pith. sign in

arxiv: 2510.14433 · v2 · submitted 2025-10-16 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· math-ph· math.MP

The Tracy-Widom distribution at large Dyson index

Alain Comtet , Pierre Le Doussal , Naftali R. Smith This is my paper

Pith reviewed 2026-05-18 06:37 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nnmath-phmath.MP
keywords Tracy-Widom distributionDyson indexlarge deviationsPainlevé IIstochastic Airy operatorAiry point processrandom matrices
0
0 comments X p. Extension

The pith

The Tracy-Widom distribution assumes a large-deviation form with a Painlevé II rate function at large Dyson index.

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

This paper investigates the behavior of the Tracy-Widom distribution for the largest eigenvalue in random matrices when the Dyson index beta becomes very large. It shows that the probability density function takes the form of an exponential of minus beta times a rate function Phi of a, where Phi solves a Painlevé II equation. This large deviation principle applies to rare events where the eigenvalue deviates by an amount of order one, which controls the higher cumulants of the distribution. The derivation uses saddle point approximations on the stochastic Airy operator, with two complementary methods based on optimal fluctuation and weak-noise theory for the associated Riccati process. The results are extended to the statistics of the entire set of edge eigenvalues in the Airy point process.

Core claim

We show that at large beta the Tracy-Widom probability density f_beta(a) takes the large deviation form f_beta(a) ~ e^{-beta Phi(a)}. The rate function Phi(a) is found as the solution to a Painlevé II equation. Explicit expressions are given for the asymptotic behavior of Phi(a) at large arguments and for the cumulants of the distribution up to fourth order. Numerical solutions for Phi(a) are computed and compared to direct evaluations of the Tracy-Widom distribution at finite beta values. These findings are generalized to the full Airy point process, providing large deviation expressions for marginals, joints and gaps.

What carries the argument

Saddle-point approximation to the stochastic Airy operator (or equivalently the Riccati diffusion) that determines the most probable noise configuration for a given ground state energy.

If this is right

  • The variance of typical fluctuations around the mean is of order 1/beta and Gaussian.
  • The cumulants of all orders are determined by derivatives of the rate function Phi(a).
  • Large deviation principles hold for the joint distribution of multiple Airy eigenvalues and for their gaps.
  • Explicit formulas allow computation of the large argument tails of the distribution.

Where Pith is reading between the lines

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

  • The validity of the leading exponential could be checked by including sub-exponential prefactors in the asymptotic analysis.
  • This method may apply to other edge statistics in random matrix ensembles with tunable beta parameter.
  • Numerical solutions of the Painlevé II equation for Phi(a) provide a practical way to estimate rare event probabilities without simulating large matrices.

Load-bearing premise

The saddle-point approximation applied to the stochastic Airy operator or Riccati diffusion remains valid and gives the exact leading exponential for order-one deviations in the limit of large beta.

What would settle it

Numerical sampling of the largest eigenvalue distribution for the Gaussian beta ensemble at beta equal to 50 or higher, checking if log of the density divided by beta approaches the solved Phi(a) for deviations of order one.

Figures

Figures reproduced from arXiv: 2510.14433 by Alain Comtet, Naftali R. Smith, Pierre Le Doussal.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Solid line: The large deviation function [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Solid line: Exact (numerically-obtained) solution [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Solid line: Exact (numerically-obtained) solution [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Solid line: The large-deviation function [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The time-dependent potential [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) is a sketch of the solution w(τ ). Note that it remains close to the parabola 2w 2 + τ = 0. The physical solution defined on the interval [−E,∞[ is shown in the solid line. The physical interpretation of the extended solution v(t) = w(t − E) [plotted in [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
read the original abstract

We study the Tracy-Widom (TW) distribution $f_\beta(a)$ in the limit of large Dyson index $\beta \to +\infty$. This distribution describes the fluctuations of the rescaled largest eigenvalue $a_1$ of the Gaussian (alias Hermite) ensemble (G$\beta$E) of (infinitely) large random matrices. We show that, at large $\beta$, its probability density function takes the large deviation form $f_\beta(a) \sim e^{-\beta \Phi(a)}$. While the typical deviation of $a_1$ around its mean is Gaussian of variance $O(1/\beta)$, this large deviation form describes the probability of rare events with deviation $O(1)$, and governs the behavior of the higher cumulants. We obtain the rate function $\Phi(a)$ as a solution of a Painlev\'{e} II equation. We derive explicit formula for its large argument behavior, and for the lowest cumulants, up to order 4. We compute $\Phi(a)$ numerically for all $a$ and compare with exact numerical computations of the TW distribution at finite $\beta$. These results are obtained by applying saddle-point approximations to an associated problem of energy levels $E=-a$, for a random quantum Hamiltonian defined by the stochastic Airy operator (SAO). We employ two complementary approaches: (i) we use the optimal fluctuation method to find the most likely realization of the noise in the SAO, conditioned on its ground-state energy being $E$ (ii) we apply the weak-noise theory to the representation of the TW distribution in terms of a Ricatti diffusion process associated to the SAO. We extend our results to the full Airy point process $a_1>a_2>\dots$ which describes all edge eigenvalues of the G$\beta$E, and correspond to (minus) the higher energy levels of the SAO, obtaining large deviation forms for the marginal distribution of $a_i$, the joint distributions, and the gap distributions.

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 manuscript studies the Tracy-Widom distribution f_β(a) of the rescaled largest eigenvalue in the Gaussian β-ensemble in the limit β → ∞. It establishes the large-deviation form f_β(a) ∼ exp(−β Φ(a)) for O(1) deviations, obtains the rate function Φ(a) as the solution of a Painlevé II equation, derives its large-argument asymptotics and the first four cumulants, and extends the results to marginals, joint distributions, and gaps of the full Airy point process. The derivations rely on saddle-point analysis of the stochastic Airy operator and weak-noise large-deviation theory applied to the associated Riccati diffusion, with numerical checks against finite-β Tracy-Widom densities.

Significance. If the leading-exponential claim is asymptotically exact, the work supplies an explicit, Painlevé-based description of rare-event probabilities and higher cumulants for the Tracy-Widom law at large β. The two complementary analytic routes and the generalization to the entire edge point process constitute clear advances; the numerical comparisons further support practical utility.

major comments (2)
  1. [Abstract and methodological sections describing the SAO saddle-point and Riccati weak-noise analysis] The central claim that the saddle-point (or weak-noise) evaluation produces the precise leading exponential rate Φ(a) for fixed a as β → ∞ rests on the assumption that the action at the optimal path dominates all other contributions at order β while Gaussian fluctuations around the saddle contribute only sub-exponential prefactors. This assumption is stated in the abstract and the two methodological paragraphs, but the manuscript does not supply explicit remainder estimates or control on the fluctuation operator that would rule out O(1) corrections to the exponent arising from the edge scaling of the stochastic Airy operator.
  2. [Derivation of the rate function (likely §3 or §4)] The derivation that the optimal fluctuation satisfies a Painlevé II equation for Φ(a) is load-bearing for the explicit results on cumulants and large-argument behavior. The boundary conditions imposed on the optimal path (or on the associated Riccati process) and the precise reduction step that yields the Painlevé II form should be displayed with equation numbers so that the reader can verify the absence of additional β-dependent terms.
minor comments (2)
  1. [Numerical results] In the numerical comparison section, state the specific finite values of β employed and the algorithm used to generate the reference Tracy-Widom densities.
  2. [Notation and setup] Clarify the notation for the rescaled eigenvalue a versus the energy level E = −a when moving between the stochastic Airy operator and the Tracy-Widom variable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and the detailed, constructive major comments. These have helped us identify points where the presentation can be clarified and strengthened. We address each comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and methodological sections describing the SAO saddle-point and Riccati weak-noise analysis] The central claim that the saddle-point (or weak-noise) evaluation produces the precise leading exponential rate Φ(a) for fixed a as β → ∞ rests on the assumption that the action at the optimal path dominates all other contributions at order β while Gaussian fluctuations around the saddle contribute only sub-exponential prefactors. This assumption is stated in the abstract and the two methodological paragraphs, but the manuscript does not supply explicit remainder estimates or control on the fluctuation operator that would rule out O(1) corrections to the exponent arising from the edge scaling of the stochastic Airy operator.

    Authors: We agree that the manuscript does not contain explicit remainder estimates or a full spectral analysis of the fluctuation operator around the saddle. The derivation relies on the standard saddle-point/large-deviation principle for the stochastic Airy operator in the weak-noise regime, where the leading exponential is given by the action evaluated at the optimal path and sub-exponential contributions arise from Gaussian fluctuations. This is consistent with the parallel Riccati-diffusion approach and is further supported by the numerical agreement shown in the paper. To address the concern, we will add a short discussion in the methodological sections (near the abstract and the two analysis paragraphs) that explicitly states the assumption, notes the absence of rigorous remainder bounds, and cites analogous results for large-deviation principles of random Schrödinger operators where the same saddle-point structure has been used. We view a complete rigorous control of the fluctuation operator as a worthwhile but technically demanding extension that lies beyond the present scope. revision: partial

  2. Referee: [Derivation of the rate function (likely §3 or §4)] The derivation that the optimal fluctuation satisfies a Painlevé II equation for Φ(a) is load-bearing for the explicit results on cumulants and large-argument behavior. The boundary conditions imposed on the optimal path (or on the associated Riccati process) and the precise reduction step that yields the Painlevé II form should be displayed with equation numbers so that the reader can verify the absence of additional β-dependent terms.

    Authors: We thank the referee for this observation. In the revised manuscript we will expand the derivation (in the section presenting the optimal-fluctuation and Riccati analyses) to include the explicit boundary conditions on the optimal path and on the Riccati process. We will number the successive equations that perform the reduction to the Painlevé II equation for Φ(a) and will add a short paragraph confirming that no additional β-dependent terms survive at the leading large-β order. This will make the derivation self-contained and allow direct verification by the reader. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation of Phi(a) from independent saddle-point on SAO/Riccati

full rationale

The paper defines the stochastic Airy operator (SAO) and its associated Riccati diffusion independently of the target large-deviation rate function Phi(a). It then applies the optimal fluctuation method and weak-noise saddle-point analysis to this external stochastic process to extract Phi(a) as the leading exponential cost for the ground-state energy to equal -a. The resulting Phi(a) is shown to obey a Painlevé II equation, but this is an output of the saddle-point calculation rather than an input; no equation is solved by assuming the final form of Phi(a) or by fitting parameters to the TW distribution itself. No self-citation is load-bearing for the central claim, no quantity is renamed as a prediction after being fitted, and the derivation chain does not reduce to a tautology. The approach is therefore self-contained against the external benchmark of the SAO definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the known representation of the TW law via the stochastic Airy operator and on the applicability of saddle-point methods in the large-beta limit; no additional free parameters or invented entities are introduced.

axioms (1)
  • domain assumption The Tracy-Widom distribution admits an exact representation in terms of the ground-state energy of the stochastic Airy operator.
    This representation is invoked to set up the saddle-point problem whose solution yields Phi(a).

pith-pipeline@v0.9.0 · 5920 in / 1265 out tokens · 34755 ms · 2026-05-18T06:37:41.505001+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

107 extracted references · 107 canonical work pages · 12 internal anchors

  1. [1]

    z(t) does not explode

    Typical fluctuations ofE 21: We have already shown that, atβ≫1, pairs of eigenvalues follow a multivariate Gaussian distribution. It follows that the gap distribution is Gaussian, and all that remains is to calculate its 19 0 1 2 3 4 5 6 7 E21 0 1 2 3 4 5 s 50 100 150 E21 0.50 0.55 0.60 0.65 0.70 s/E3/2 21 FIG. 4. Solid line: The large-deviation functions...

    work page

  2. [2]

    ForE >0 , at timet= 0,U(z,0) is a smooth monotonous function. One therefore expects that a particle which is released from +∞at timet= 0 will rapidly fall in the potential well but, as time goes on, a barrier is formed which prevents the particle to escape from the unstable pointz 1 =− √ t−E

    work page

  3. [3]

    This observation qualitatively accounts for the sharp asymmetry of the tails of the large deviation function discussed in section IV

    ForE <0,U(z, t) exhibits a potential barrier∀t >0, therefore the escape mechanism is similar to a standard activation process with an Arrhenius behaviour. This observation qualitatively accounts for the sharp asymmetry of the tails of the large deviation function discussed in section IV. The transition probabilityP(z 1, t1|z0, t0) of the process is given ...

    work page

  4. [4]

    For smallt, we solve Eq. (110) through a Laurent expansion, and the result is q(t) = 1 t − Et 3 + t2 4 +ct 3 + E 36 t4 +O(t 5) (120) 6 More generally, the Painlev´ e 2 equation may contain an additional additive constant term which leads to additional types of solutions, see e.g. [92], but we do not encounter this term here. 22 wherecis an arbitrary const...

    work page

  5. [5]

    mechanical energy

    Fort→+∞the linearization of Eq. (118) gives the Airy equation, whose solution isv(t)≃kAi(t−E) for some kwhich depends onE. The tail of the Airy functionv(t) = ϕ1(t)√ 2 ∝e − 2 3 (t−E)3/2 can be seen in Fig. 3 fort > E 1 (tis calledxthere). This solution implies thatq(t) satisfies the required boundary condition att→+∞, since q(t) = ˙v(t) v(t) ≃ −(t−E) 1/2....

    work page

  6. [6]

    The second cumulant was obtained by this method in Appendix F of [42]

    Third cumulant In this Appendix we obtain the third and fourth cumulants by direct perturbation theory of the eigenvalues of the SAO. The second cumulant was obtained by this method in Appendix F of [42]. Let us use compact notations and denoteψ (0) i (x) =A i(x) := Ai(x+ζi) Ai′(ζi) . We use here standard quantum mechanics perturbation theory, with explic...

    work page

  7. [7]

    Fourth cumulant To compute the leading contribution to the fourth cumulant we need the next order in perturbation theory. The termδ (3)ai =O(w 3) in (B1) can be written explicitly as (see Wikipedia [100]) δ(3)ai =− 8 β3/2 X i′̸=i X i′′̸=i ⟨AiAi′, w⟩⟨Ai′Ai′′ , w⟩⟨Ai′′ Ai, w⟩ (ζi −ζ i′)(ζi −ζ i′′) + 8 β3/2 A2 i , w X i′̸=i ⟨AiAi′, w⟩2 (ζi −ζ i′)2 .(B13) The...

    work page

  8. [8]

    (78) and (76) of the main text

    Finiteβcorrection to the variance As reported in the main text, the variance ofa i is, in the leading order,O(1/β), with the coefficient given in Eqs. (78) and (76) of the main text. The next order correction to the variance isO(1/β 2). It is the sum of terms of the form w2w2c and of the form w3w c . Using a similar derivation to that of the calculation o...

    work page

  9. [9]

    Wishart,The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A, 32 (1928)

    J. Wishart,The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A, 32 (1928). 35

    work page 1928

  10. [10]

    P. J. Forrester,Log-gases and random matrices, Princeton University Press, Princeton, New Jersey, 2010

    work page 2010

  11. [11]

    M. L. Mehta,Random matrices, Vol. 142 (Elsevier, 2004)

    work page 2004

  12. [12]

    Akemann, J

    G. Akemann, J. Baik, and P. Di Francesco,The Oxford handbook of random matrix theory(Oxford University Press, 2011)

    work page 2011

  13. [13]

    Potters, J

    M. Potters, J. P. Bouchaud,A first course in random matrix theory: for physicists, engineers and data scientists, Cam- bridge University Press (2020)

    work page 2020

  14. [14]

    E. P. Wigner,On the Distribution of the Roots of Certain Symmetric Matrices, Ann. Math67, 325 (1958)

    work page 1958

  15. [15]

    Johansson,Shape fluctuations and random matrices

    K. Johansson,Shape fluctuations and random matrices. Comm. Math. Phys. 209, 437 (2000)

    work page 2000

  16. [16]

    Kriecherbauer and J

    T. Kriecherbauer and J. Krug,A pedestrian’s view on interacting particle systems, KPZ universality and random matrices, J. Phys. A.: Math. Theor.43, 403001 (2010)

    work page 2010

  17. [17]

    S. N. Majumdar, in Complex Systems: Lecture Notes of the Les Houches Summer School: July 2007, Vol. 85, edited by J.-P. Bouchaud, M. Mezard, and J. Dalibard (2007), arXiv:cond-mat/0701193

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  18. [18]

    Marino, S

    R. Marino, S. N. Majumdar, G. Schehr, and P. Vivo,Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices, Phys. Rev. Lett.112, 254101 (2014)

    work page 2014

  19. [19]

    I. P. Castillo,Spectral order statistics of Gaussian random matrices: Large deviations for trapped fermions and associated phase transitions, Phys. Rev. E90, 040102 (2014)

    work page 2014

  20. [20]

    D. S. Dean, P. Le Doussal, S. N. Majumdar and G. Schehr,Noninteracting fermions in a trap and random matrix theory J. Phys. A: Math. Theor.52, 144006 (2019)

    work page 2019

  21. [21]

    C. A. Tracy and H. Widom,Level spacing distributions and the Airy kernel, Commun. Math. Phys.159, 151 (1994), hep-th/9211141

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  22. [22]

    C. A. Tracy and H. Widom,On orthogonal and symplectic matrix ensembles, Commun. Math. Phys.177, 727 (1996), solv-int/9509007

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  23. [23]

    Distribution functions for largest eigenvalues and their applications

    C.A. Tracy and H. Widom,Distribution functions for largest eigenvalues and their applications, Proceedings of the ICM 1, 587 (2002),math-ph/0210034

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  24. [24]

    P. Le Doussal,Dynamic scaling of growing interfaces, Contribution to the volume From Quantum Fields to Spin Glasses: A journey through the contributions of Giorgio Parisi to theoretical Physics, arXiv preprint arXiv:2507.08341, (2025)

    work page arXiv 2025

  25. [25]

    Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals

    K. Takeuchi and M. Sano,Universal fluctuations of growing interfaces: evidence in turbulent liquid crystals, Phys. Rev. Lett.104, 230601 (2010),cond-mat.stat-mech/1001.5121

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  26. [26]

    Pr¨ ahofer and H

    M. Pr¨ ahofer and H. Spohn,Universal distributions for growth processes in 1+1 dimensions and random matrices, Phys. Rev. Lett.84, 4882 (2000)

    work page 2000

  27. [27]

    Dumitriu and A

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

    work page 2002

  28. [28]

    Ramirez, B

    J. Ramirez, B. Rider and B. Virag,Beta ensembles, stochastic Airy spectrum and a diffusion, J. Amer. Math. Soc.24, 919 (2011)

    work page 2011

  29. [29]

    Dumaz, B

    L. Dumaz, B. Vir´ ag,The right tail exponent of the Tracy–Widomβdistribution, Ann. Inst. H. Poincar´ e Probab. Statist. 49, 915 (2013)

    work page 2013

  30. [30]

    Borot, G., and Nadal, C.,Right tail asymptotic expansion of Tracy–Widom beta laws, Random Matrices: Theory and Applications,1, 1250006 (2012)

    work page 2012

  31. [31]

    Borot, B

    G. Borot, B. Eynard, S.N. Majumdar, and C. Nadal,Large deviations of the maximal eigenvalue of random matrices, J. Stat. Mech. P11024 (2011)

    work page 2011

  32. [32]

    J. Baik, R. Buckingham, and J. DiFranco,Asymptotics of Tracy-Widom distributions and the total integral of a Painlev´ e II function, Commun. Math. Phys.280, 463 (2008)

    work page 2008

  33. [33]

    PhD thesis of C´ eline Nadal,https://theses.hal.science/tel-00633266v1/file/VA2_NADAL_CELINE_21062011.pdf

    work page

  34. [34]

    Pr¨ ahofer, H

    M. Pr¨ ahofer, H. Spohn,Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys.108, 1071 (2002)

    work page 2002

  35. [35]

    Johansson,Discrete polynuclear growth and determinantal processes, Comm

    K. Johansson,Discrete polynuclear growth and determinantal processes, Comm. Math. Phys.242, 277 (2003)

    work page 2003

  36. [36]

    Corwin, A

    I. Corwin, A. Hammond,Brownian Gibbs property for Airy line ensembles, Inventiones mathematicae,195, 441 (2014)

    work page 2014

  37. [37]

    Osada, H

    H. Osada, H. Tanemura,Infinite-dimensional stochastic differential equations arising from Airy random point fields, Stoch PDE: Anal Comp13, 770–886 (2025)

    work page 2025

  38. [38]

    Huang, L

    J. Huang, L. Zhang, (2024).A convergence framework for Airy β line ensemble via pole evolution. arXiv preprint arXiv:2411.10586

    work page arXiv 2024

  39. [39]

    Landon,Edge scaling limit of Dyson Brownian motion at equilibrium for generalβ≥1, arXiv preprint arXiv:2009.11176, (2020)

    B. Landon,Edge scaling limit of Dyson Brownian motion at equilibrium for generalβ≥1, arXiv preprint arXiv:2009.11176, (2020)

    work page arXiv 2009

  40. [40]

    Edelman and B

    A. Edelman and B. D. Sutton,From random matrices to stochastic operators, J. Stat. Phys.127, 1121 (2007)

    work page 2007

  41. [41]

    Limits of spiked random matrices I

    Bloemendal A., Virag B.,Limits of spiked random matrices I, Probab. Theory Related Fields156, 795 (2013), arXiv:1011.1877

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  42. [42]

    Operator limits of random matrices

    B. Virag,Operator limits of random matrices, arXiv:1804.06953 (2018), Proceedings of the International Congress of Mathematicians, Seoul 2014. Volume 4, 247-272

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  43. [43]

    Edelman, B

    A. Edelman, B. D. Sutton, and Y. Wang,Random Matrix Theory, Numerical Computation and Applications, Proc. Symp. Appl72(2014)

    work page 2014

  44. [44]

    Sutherland,Beautiful models: 70 years of exactly solved quantum many-body problems, World Scientific Publishing Company (2004), doi:10.1142/5552

    B. Sutherland,Beautiful models: 70 years of exactly solved quantum many-body problems, World Scientific Publishing Company (2004), doi:10.1142/5552

    work page doi:10.1142/5552 2004

  45. [45]

    Calogero,One-dimensional many-body problems with pair interactions whose exact ground-state wave function is of product type, Lett

    F. Calogero,One-dimensional many-body problems with pair interactions whose exact ground-state wave function is of product type, Lett. Nuovo Cimento13, 507 (1975), doi:10.1007“%2FBF02753857?LI=true

    work page 1975

  46. [46]

    N. R. Smith, P. Le Doussal, S. N. Majumdar and G. Schehr,Full counting statistics for interacting trapped fermions, SciPost Phys.11, 110 (2021), doi:10.21468/SciPostPhys.11.6.110 36

    work page doi:10.21468/scipostphys.11.6.110 2021

  47. [47]

    Dumitriu and A

    I. Dumitriu and A. Edelman,Eigenvalues of Hermite and Laguerre ensembles: large beta asymptotics, Annales de l’Institut Henri Poincar´ e. Probabilit´ es et Statistiques41, 1083 (2005)

    work page 2005

  48. [48]

    Jiang and S

    H. Jiang and S. Wang,Moderate deviation principles for eigenvalues ofβ-Hermite andβ-Laguerre ensembles withβ→ ∞, Statistics & Probability Letters118, 50 (2016)

    work page 2016

  49. [49]

    Gorin, V

    V. Gorin, V. Kleptsyn,Universal objects of the infinite beta random matrix theory, Eur. Math. Soc.26, 3429 (2023)

    work page 2023

  50. [50]

    Touzo, P

    L. Touzo, P. Le Doussal,G. Schehr,Fluctuations in the active Dyson Brownian motion and the overdamped Calogero- Moser model, arXiv:2307.14306, Phys. Rev. E,109, 014136 (2024). See Section VII and Appendix F (published version) or Appendix I (arXiv version)

    work page arXiv 2024

  51. [51]

    Edelman, P.-O

    A. Edelman, P.-O. Persson, B. D. Sutton,Low-temperature random matrix theory at the soft edge, J. Math. Phys.55, 063302 (2014)

    work page 2014

  52. [52]

    Andraus, K

    S. Andraus, K. Hermann and M. Voit,Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials, J. Math. Phys.62, 083303 (2021)

    work page 2021

  53. [53]

    N. R. Smith,Full distribution of the ground-state energy of potentials with weak disorder, Phys. Rev. E110, 064129 (2024)

    work page 2024

  54. [54]

    Perret, G

    A. Perret, G. Schehr,Near-extreme eigenvalues and the first gap of Hermitian random matrices. J. Stat. Phys.156, 843 (2014)

    work page 2014

  55. [55]

    S. N. Majumdar and M. Vergassola,Large Deviations of the Maximum Eigenvalue for Wishart and Gaussian Random Matrices, Phys. Rev. Lett.102, 060601 (2009)

    work page 2009

  56. [56]

    S. N. Majumdar, C. Nadal, A. Scardicchio, and P. Vivo,Index Distribution of Gaussian Random Matrices, Phys. Rev. Lett.103, 220603 (2009);How many eigenvalues of a Gaussian random matrix are positive?, Phys. Rev. E83, 041105 (2011)

    work page 2009

  57. [57]

    D. S. Dean and S. N. Majumdar,Large deviations of extreme eigenvalues of random matrices, Phys. Rev. Lett.97, 160201 (2006)

    work page 2006

  58. [58]

    D. S. Dean and S. N. Majumdar.Extreme value statistics of eigenvalues of Gaussian random matrices, Phys. Rev. E77, 041108 (2008)

    work page 2008

  59. [59]

    Ben Arous, A

    G. Ben Arous, A. Dembo, A. Guionnet,Aging of spherical spin glasses, Probab. Theory Relat. Fields120, 1 (2001)

    work page 2001

  60. [60]

    Y. V. Fyodorov, P. Le Doussal,Topology trivialization and large deviations for the minimum in the simplest random optimization. J. Stat. Phys.154, 466 (2014)

    work page 2014

  61. [61]

    Borodin, V

    A. Borodin, V. Gorin,Moments match between the KPZ equation and the Airy point process, SIGMA. Symmetry, Inte- grability and Geometry: Methods and Applications12, 102 (2016)

    work page 2016

  62. [62]

    The KPZ equation and moments of random matrices

    Gorin, V., Sodin, S. (2018).The KPZ equation and moments of random matrices. arXiv preprint arXiv:1801.02574, Journal of Mathematical Physics, Analysis, Geometry14, 286 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  63. [63]

    The Annals of Probability46, 2287 (2018)

    Gorin, V., Shkolnikov, M.,Stochastic Airy semigroup through tridiagonal matrices. The Annals of Probability46, 2287 (2018)

    work page 2018

  64. [64]

    Krajenbrink and P

    A. Krajenbrink and P. Le Doussal,Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations, Europhys. Lett.125, 20009 (2019)

    work page 2019

  65. [65]

    Coulomb-gas electrostatics controls large fluctuations of the KPZ equation

    I. Corwin, P. Ghosal, A. Krajenbrink, P. Le Doussal and L.-C. Tsai,Coulomb-gas electrostatics controls large fluctuations of the KPZ equation, arXiv:1803.05887, Phys. Rev. Lett.121, 060201 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  66. [66]

    Krajenbrink, P

    A. Krajenbrink, P. Le Doussal, S. Prolhac,Systematic time expansion for the Kardar–Parisi–Zhang equation, linear statistics of the GUE at the edge and trapped fermions. Nuclear Physics B936, 239 (2018)

    work page 2018

  67. [67]

    J. A. Ram´ ırez and B. Rider,Diffusion at the random matrix hard edge, Commun. Math. Phys.288, 887 (2009)

    work page 2009

  68. [68]

    Dumaz, Y

    L. Dumaz, Y. Li, B. Valko,Operator level hard-to-soft transition forβ-ensembles, Electron. J. Probab.26, 1 (2021)

    work page 2021

  69. [69]

    Bloemendal and B

    A. Bloemendal and B. Virag,Limits of spiked random matrices II, Ann. Probab.44, 2726 (2016)

    work page 2016

  70. [70]

    Rider, B

    B. Rider, B. Valko,Solvable families of random block tridiagonal matrices, arXiv preprint arXiv:2412.04579. (2024)

    work page internal anchor Pith review arXiv 2024

  71. [71]

    Gorin,, J

    V. Gorin,, J. Xu, L. Zhang,Airy β line ensemble and its Laplace transform. arXiv preprint arXiv:2411.10829. (2024)

    work page arXiv 2024

  72. [72]

    Allez, J

    R. Allez, J. P. Bouchaud, A. Guionnet,Invariant beta ensembles and the Gauss-Wigner crossover, Phys. Rev. Lett.109, 094102 (2012)

    work page 2012

  73. [73]

    Dumaz, C

    L. Dumaz, C. Labb´ e,The stochastic Airy operator at large temperature, Ann. Appl. Probab.32, 4481 (2022)

    work page 2022

  74. [74]

    Bloemendal,Finite Rank Perturbations of Random Matrices and Their Continuum Limits,https://utoronto

    A. Bloemendal,Finite Rank Perturbations of Random Matrices and Their Continuum Limits,https://utoronto. scholaris.ca/items/3023d5e6-04ec-4883-a462-ce00f11bf037, Thesis (Ph.D.)–University of Toronto (Canada) (2011)

    work page 2011

  75. [75]

    B. I. Halperin,Green’s Functions for a Particle in a One-Dimensional Random Potential, Phys. Rev.139, A104 (1965)

    work page 1965

  76. [76]

    Vorov and A.V

    O.K. Vorov and A.V. Vagov,Problem of a quantum particle in a random potential on a line revisited, Phys. Lett. A205, 301 (1995)

    work page 1995

  77. [77]

    F. D. Cunden, P. Facchi, and P. Vivo,A shortcut through the Coulomb gas method for spectral linear statistics on random matrices, J. Phys. A: Math. Theor.49, 135202 (2016)

    work page 2016

  78. [78]

    Stoer and R

    J. Stoer and R. Bulirsch,Introduction to Numerical Analysis(Springer-Verlag, New York, 1980)

    work page 1980

  79. [79]

    Fornberg and J

    B. Fornberg and J. A. C. Weideman,A Computational Exploration of the Second Painlev´ e Equation, Foundations of Computational Mathematics14, 985 (2014);A computational overview of the solution space of the imaginary Painlev´ e II equation, Physica D309, 108 (2015)

    work page 2014

  80. [80]

    SIGMA20, 005 (2024)

    Trogdon, T., Zhang, Y.,Computing the Tracy-Widom Distribution for Arbitrary beta. SIGMA20, 005 (2024)

    work page 2024

Showing first 80 references.