Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics

Nakano, Fumihiko; Trinh, Khanh Duy

doi:10.1007/s10955-018-2131-9

Cited by 34 publications

(29 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As to be explained in Section 6 below, also the Gaussian fluctuations are of interest in connection with hydrodynamics linearized around thermal equilibrium. The CLT for the Lax matrix is studied in [32] with a somewhat different perspective. Our goal is to have reasonably explicit expressions for the covariance matrix.…”

Section: Toda Chain In Thermal Equilibriummentioning

confidence: 99%

Generalized Gibbs Ensembles of the Classical Toda Chain

Spohn

2019

J Stat Phys

View full text Add to dashboard Cite

Version 3 of this preprint is published online by Journal of Statistical Physics. Unfortunately, on the right hand side of the two equations in (2.27) a factor 2 is missing. To properly correct, factors of 2 dis-and reappear in various formulas throughout Sections 2 and 3. In this version the text, including references, remains untouched, only the factors of 2 are properly taken care of.Abstract. The Toda chain is the prime example of a classical integrable system with strictly local conservation laws. Relying on the Dumitriu-Edelman matrix model, we obtain the generalized free energy of the Toda chain and thereby establish a mapping to the one-dimensional log-gas with an interaction strength of order 1/N. The (deterministic) local density of states of the Lax matrix is identified as the object, which should evolve according to generalized hydrodynamics.It is my great pleasure to dedicate this article to Joel Lebowitz as teacher, as guide to yet unexplored scientific territories, and with gratitude for a lasting friendship. 25.11.2019

show abstract

Section: Toda Chain In Thermal Equilibriummentioning

confidence: 99%

Generalized Gibbs Ensembles of the Classical Toda Chain

Spohn

2019

J Stat Phys

View full text Add to dashboard Cite

show abstract

“…[ABG12, ABMV13,TT20] prove that for all classical ensembles of random matrices (Gaussian/Wigner, Laguerre/Wishart, and Jacobi/MANOVA) there is a different Law of Large Numbers in the high-temperature regime, and the resulting limit shapes non-trivially depend on the γ parameter. A subsequent wave produced many more results in the βN → 2γ asymptotic regime, such as the study of local statistics in [KS09, BP15,Pa18], or of central limit theorems in [NT18,HL21], or of the loop equations in [FM21], or of the spherical integrals in [MP21], or of the 2D systems in [AB19], or connections to Toda chain in [S19], or of dynamic versions in [NTT21]; this is very far from the complete list of results and we refer to the previously mentioned articles for further references.…”

Section: Introduction 1overviewmentioning

confidence: 99%

Matrix addition and the Dunkl transform at high temperature

Benaych-Georges,

Cuenca,

Gorin

2021

Preprint

View full text Add to dashboard Cite

We develop a framework for establishing the Law of Large Numbers for the eigenvalues in the random matrix ensembles as the size of the matrix goes to infinity simultaneously with the beta (inverse temperature) parameter going to zero. Our approach is based on the analysis of the (symmetric) Dunkl transform in this regime. As an application we obtain the LLN for the sums of random matrices as the inverse temperature goes to 0. This results in a one-parameter family of binary operations which interpolates between classical and free convolutions of the probability measures. We also introduce and study a family of deformed cumulants, which linearize this operation.

show abstract

“…Such transition from Gaussian to Wigner distribution is furthermore investigated in [1,12]. The limiting empirical eigenvalue density in the double limit nβ n − → 2γ ≥ 0 is derived as a family of densities with parameter γ ≥ 0.…”

Section: Background and Preliminariesmentioning

confidence: 99%

“…They alluded to the idea that one can achieve Poissonian statistics for β-ensemble using the same techniques as [13,8], at high temperature within the regime nβ − −− → n→∞ +∞. Concerning the bulk statistics, such work has been accomplished in the regime β ∼ n −1 , that is Poisson convergence of the point process n i=1 δ n(λ i −E) with E ∈ (−2, 2) an energy level in the Wigner sea (see [12,15]). This was achieved in [4] by means of correlation functions, which is also our method.…”

Section: Introductionmentioning

confidence: 99%

Poisson statistics at the edge of Gaussian beta-ensemble at high temperature

Pakzad¹

2019

ALEA

View full text Add to dashboard Cite

We study the asymptotic edge statistics of the Gaussian β-ensemble, a collection of n particles, as the inverse temperature β tends to zero as n tends to infinity. In a certain decay regime of β, the associated extreme point process is proved to converge in distribution to a Poisson point process as n → +∞. We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.

show abstract

Gaussian Beta Ensembles at High Temperature: Eigenvalue Fluctuations and Bulk Statistics

Cited by 34 publications

References 15 publications

Generalized Gibbs Ensembles of the Classical Toda Chain

Generalized Gibbs Ensembles of the Classical Toda Chain

Matrix addition and the Dunkl transform at high temperature

Poisson statistics at the edge of Gaussian beta-ensemble at high temperature

Contact Info

Product

Resources

About