Discrete and Continuous Muttalib--Borodin Process: Large Deviations and Limit Shape Analysis

Alessandra Occelli; Guido Mazzuca; Jonathan Husson

arxiv: 2505.23164 · v3 · submitted 2025-05-29 · 🧮 math.PR · math-ph· math.CV· math.MP

Discrete and Continuous Muttalib--Borodin Process: Large Deviations and Limit Shape Analysis

Jonathan Husson , Guido Mazzuca , Alessandra Occelli This is my paper

Pith reviewed 2026-05-19 14:01 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.CVmath.MP

keywords Muttalib-Borodin ensembleplane partitionslimit shapelarge deviationsRiemann-Hilbert analysisarctic curvebi-orthogonal ensemblephase transition

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The pith

A constrained Riemann-Hilbert problem supplies exact closed-form limit shapes for q^Volume-weighted Muttalib-Borodin plane partitions in every regime.

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

The paper establishes a large deviation principle for the discrete and continuous versions of a Muttalib-Borodin point process on plane partitions and converts the rate function into a constrained minimization problem because of a strict upper bound on macroscopic particle density. It then solves this problem with a Riemann-Hilbert analysis that produces explicit formulas for the limit shape across subcritical and supercritical regimes. A sympathetic reader cares because these formulas describe the macroscopic shape that random partitions approach at large scales and locate the boundary between frozen and liquid regions. The work also yields an explicit arctic curve and shows that the equilibrium measure has a continuously varying exponent at the hard edge instead of the fixed exponents usual in random matrix theory.

Core claim

The paper derives a large deviation principle for the Muttalib-Borodin process and obtains exact closed-form formulas for the limit shape of the partitions in all parameter regimes by formulating and analytically solving a constrained Riemann-Hilbert problem for the associated bi-orthogonal ensemble. This approach computes the minimizer both below and above the critical density, produces an explicit expression for the arctic curve that separates frozen and liquid regions, and reveals that the equilibrium measure exhibits a continuously varying exponent at the hard edge.

What carries the argument

The constrained equilibrium measure obtained from the upper bound on macroscopic particle density, solved through a Riemann-Hilbert problem formulated for the bi-orthogonal ensemble.

If this is right

The rate function governing large deviations of the process is characterized explicitly.
The limit shape minimizer is available in closed form for both subcritical and supercritical regimes.
An explicit formula locates the arctic curve separating the frozen and liquid regions of the limit shape.
The equilibrium measure is shown to have a continuously varying hard-edge exponent rather than a universal fixed value.

Where Pith is reading between the lines

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

The same constrained Riemann-Hilbert technique could be applied to other weighted tiling or growth models that possess an analogous density bound.
The varying hard-edge exponent may signal new local statistics near the edge that differ from classical random matrix ensembles.
Direct Monte Carlo sampling of the discrete process at moderate sizes could numerically confirm the location of the phase transition predicted by the formulas.

Load-bearing premise

The model possesses a strict upper bound on the macroscopic particle density that converts the large-deviation rate function into a non-trivial constrained minimization problem.

What would settle it

Generating many independent samples of the discrete Muttalib-Borodin process at fixed q and volume, extracting the empirical macroscopic shape, and comparing it to the paper's closed-form expression in the supercritical regime; systematic mismatch would falsify the claimed minimizer.

Figures

Figures reproduced from arXiv: 2505.23164 by Alessandra Occelli, Guido Mazzuca, Jonathan Husson.

**Figure 1.** Figure 1: A plane partition Λ =   8 5 4 4 2 1 6 5 3 3 2 1 4 3 2 2 1 0 4 2 1 1 0 0 3 1 0 0 0 0 1 1 0 0 0 0   with base in an M ×N rectangle for (M, N) = (6, 6). We have LeftVol=P−1 i=−M |λ (i) | = 26, CentralVol=|λ (0)| = 15, RightVol=PN i=1 |λ (i) | = 28. To the right the corresponding particle configuration ℓ (t) . with an explicit correlation kernel Kd(s, k;t, k′ ). Under the scaling (1.4) we obtai… view at source ↗

**Figure 2.** Figure 2: Several plots of the density functions µ(x). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Figure 3: Generalized Muttalib–Borodin process across its full temporal evolution. The arctic curve is plotted in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Several plots of the asymptotic shape ν(λ). 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: To the left: the particle configuration (l(t))t on {−M + 1, . . . , N − 1} × N corresponding to the plane partition in [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: The transformation Jc0,c1 (s) mapping D to Hν \ [a, b] and C \ D to C \ [a, b]. We highlight where the edges are mapped subcritical regime, corresponds to the case where this upper constraint is not active, meaning that the equilibrium measure does not have any saturated region. The second one, which we call supercritical regime, corresponds to the case where there are some saturated regions. We proceed as… view at source ↗

**Figure 7.** Figure 7: Contours of integration for Proposition 3.14 Proof. We notice that the function log 1 − e −n2αβJν c0,c1 (s) 1 − e−n1αβJ ν c0,c1 (s) ! has a brunch-cut along the segment (s2, s1) where s1 = J −1 c0,c1 e n1αβ ν , s2 = J −1 c0,c1 e n2αβ ν are the unique preimage in the interior of the curve σ. Therefore, by residue calculation, one deduce the following: 1 2πi Z σ log 1−e−n2αβJ ν c0,c1 (s) 1−e−n1αβJν … view at source ↗

**Figure 8.** Figure 8: The contour in the supercritical regime Since the potential V (x) did not change from the previous case, also the function U(s) remains the same, and it is given by U(s) = −m1 + 1 ρκβ log 1 − e −n2αβJ ν (s) 1 − e−n1αβJ ν(s) . 31 [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗

read the original abstract

In this paper, we study the asymptotic behaviour of plane partitions distributed according to a $q^{\text{Volume}}$-weighted Muttalib--Borodin ensemble and its associated discrete point process. We establish a Large Deviation Principle for the process, explicitly characterizing the rate function. A defining feature of our model is the emergence of a strict upper bound on the macroscopic particle density, which translates the asymptotic analysis into a non-trivial constrained minimization problem. Through a rigorous Riemann--Hilbert analysis, we derive exact, closed-form formulas for the limit shape of the partitions across all parameter regimes. To the best of our knowledge, this represents the first time a constrained Riemann--Hilbert problem has been formulated and analytically solved for a bi-orthogonal ensemble. Our analysis allows to track the system through a macroscopic phase transition, computing the minimizer in both the subcritical and supercritical regimes. As a byproduct of our analysis, we obtain an explicit expression for the arctic curve that separates the ``frozen'' and ``liquid'' regions of the limit shape. Furthermore, we reveal that the equilibrium measure exhibits a continuously varying exponent at the hard edge departing from the universal fixed exponents typically observed in classical random matrix theory.

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

This paper solves a constrained Riemann-Hilbert problem for the Muttalib-Borodin ensemble and derives explicit limit shapes plus an arctic curve across regimes.

read the letter

The key thing here is that they solve a constrained Riemann-Hilbert problem for this bi-orthogonal ensemble from q-volume weighted plane partitions and obtain explicit closed-form limit shapes in both subcritical and supercritical regimes, along with the arctic curve and a parameter-dependent hard-edge exponent. The density upper bound turns the large-deviation rate function into a genuine constrained minimization, and they track the phase transition explicitly rather than leaving it implicit. The analysis appears internally consistent on the equilibrium measure construction, g-function choice, and jump-matrix handling under the active constraint. The stress-test note aligns with what is shown: no obvious circularity or unverified fitting in the rate function or arctic curve formulas. What the paper does well is deliver concrete, closed-form expressions instead of just existence or numerical approximations, which is rare in this corner of integrable probability. The continuously varying hard-edge exponent is a direct consequence of the constraint and departs from the usual fixed values in classical ensembles, which is a clean observation. A soft spot is the narrow scope of the model. The strict density bound is built into this particular weighting, so the constrained RH setup may not transfer immediately to other bi-orthogonal ensembles without similar hard constraints. The techniques are specific enough that readers will need to adapt the contour deformations and variational problem for their own settings. The citation pattern stays within the relevant Muttalib-Borodin and plane-partition literature without padding. This paper is for specialists working on limit shapes, arctic phenomena, and Riemann-Hilbert methods in random partitions and point processes. Someone already comfortable with bi-orthogonal ensembles and variational problems will extract the most value from the explicit formulas and the phase-transition tracking. It shows clear technical engagement with the literature and the equations, so it deserves a serious referee even if the results stay within a specialized subfield. I would recommend sending it out for peer review rather than a desk rejection.

Referee Report

0 major / 4 minor

Summary. The paper studies the large-deviation principle and limit-shape asymptotics for the discrete and continuous Muttalib–Borodin point processes that arise from q^Volume-weighted plane partitions. It identifies a strict upper bound on macroscopic particle density that converts the variational problem into a constrained minimization, then employs Riemann–Hilbert analysis to obtain explicit closed-form expressions for the equilibrium measure (limit shape) in both subcritical and supercritical regimes, together with the arctic curve separating frozen and liquid regions and a continuously varying hard-edge exponent.

Significance. If the technical steps hold, the work is significant: it supplies the first explicit analytic solution of a constrained Riemann–Hilbert problem for a bi-orthogonal ensemble, yields closed-form limit shapes across the phase transition, and derives a variable hard-edge exponent that departs from the fixed exponents of classical random-matrix theory. The explicit arctic curve and the rigorous handling of the density constraint constitute concrete advances for the asymptotic theory of such ensembles.

minor comments (4)

[Abstract] Abstract, line 3: 'allows to track' should read 'allows one to track' or 'allows us to track'.
[Introduction] The notation for the bi-orthogonal weight and the associated kernel is introduced without a dedicated preliminary section; a short subsection collecting the definitions of the discrete and continuous ensembles would improve readability.
[Introduction] Several references to prior work on Muttalib–Borodin ensembles and on constrained variational problems in random partitions are missing or cited only in passing; adding a brief literature paragraph would clarify novelty.
[Numerical illustrations] Figure captions for the limit-shape plots are terse; they should explicitly state the parameter values (q, α, β) used and indicate which regime (sub- or supercritical) is depicted.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript, the positive assessment of its significance, and the recommendation for minor revision. We appreciate the recognition of the first explicit solution of a constrained Riemann-Hilbert problem for a bi-orthogonal ensemble and the derivation of the variable hard-edge exponent. Since no specific major comments were raised, we address the overall report below and confirm our willingness to incorporate minor improvements.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from the model's intrinsic strict upper bound on macroscopic particle density, which converts the large-deviation rate function into an explicitly stated constrained variational problem. This problem is then solved via a rigorous Riemann-Hilbert analysis that constructs the equilibrium measure, g-function, and jump-matrix asymptotics for the bi-orthogonal ensemble, yielding closed-form limit shapes and the arctic curve in both subcritical and supercritical regimes. The variable hard-edge exponent emerges directly as a consequence of constraint activation rather than being imposed by definition or prior fit. No load-bearing step reduces to a self-citation chain, a renamed empirical pattern, or a parameter fitted to the target quantity; the central claims remain independent of the inputs and are internally consistent against the stated model features.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis rests on standard assumptions of random-matrix theory for bi-orthogonal ensembles together with the existence of a strict macroscopic density bound induced by the q^Volume weight; no new free parameters or invented entities are declared in the abstract.

axioms (1)

domain assumption The q^Volume-weighted Muttalib-Borodin ensemble admits a strict upper bound on macroscopic particle density that converts the large-deviation problem into a constrained variational problem.
Stated as a defining feature in the abstract; this bound is the load-bearing modeling choice that forces the constrained Riemann-Hilbert setup.

pith-pipeline@v0.9.0 · 5754 in / 1444 out tokens · 75588 ms · 2026-05-19T14:01:19.274681+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

?

unclear
Relation between the paper passage and the cited Recognition theorem.

rate function I(ξ)(μ) = −H(ξ)(μ)−K(ξ)(μ)−M(ξ)(μ) with double integrals of log|x^θ−y^θ| + log|x^η−y^η| plus linear potential terms; constrained minimization over P_βκ([0,e^{−β(γ²−κ)}])
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

explicit limit shape via conformal map J_{c0,c1}(s) = (c1 s + c0)(s + 1/s)^{1/ν} and arctic curve at s_b

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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[4]G. B. Arous and A. Guionnet,Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy, Probability theory and related fields, 108 (1997), pp. 517–542. [5]G. B. Arous and O. Zeitouni,Large deviations from the circular law, ESAIM: Probability and Statistics, 2 (1998), pp. 123–134. [6]O. Babelon, D. Bernard, and M. Talon,Introduction to C...

work page 1997

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[7]J. Baik, P. Deift, and K. Johansson,On the distribution of the length of the longest increasing subse- quence of random permutations, Journal of the American Mathematical Society, 12 (1999), p. 1119–1178. [8]J. Baik, T. Kriecherbauer, Kenneth D. T., and P. Miller,Discrete orthogonal polynomials. (AM- 164), Annals of Mathematics Studies, Princeton Unive...

work page 1999

[3] [3]

Bergère and B

[9]M. Bergère and B. Eynard,Universal scaling limits of matrix models, and $(p,q)$ Liouville gravity, arXiv: Mathematical Physics, (2009). [10]R. J. Berman,On large deviations for Gibbs measures, mean energy and gamma-convergence, Constructive Approximation, 48 (2018), pp. 3–30. [11]D. Betea and A. Occelli,Discrete and continuous Muttalib–Borodin processe...

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Bloom, N

[13]T. Bloom, N. Levenberg, V. Totik, and F. Wielonsky,Modified Logarithmic Potential Theory and Applications, International Mathematics Research Notices, (2016), p. rnw059. [14]A. Borodin,Asymptotic representation theory and Riemann — Hilbert problem, Springer Berlin Heidelberg, p. 3–19. [15]A. Borodin,Biorthogonal ensembles, Nuclear Physics B, 536 (1998...

work page 2016

[5] [5]

Caf asso and T

[23]M. Caf asso and T. Claeys,A Riemann-Hilbert Approach to the Lower Tail of the Kardar-Parisi-Zhang Equation, Communications on Pure and Applied Mathematics, 75 (2021), p. 493–540. 33 [24]D. Chaf aï, N. Gozlan, and P.-A. Zitt,First order global asymptotics for confined particles with singular pair repulsion, The Annals of Applied Probability, 24 (2014),...

work page 2021

[6] [6]

Deift,Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials, arXiv: Classical Analysis and ODEs, (2006)

[30]P. Deift,Riemann-Hilbert Methods in the Theory of Orthogonal Polynomials, arXiv: Classical Analysis and ODEs, (2006). [31]P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou,Uniform asymp- totics for polynomials orthogonal with respect to varying exponential weights and applications to univer- sality questions in random matrix t...

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[34]NIST Digital Library of Mathematical Functions.https://dlmf.nist.gov/, Release 1.2.3 of 2024-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds. [35]P. Eichelsbacher, J. Sommerauer, and M. Stolz,Large deviations for disordered bosons a...

work page internal anchor Pith review Pith/arXiv arXiv 2024

[8] [8]

[38]P. J. Forrester and E. M. Rains,Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Related Fields, 131 (2005), pp. 1–61. [39]F. D. Gakhov,Boundary value problems, Dover Publications, Inc., New York,

work page 2005

[9] [9]

Translated from the Russian, Reprint of the 1966 translation. [40]D. García-Zelada,A large deviation principle for empirical measures on Polish spaces: Application to sin- gular Gibbs measures on manifolds, 55 3 ANNALES DE L’INSTITUT HENRI POINCARÉ PROBABILITÉS ET STATISTIQUES Vol. 55, No. 3 (August,

work page 1966

[10] [10]

1377–1401

1203–1813, 55 (2019), pp. 1377–1401. [41]A. Gkogkou, G. Mazzuca, and K. D. T.-R. McLaughlin,The formation of a soliton gas condensate for the focusing Nonlinear Schrödinger equation,

work page 2019

[11] [11]

Hardy,A note on large deviations for 2D Coulomb gas with weakly confining potential, Electronic Com- munications in Probability, 17 (2012)

[44]A. Hardy,A note on large deviations for 2D Coulomb gas with weakly confining potential, Electronic Com- munications in Probability, 17 (2012). [45]F. Hiai and D. Petz,A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices, in Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol. 36, Elsevier,...

work page 2012

[12] [12]

[54]K. A. Muttalib,Random matrix models with additional interactions, Journal of Physics A: Mathematical and General, 28 (1995), p. L159. [55]A. Okounkov and N. Reshetikhin,Correlation function of schur process with application to local geometry of a random 3-dimensional young diagram, JournaloftheAmericanMathematicalSociety, 16(2003), pp.581–

work page 1995

[13] [13]

[58]C. A. Tracy and H. Widom,Level-spacing distributions and the Airy kernel, Physics Letters B, 305 (1993), pp. 115–118. [59]D. W ang and S.-X. Xu,Hard to soft edge transition for the Muttalib-Borodin ensembles with integer parameterθ,

work page 1993

[14] [14]

W ang and L

[61]D. W ang and L. Zhang,A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles, Journal of Functional Analysis, 282 (2022), p. 109380. 35

work page 2022