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arxiv: 2605.05431 · v1 · submitted 2026-05-06 · ❄️ cond-mat.stat-mech · cond-mat.quant-gas· math-ph· math.MP· math.PR

A transition in the hole probability at finite temperature for free fermions in d dimensions

Giuseppe Del Vecchio Del Vecchio , Pierre Le Doussal , Gregory Schehr This is my paper

Pith reviewed 2026-05-08 15:30 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.quant-gasmath-phmath.MPmath.PR
keywords hole probabilityfree fermionsfinite temperatureCoulomb gasscaling functionrare fluctuationsFermi gasFredholm determinant
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The pith

The hole probability for free fermions in d dimensions follows an exact scaling function Φ_d(u) with a transition at u_c=2/π due to a gap in the optimal Coulomb gas density.

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

The paper establishes that the probability of an empty spherical region of radius R in a d-dimensional free Fermi gas at temperature T takes the scaling form exp[-(k_F R)^{d+1} Φ_d(2 R T / k_F)]. By mapping this to an effective Coulomb gas, the scaling function Φ_d(u) is computed exactly for any dimension. This function exhibits a transition of order 3/2(d+1) at the universal point u=2/π, where the mechanism of rare fluctuations changes due to the appearance of a macroscopic gap in the optimal particle density. This provides a precise description of the crossover from quantum Pauli statistics at low temperature to classical Poissonian behavior at high temperature.

Core claim

By mapping the hole probability to the partition function of an effective Coulomb gas, we compute exactly the scaling function Φ_d(u) in any dimension d. This function exhibits a transition of order 3/2(d+1) at the universal critical value u_c=2/π, signaling a sharp change in the mechanism of rare fluctuations associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas.

What carries the argument

The effective Coulomb gas obtained from the mapping of the hole probability, whose saddle-point density determines the scaling function Φ_d(u) and develops a gap at the transition.

Load-bearing premise

The hole probability for free fermions can be mapped exactly onto the partition function of an effective Coulomb gas whose saddle-point density yields the scaling function.

What would settle it

Numerical computation of the Fredholm determinant for the hole probability at u near 2/π, checking whether the scaling function or its derivatives change behavior with the predicted order 3/2(d+1).

Figures

Figures reproduced from arXiv: 2605.05431 by Giuseppe Del Vecchio Del Vecchio, Gregory Schehr, Pierre Le Doussal.

Figure 1
Figure 1. Figure 1: FIG. 1. Scaling function Φ view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plots of view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of the scaled density ˜r view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Scaling functions view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plot of view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Scaling functions view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of the view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Percentage errors between the theoretical prediction for Φ view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Scaling collapse of the functions view at source ↗
read the original abstract

In a free Fermi gas at temperature $T$ much higher than the Fermi temperature one expects that the fluctuations of the number of particles in a given region has Poissonian/classical statistics. On the other hand at low temperature the Pauli exclusion principle leads to non trivial counting statistics. It is of great interest from a theoretical and experimental point of view to characterize the crossover between these two limits. Here we focus on the hole probability $P(R,T)$, i.e. the probability that a region of size $R$ is devoid of particles, in dimension $d$, and on the case of a spherical region of large radius $R$. We show that at low temperature it takes the scaling form $P(R,T)\sim \exp\big[-(k_F R)^{d+1}\Phi_d(u=2R\,T/k_F)\big],$ where $k_F$ is the Fermi momentum. By mapping the problem to an effective Coulomb gas, we compute exactly the scaling function $\Phi_d(u)$ in any dimension. Remarkably, it exhibits a transition of order $\tfrac{3}{2}(d+1)$ at the universal critical value $u_c=2/\pi$, signaling a sharp change in the mechanism of rare fluctuations, associated with the emergence of a macroscopic gap in the optimal density of the associated Coulomb gas. Our analytical predictions are supported by precise numerical evaluations of the corresponding Fredholm determinants.

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 paper studies the hole probability P(R,T) for a spherical region of radius R in a d-dimensional free Fermi gas at temperature T. It shows that at low temperature, this probability scales as exp[-(k_F R)^{d+1} Φ_d(u)] where u = 2 R T / k_F. By mapping the fermionic determinant to an effective Coulomb gas, the authors compute the scaling function Φ_d(u) exactly for any dimension d. They report a transition in Φ_d(u) of order 3/2(d+1) at the universal critical value u_c = 2/π, linked to the appearance of a macroscopic gap in the optimal Coulomb gas density. The analytical results are supported by numerical evaluations of the associated Fredholm determinants.

Significance. If the exact mapping and saddle-point calculation hold, this provides a notable exact result for the finite-temperature hole probability in free fermions across dimensions. The universal transition and the change in fluctuation mechanism are significant findings that could impact understanding of rare events in quantum gases. The use of Coulomb gas mapping combined with Fredholm determinant numerics is a strength, offering both analytic insight and verification. This could serve as a benchmark for more complex systems.

major comments (2)
  1. The central claim relies on an exact mapping of log det(1 - K_T) to the minimum of the Coulomb gas energy functional without subleading corrections to the (k_F R)^{d+1} term. The manuscript should explicitly demonstrate why the saddle-point density ρ* yields the exact leading exponential behavior for all u, particularly around the transition at u_c.
  2. The derivation of the transition order 3/2(d+1) and the universality of u_c = 2/π independent of d should be detailed with the explicit form of the effective potential and the saddle equation. It is not clear how the d-dependence cancels in u_c while the order depends on d.
minor comments (2)
  1. Clarify the definition of u = 2 R T / k_F; ensure consistency with the Fermi-Dirac distribution in the kernel.
  2. Provide more details on the precision of the Fredholm determinant evaluations and how they corroborate the analytic transition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: The central claim relies on an exact mapping of log det(1 - K_T) to the minimum of the Coulomb gas energy functional without subleading corrections to the (k_F R)^{d+1} term. The manuscript should explicitly demonstrate why the saddle-point density ρ* yields the exact leading exponential behavior for all u, particularly around the transition at u_c.

    Authors: The mapping of log det(1 - K_T) to the Coulomb gas energy functional is exact at the level of the large-deviation rate function. In the limit k_F R ≫ 1 the logarithm of the determinant is given by the minimum of the energy functional plus subleading corrections that are o((k_F R)^{d+1}); these corrections arise from Gaussian fluctuations around the saddle and are of order (k_F R)^d log(k_F R) or lower. The saddle-point density ρ* therefore controls the leading exponential behavior for every fixed u, including at the transition. We will add a new subsection that recalls the large-deviation principle for the underlying determinantal process and shows that the approximation remains uniform in u across the transition point. revision: yes

  2. Referee: The derivation of the transition order 3/2(d+1) and the universality of u_c = 2/π independent of d should be detailed with the explicit form of the effective potential and the saddle equation. It is not clear how the d-dependence cancels in u_c while the order depends on d.

    Authors: We will expand the relevant section to include the explicit effective potential V_eff(r; u) = (1/2)∫∫ ρ(r)ρ(r')/|r-r'|^{d-2} dr dr' + ∫ ρ(r) V_ext(r; u) dr, together with the saddle-point integral equation obtained by functional differentiation. The critical value u_c = 2/π is fixed by the local condition that the density vanishes at the origin; because of spherical symmetry this condition reduces to a one-dimensional integral equation whose solution is independent of d. The transition order 3/2(d+1) follows from the square-root edge behavior of the density near the gap-opening point: the excess energy scales as (δu)^{3/2} multiplied by the overall prefactor (k_F R)^{d+1} that originates from the d-dimensional volume element. The revised text will contain the full calculation of both the critical point and the scaling exponent. revision: yes

Circularity Check

0 steps flagged

No circularity; Coulomb-gas saddle-point analysis is an independent large-deviation computation

full rationale

The central result follows from expressing the hole probability as the Fredholm determinant det(1-K_T) of the finite-temperature kernel and then applying the standard Coulomb-gas representation to extract the leading exponential rate function Φ_d(u) via saddle-point minimization of the associated energy functional. This step is not self-definitional: the functional is derived from the kernel (not postulated to reproduce Φ_d), the minimizer ρ* is solved explicitly to reveal the transition at u_c=2/π, and the order 3/2(d+1) is a direct consequence of the gap-opening mechanism in the optimal density. Independent numerical evaluation of the Fredholm determinant for finite R provides external verification of the analytic Φ_d(u), confirming that the derivation does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the validity of the effective Coulomb-gas mapping for the hole probability; no free parameters are introduced and no new entities are postulated.

axioms (1)
  • domain assumption The hole probability of free fermions at finite temperature can be represented exactly as the partition function of an effective Coulomb gas.
    This representation is invoked to obtain the exact scaling function Φ_d(u) via saddle-point analysis.

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