random walk loop measures , author=. Stochastic Processes and their Applications , volume=. 2020 , publisher=

work page 2020
[27]

Probability Theory and Related Fields , pages=

Critical one-arm probability for the metric Gaussian free field in low dimensions , author=. Probability Theory and Related Fields , pages=. 2025 , publisher=

work page 2025
[28]

Inventiones mathematicae , volume=

Critical exponents for a percolation model on transient graphs , author=. Inventiones mathematicae , volume=. 2023 , publisher=

work page 2023
[29]

Communications in Mathematical Physics , volume=

Gaussian random permutation and the boson point process , author=. Communications in Mathematical Physics , volume=. 2021 , publisher=

work page 2021
[30]

and Singer, I

McKean Jr, H. and Singer, I. , journal=. Curvature and the eigenvalues of the. 1967 , publisher=

work page 1967
[31]

Physics reports , volume=

Heat kernel expansion: user's manual , author=. Physics reports , volume=. 2003 , publisher=

work page 2003
[32]

Off-diagonal long-range order for the free

K. Off-diagonal long-range order for the free. The Annals of Applied Probability , volume=. 2025 , publisher=

work page 2025
[33]

Bai, T. and K. Proof of off-diagonal long-range order in a mean-field trapped. Electronic Journal of Probability , volume=. 2025 , publisher=

work page 2025
[34]

2018 , publisher=

Invariance theory: the heat equation and the Atiyah-Singer index theorem , author=. 2018 , publisher=

work page 2018