pith. sign in

arxiv: 2605.18198 · v1 · pith:MAIBULGKnew · submitted 2026-05-18 · 🧮 math.PR

Large deviations of crowding in finite β-ensembles

Kartick Adhikari , Sitanath Majumder This is my paper

Pith reviewed 2026-05-20 00:25 UTC · model grok-4.3

classification 🧮 math.PR
keywords large deviationsbeta ensemblescrowdingrandom point processesempirical measurescontraction principle
0
0 comments X

The pith

The fraction of points from a finite β-ensemble that fall inside a fixed bounded region obeys a large deviation principle with speed n² and a good rate function.

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

Finite β-ensembles place n points on the real line or complex plane according to a joint law whose repulsion term is controlled by the parameter β. For a bounded test set U whose boundary meets a mild geometric condition, the paper shows that the random fraction n^{-1} X_n(U) of points lying in U satisfies a large deviation principle: the probability that this fraction deviates from its typical value decays at speed n² times a rate function that is good (lower semicontinuous with compact level sets). On the real line the result follows from the contraction principle applied to the known large deviation principle for the empirical measure; on the complex plane a direct argument is required because contraction does not close the estimate.

Core claim

The sequence of laws of {n^{-1} X_{n,β}^F(U); n ≥ 1} satisfies the large deviation principle with speed n² and a good rate function, where X_{n,β}^F(U) counts the number of points of the finite β-ensemble lying in U.

What carries the argument

The scaled counting functional n^{-1} X_{n,β}^F(U) that maps each point configuration to the empirical fraction inside U; this functional is shown to obey a large deviation principle by contraction on the line and by direct exponential-moment estimates in the plane.

If this is right

  • The probability of any fixed atypical crowding fraction decays exponentially with speed n².
  • The same large deviation statement holds for both real and complex β-ensembles once the boundary regularity of U is satisfied.
  • The good rate function supplies the exponential cost of every possible deviation and therefore identifies the most likely atypical configurations.

Where Pith is reading between the lines

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

  • The direct argument developed for the complex plane may apply to other non-Hermitian point processes whose empirical measures lack a simple contraction map.
  • The speed n² and the associated rate function suggest that similar large-deviation control should hold for smooth linear statistics or for the empirical measure restricted to slightly larger classes of test sets.
  • The result supplies quantitative tail bounds that could be used to study moderate-deviation regimes or to justify numerical sampling of rare crowding events.

Load-bearing premise

The boundary of U must be polar when the ensemble is real and must be a closed 1-rectifiable set of finite one-dimensional Hausdorff measure when the ensemble is complex.

What would settle it

A concrete bounded set U satisfying the stated boundary condition for which the probability P(n^{-1} X_n(U) > a) fails to decay at rate exactly n² for some a away from the typical value, or for which the putative rate function is not lower semicontinuous.

Figures

Figures reproduced from arXiv: 2605.18198 by Kartick Adhikari, Sitanath Majumder.

Figure 1
Figure 1. Figure 1: Dictionary-ordered partitioning of square D at the n = m2 -th stage. Case-II: Next we construct n points when n is not a perfect square. That is, there exists m ∈ N such that m2 < n < (m + 1)2 . There exists 1 ≤ k ≤ 2m such that n = m2 + k. In this case, we choose 0 = x0, x1, . . . , xm < L such that ν(Ri) = 1 √ n , where Ri = [xi−1, xi ] × [0, L] for i = 1, . . . , m [PITH_FULL_IMAGE:figures/full_fig_p02… view at source ↗
Figure 2
Figure 2. Figure 2: Grid partition of D divided into m2 perfect square portions and remaining non-square portions mapped to the upper and right boundaries. Let bi be the side length of the sub-rectangle Ri,n, for i = 1, . . . , m2 . Then, by (27) and the same arguments as before, we have L √ nC2 ≤ bi ≤ L √ n , for i = 1, . . . , m2 . Thus the side lengths of the rectangle Ri,n is O(1/ √ n), for i = 1, . . . , m2 . Case-II. Su… view at source ↗
read the original abstract

We consider finite $\beta$-ensembles $\mathcal X_{n,\beta}^{\mathbb F}$ with $n$ points on $\mathbb F$, where $\mathbb F$ denotes either the real line or the complex plane. Let $U$ be a bounded subset of $ \mathbb F$ such that $\partial U$ (the boundary of $U$) is polar for $\mathbb F=\mathbb R$ and $\partial U$ is a closed $1$--rectifiable set with finite $1$-dimensional Hausdorff measure. Suppose $\mathcal X_{n,\beta}^{\mathbb F}(U)$ denotes the number of points in the region $U$. We show that the sequence of laws of $\{n^{-1}\mathcal X_{n,\beta}^{\mathbb F}(U); n\ge 1\}$ satisfies the large deviation type bound with speed $n^2$ and with a good rate function. For $\mathbb{F} = \mathbb{R}$, this result can be derived using the contraction principle. However, when $\mathbb{F} = \mathbb{C}$, the contraction principle does not yield the desired outcome. Therefore, we adopt a direct approach to establish our results.

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

0 major / 2 minor

Summary. The paper considers finite β-ensembles with n points on the real line or the complex plane. For a bounded set U whose boundary satisfies a polar condition when F=R and is closed 1-rectifiable with finite 1-Hausdorff measure when F=C, it proves that the sequence of laws of the normalized crowding random variable n^{-1} X_{n,β}^F(U) obeys a large-deviation principle with speed n² and a good rate function. The real-line case is obtained by contraction from the known n²-speed LDP for the empirical measure; the complex-plane case requires a direct argument establishing exponential tightness together with matching Laplace upper and lower bounds.

Significance. If the central claims hold, the work supplies a useful extension of large-deviation theory from global empirical measures to local point counts in β-ensembles. The explicit separation of the contraction route from the direct potential-theoretic argument, together with the precise geometric hypotheses on ∂U, clarifies when standard tools suffice and when boundary-layer control is needed. The result is in principle testable by Monte-Carlo sampling of the ensembles and could inform rare-event analysis in log-gases.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'large deviation type bound' is imprecise; the main theorem statement should explicitly declare a large deviation principle (upper and lower bounds with the same good rate function) rather than a 'type bound'.
  2. [Section 4 (direct approach)] The direct proof for F=C invokes uniform estimates on the logarithmic potential near ∂U; a short paragraph recalling the relevant potential-theoretic lemma (with a precise citation) would make the role of the 1-rectifiable finite-Hausdorff-measure hypothesis transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation of minor revision. We are pleased that the separation between the contraction argument on the line and the direct potential-theoretic argument in the plane, together with the precise geometric hypotheses on the boundary, is viewed as clarifying the scope of standard tools.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the standard contraction principle applied to the known n²-speed LDP for the empirical measure of the β-ensemble (real case) together with a direct exponential-tightness plus Laplace-principle argument (complex case). Both routes invoke only classical potential-theoretic estimates and the stated geometric hypotheses on ∂U to guarantee continuity of the map μ ↦ μ(U) or uniform control on the logarithmic potential; no step reduces by definition to a fitted parameter, renames a known result as a new derivation, or depends on a load-bearing self-citation whose content is itself unverified within the paper. The central claim therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of beta-ensembles as determinantal or Pfaffian point processes and on large-deviation theory; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Finite beta-ensembles are well-defined probability measures on configurations of n points with the given repulsion kernel.
    Invoked implicitly when defining X_{n,β}^F(U).
  • standard math Large-deviation theory supplies the contraction principle and the notion of a good rate function.
    Used explicitly for the real-line case and as background for the direct proof.

pith-pipeline@v0.9.0 · 5737 in / 1261 out tokens · 51456 ms · 2026-05-20T00:25:05.574811+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

36 extracted references · 36 canonical work pages · 1 internal anchor

  1. [1]

    G. W. Anderson, A. Guionnet, and O. Zeitouni.An introduction to random matrices. Number

    work page

  2. [2]

    Cambridge university press, 2010

    work page 2010

  3. [3]

    G. B. Arous and A. Guionnet. Large deviations for wigner’s law and voiculescu’s non- commutative entropy.Probability theory and related fields, 108:517–542, 1997

    work page 1997

  4. [4]

    F. Augeri. Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails.Electron. J. Probab., 21:Paper No. 32, 49, 2016

    work page 2016

  5. [5]

    Ben Arous and O

    G. Ben Arous and O. Zeitouni. Large deviations from the circular law.ESAIM: Probability and Statistics, 2:123–134, 1998

    work page 1998

  6. [6]

    Billingsley.Convergence of probability measures

    P. Billingsley.Convergence of probability measures. John Wiley & Sons, 2013

    work page 2013

  7. [7]

    T. Bloom. Large deviation for outlying coordinates inβensembles.Journal of Approximation Theory, 180:1–20, 2014. 26 KARTICK ADHIKARI AND SITANATH MAJUMDER

    work page 2014

  8. [8]

    Bordenave and P

    C. Bordenave and P. Caputo. A large deviation principle for Wigner matrices without Gauss- ian tails.Ann. Probab., 42(6):2454–2496, 2014

    work page 2014

  9. [9]

    Chafa¨ ı, N

    D. Chafa¨ ı, N. Gozlan, and P.-A. Zitt. First-order global asymptotics for confined particles with singular pair repulsion.Ann. Appl. Probab., 24(6):2371–2413, 2014

    work page 2014

  10. [10]

    N. A. Cook, R. Ducatez, and A. Guionnet. Full large deviation principles for the largest eigenvalue of sub-gaussian wigner matrices.arXiv preprint arXiv:2302.14823, 2023

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  11. [11]

    Cram´ er

    H. Cram´ er. Sur un nouveau th´ eoreme-limite de la th´ eorie des probabilit´ es. InCollected Works II, pages 895–913. Springer, 1994

    work page 1994

  12. [12]

    Dembo and O

    A. Dembo and O. Zeitouni.Large deviations techniques and applications, volume 38. Springer Science & Business Media, 2009

    work page 2009

  13. [13]

    Deuschel and D

    J. Deuschel and D. Stroock. Large deviations, acad.Press, Boston, 1989

    work page 1989

  14. [14]

    Deuschel and D

    J.-D. Deuschel and D. W. Stroock.Large deviations, volume 342. American Mathematical Soc., 2001

    work page 2001

  15. [15]

    Dumitriu and A

    I. Dumitriu and A. Edelman. Matrix models for beta ensembles.J. Math. Phys., 43(11):5830– 5847, 2002

    work page 2002

  16. [16]

    F. J. Dyson. Statistical theory of the energy levels of complex systems. i.Journal of Mathe- matical Physics, 3(1):140–156, 1962

    work page 1962

  17. [17]

    Federer.Geometric measure theory

    H. Federer.Geometric measure theory. Springer, 2014

    work page 2014

  18. [18]

    P. J. Forrester.Log-gases and random matrices (LMS-34). Princeton university press, 2010

    work page 2010

  19. [19]

    J. Ginibre. Statistical ensembles of complex, quaternion, and real matrices.Journal of Mathematical Physics, 6(3):440–449, 1965

    work page 1965

  20. [20]

    Guionnet

    A. Guionnet. Large deviations and stochastic calculus for large random matrices.Probab. Surv., 1:72–172, 2004

    work page 2004

  21. [21]

    A. Hardy. A note on large deviations for 2D Coulomb gas with weakly confining potential. Electron. Commun. Probab., 17:no. 19, 12, 2012

    work page 2012

  22. [22]

    Hedenmalm and N

    H. Hedenmalm and N. Makarov. Coulomb gas ensembles and laplacian growth.Proceedings of the London Mathematical Society, 106(4):859–907, 2013

    work page 2013

  23. [23]

    Holcomb and B

    D. Holcomb and B. Valk´ o. Overcrowding asymptotics for the sine(beta) process.Annales De L Institut Henri Poincare-probabilites Et Statistiques, 53:1181–1195, 2017

    work page 2017

  24. [24]

    J. Husson. Large deviations for the largest eigenvalue of matrices with variance profiles. Electronic Journal of Probability, 27:1–44, 2022

    work page 2022

  25. [25]

    Kinderlehrer and G

    D. Kinderlehrer and G. Stampacchia.An introduction to variational inequalities and their applications. SIAM, 2000

    work page 2000

  26. [26]

    Krishnapur

    M. Krishnapur. Overcrowding estimates for zeroes of planar and hyperbolic gaussian analytic functions.Journal of statistical physics, 124(6):1399–1423, 2006

    work page 2006

  27. [27]

    Lacroix-A-Chez-Toine, J

    B. Lacroix-A-Chez-Toine, J. A. M. Garz´ on, C. S. H. Calva, I. P. Castillo, A. Kundu, S. N. Majumdar, and G. Schehr. Intermediate deviation regime for the full eigenvalue statistics in the complex ginibre ensemble.Physical Review E, 100(1):012137, 2019

    work page 2019

  28. [28]

    S. N. Majumdar and M. Vergassola. Large deviations of the maximum eigenvalue for wishart and gaussian random matrices.Physical review letters, 102(6):060601, 2009

    work page 2009

  29. [29]

    B. McKenna. Large deviations for extreme eigenvalues of deformed Wigner random matrices. Electron. J. Probab., 26:Paper No. 34, 37, 2021

    work page 2021

  30. [30]

    G. L. O’brien and W. Vervaat. Capacities, large deviations and loglog laws. InStable Pro- cesses and Related Topics: A Selection of Papers from the Mathematical Sciences Institute Workshop, January 9–13, 1990, pages 43–83. Springer, 1991

    work page 1990

  31. [31]

    A. A. Pukhalskii. On the theory of large deviations.Theory of Probability & Its Applications, 38(3):490–497, 1994

    work page 1994

  32. [32]

    E. B. Saff, V. Totik, et al.Logarithmic potentials with external fields, volume 316. Springer, 1997

    work page 1997

  33. [33]

    Sandier and S

    E. Sandier and S. Serfaty. 2D Coulomb gases and the renormalized energy.Ann. Probab., 43(4):2026–2083, 2015

    work page 2026

  34. [34]

    S. Serfaty. Lectures on coulomb and riesz gases.arXiv preprint arXiv:2407.21194, 2024

    work page arXiv 2024

  35. [35]

    T. Shirai. Ginibre-type point processes and their asymptotic behavior.Journal of the Math- ematical Society of Japan, 67(2):763–787, 2015

    work page 2015

  36. [36]

    S. S. Varadhan. Asymptotic probabilities and differential equations.Communications on Pure and Applied Mathematics, 19(3):261–286, 1966. LARGE DEVIATIONS OF CROWDING IN FINITEβ-ENSEMBLES 27 Department of Mathematics, Indian Institute of Science Education and Research, Bhopal 462066 Email address:kartickmath [at] iiserb.ac.in Department of Mathematics, Ind...

    work page 1966