pith. sign in

arxiv: 2605.03647 · v1 · submitted 2026-05-05 · 🧮 math.PR · math.CO

On the partition function of a class of Mallows model

Raghavendra Tripathi This is my paper

Pith reviewed 2026-05-07 14:25 UTC · model grok-4.3

classification 🧮 math.PR math.CO
keywords Mallows modelpartition functionFredholm determinantoptimal transportpermutationsasymptoticsGibbs measures
0
0 comments X

The pith

The partition function of a class of Mallows models converges to a Fredholm determinant after exponential rescaling.

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

This paper gives a complete proof that the partition function L_n of the Gibbs measure on permutations defined by a twice differentiable cost c satisfies lim e^{-n Γ_0} L_n equals the Fredholm determinant of an integral operator from entropy-regularized optimal transport. Mukherjee previously established the leading exponential rate -Γ_0 for (1/n) log L_n. Pal had conjectured the exact constant factor and given a partial proof; this work finishes it. A reader would care because it supplies the missing multiplicative constant in the asymptotic count of permutations under this probability model, which arises in ranking statistics and assignment problems.

Core claim

The author establishes Pal's conjecture by showing that for twice continuously differentiable c, the limit lim_{n→∞} e^{-n Γ_0} L_n is equal to the Fredholm determinant of the integral operator associated with the entropy-regularized optimal transport problem.

What carries the argument

Fredholm determinant of the integral operator associated with the entropy-regularized optimal transport problem

If this is right

  • If the claim holds, then L_n is asymptotically equivalent to the determinant times e^{n Γ_0}.
  • The connection between the Mallows model and entropy-regularized transport is made rigorous at the level of the partition function.
  • No further regularity on the operator is needed beyond the twice differentiability of c.
  • Exact asymptotics become available for this class of models on the symmetric group.

Where Pith is reading between the lines

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

  • The result suggests that similar Fredholm determinants may appear in asymptotics of other discrete models with continuous relaxations.
  • Practical computation of the determinant could approximate L_n for large n in applications.
  • Extensions might include higher-order terms in the asymptotic expansion or other cost functions.

Load-bearing premise

The twice continuous differentiability of the cost function c is sufficient to complete the proof of the limit without additional conditions.

What would settle it

Direct computation of lim e^{-n Γ_0} L_n for increasing n using Monte Carlo or exact methods for a chosen c, compared against numerical approximation of the Fredholm determinant.

read the original abstract

Let $\Sym{n}$ denote the set of all permutations on $n$ labels. Let $c:[0, 1]^2\to [0, \infty)$ be a twice continuously differentiable function. A subfamily of the Mallows model is the Gibbs probability measures on $\Sym{n}$ such that $\mathbb{P}(X=\sigma)=L_n^{-1} \prod_{i=1}^{n}\exp(-c(i/n, \sigma(i)/n))$. Mukherjee [Ann. Stat., Vol. 44(2), pp 853--875 (2016)] computed the limit of the log partition function and showed that $\lim_{n\to \infty}\frac{1}{n}\log L_n=-\Gamma_0$ where $\Gamma_0$ is the optimal cost associated with an entropy regularized optimal transport problem. In the KRP Memorial Volume of the Indian Journal of Pure and Applied Math, Pal conjectured an exact value for the limit $\lim_{n\to \infty} e^{-n\Gamma_0}L_n$ in terms of the Fredholm determinant of an integral operator and provided a partial proof. We give a complete proof of Pal's conjecture.

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 claims to provide a complete proof of Pal's conjecture: for a twice continuously differentiable cost c:[0,1]^2 → [0,∞), the partition function L_n of the associated Gibbs measure on S_n satisfies lim_{n→∞} e^{-n Γ_0} L_n = det(I+K), where Γ_0 is the entropy-regularized OT cost from Mukherjee's theorem and K is the integral operator on L^2[0,1] whose kernel is constructed from the optimal plan (or its derivatives) for that OT problem.

Significance. If the proof holds, the result supplies the precise multiplicative constant in the asymptotic expansion of the Mallows partition function, completing the large-deviation analysis of Mukherjee and the partial argument of Pal. The explicit identification with a Fredholm determinant from entropy-regularized OT is a concrete analytic advance that may permit further exact asymptotics or extensions to other permutation models.

major comments (2)
  1. [§3] §3 (completion of Pal's argument): the claim that the integral operator K is trace-class (hence that the Fredholm determinant is defined without further conditions) rests only on the C^2 regularity of c. A C^2 kernel yields a Hilbert-Schmidt operator, but trace-class membership requires ∑|λ_i| < ∞, which does not follow automatically. The manuscript must supply an explicit verification—either by exploiting the special structure of the OT plan or by deriving a stronger decay estimate on the eigenvalues—otherwise the identification with det(I+K) remains formally incomplete.
  2. [Definition of K] Definition of K (immediately after the statement of the main theorem): the kernel is described as 'associated with' the entropy-regularized OT problem, but the precise formula (involving the optimal density or its derivatives) is not written out. Without this explicit expression it is impossible to check the trace-class property or to confirm that the operator is independent of auxiliary choices in the OT formulation.
minor comments (2)
  1. [Introduction] The sign convention for the Fredholm determinant (I+K versus I-K) should be stated explicitly and matched to the standard literature on regularized OT.
  2. [§2] A short remark on the domain of the operator (L^2[0,1] with Lebesgue measure) and the precise normalization of the kernel would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments highlight two points where additional explicit details will strengthen the manuscript. We address each below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (completion of Pal's argument): the claim that the integral operator K is trace-class (hence that the Fredholm determinant is defined without further conditions) rests only on the C^2 regularity of c. A C^2 kernel yields a Hilbert-Schmidt operator, but trace-class membership requires ∑|λ_i| < ∞, which does not follow automatically. The manuscript must supply an explicit verification—either by exploiting the special structure of the OT plan or by deriving a stronger decay estimate on the eigenvalues—otherwise the identification with det(I+K) remains formally incomplete.

    Authors: We agree that C^2 regularity of the cost alone yields a Hilbert-Schmidt operator and that an explicit verification of trace-class membership is required. The kernel K is constructed from the optimal entropy-regularized transport plan, whose density and derivatives inherit infinite smoothness from the C^2 cost and the strictly convex entropy term. This structure produces a kernel whose eigenvalues decay exponentially, which we will verify in a new lemma in §3 by combining the analyticity of the optimal plan with standard trace-class criteria for integral operators on L^2[0,1]. revision: yes

  2. Referee: [Definition of K] Definition of K (immediately after the statement of the main theorem): the kernel is described as 'associated with' the entropy-regularized OT problem, but the precise formula (involving the optimal density or its derivatives) is not written out. Without this explicit expression it is impossible to check the trace-class property or to confirm that the operator is independent of auxiliary choices in the OT formulation.

    Authors: We accept that the current description is insufficiently explicit. Immediately after the main theorem we will insert the precise formula for the kernel K(x,y) in terms of the optimal density ρ^* and its first derivatives arising from the entropy-regularized OT problem (as obtained from the Euler-Lagrange equation for the dual potentials). This explicit expression makes the independence from auxiliary choices manifest and supplies the starting point for the trace-class argument. revision: yes

Circularity Check

0 steps flagged

Derivation completes Pal's conjecture via external analytic tools without self-referential reduction

full rationale

The paper's central result is a completion of Pal's partial proof for the exact limit of the normalized partition function as a Fredholm determinant, building directly on Mukherjee's independently established limit for (1/n) log L_n = -Γ_0. No step in the provided abstract or described chain defines the target determinant or the limit value in terms of quantities fitted or constructed inside this manuscript; the argument invokes standard properties of entropy-regularized OT and C^2 regularity to close the operator analysis left open by Pal. This is a standard mathematical completion relying on external results rather than any self-definitional, fitted-input, or self-citation load-bearing reduction. The derivation chain therefore remains self-contained against the cited benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard results from analysis, optimal transport, and Fredholm theory with no new free parameters or invented entities; the cost function c is an arbitrary twice-differentiable input.

axioms (2)
  • domain assumption The entropy-regularized optimal transport problem admits a unique optimal plan whose cost defines Γ_0
    Invoked to identify the leading exponential rate
  • standard math The integral operator arising from the model is trace-class so that its Fredholm determinant is well-defined and finite
    Required for the conjectured limit expression to make sense

pith-pipeline@v0.9.0 · 5505 in / 1301 out tokens · 64516 ms · 2026-05-07T14:25:47.411592+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

  1. [1]

    Yeganeh Alimohammadi and Kiana Asgari, Mallows model with learned distance metrics: Sampling and maximum likelihood estimation, arXiv:2507.08108, 2025

    work page arXiv 2025

  2. [2]

    8, 3013--3029

    Itai Benjamini, Noam Berger, Christopher Hoffman, and Elchanan Mossel, Mixing times of the biased card shuffling and the asymmetric exclusion process, Transactions of the American Mathematical Society 357 (2005), no. 8, 3013--3029

    work page 2005

  3. [3]

    Csisz\'ar, I -divergence geometry of probability distributions and minimization problems , Ann

    I. Csisz\'ar, I -divergence geometry of probability distributions and minimization problems , Ann. Probability 3 (1975), 146--158. 365798

    work page 1975

  4. [4]

    11, Institute of Mathematical Statistics, Hayward, CA, 1988

    Persi Diaconis, Group representations in probability and statistics, Institute of Mathematical Statistics Lecture Notes---Monograph Series, vol. 11, Institute of Mathematical Statistics, Hayward, CA, 1988. 964069

    work page 1988

  5. [5]

    1, 157--190

    Persi Diaconis and Arun Ram, Analysis of systematic scan metropolis algorithms using iwahori-hecke algebra techniques., Michigan Mathematical Journal 48 (2000), no. 1, 157--190

    work page 2000

  6. [6]

    Christian L \'e onard, From the S chr\"odinger problem to the M onge- K antorovich problem , J. Funct. Anal. 262 (2012), no. 4, 1879--1920. 2873864

    work page 2012

  7. [7]

    thesis, University of Washington, Seattle, 2017

    Avi William Levy, Novel uses of the mallows model in coloring and matching, Ph.D. thesis, University of Washington, Seattle, 2017

    work page 2017

  8. [8]

    C. L. Mallows, Non-null ranking models. I , Biometrika 44 (1957), no. 1/2, 114--130

    work page 1957

  9. [9]

    2, 404--414

    Peter McCullagh, An asymptotic approximation for the permanent of a doubly stochastic matrix, Journal of Statistical Computation and Simulation 84 (2014), no. 2, 404--414

    work page 2014

  10. [10]

    Sumit Mukherjee, Estimation in exponential families on permutations, Ann. Statist. 44 (2016), no. 2, 853--875. 3476619

    work page 2016

  11. [11]

    Soumik Pal, Limiting partition function for the mallows model: a conjecture and partial evidence, To appear in Indian Journal of Pure and Applied Mathematics (2024)

    work page 2024

  12. [12]

    uschendorf and W. Thomsen, Note on the S chr\

    L. R\"uschendorf and W. Thomsen, Note on the S chr\"odinger equation and I -projections , Statist. Probab. Lett. 17 (1993), no. 5, 369--375. 1237783

    work page 1993

  13. [13]

    120, American Mathematical Society, Providence, RI, 2005

    Barry Simon, Trace ideals and their applications, second ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. 2154153

    work page 2005

  14. [14]

    58, American Mathematical Soc., 2021

    C \'e dric Villani, Topics in optimal transportation, vol. 58, American Mathematical Soc., 2021

    work page 2021

  15. [15]

    201, Cambridge University Press, Cambridge, 2022

    Jan van Neerven, Functional analysis, Cambridge Studies in Advanced Mathematics, vol. 201, Cambridge University Press, Cambridge, 2022. 4425670

    work page 2022