Rigidity and Edge Universality of Discrete <i>β</i>‐Ensembles

Guionnet, Alice; Huang, Jiaoyang

doi:10.1002/cpa.21818

Cited by 21 publications

(15 citation statements)

References 102 publications

(173 reference statements)

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“…The main contribution of [15] is that it establishes general conditions on the potential V (x) that lead to the asymptotic Gaussianity of (1.4). Similarly to the continuous case, discrete loop equations have become a valuable tool to study not only global fluctuations [15] but also edge universality for discrete β-ensembles [33].…”

Section: )mentioning

confidence: 99%

“…They manage to obtain results about the global fluctuations of these particle systems and their analysis is based on appropriate discrete versions of the Schwinger-Dyson equations, which they also call the Nekrasov's equations. More recently, in [33] the same Nekrasov's equations were used to prove rigidity and edge universality for the models in [15].…”

Section: 1mentioning

confidence: 99%

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Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Dimitrov

Knizel

2019

Journal of Functional Analysis

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We study a general class of log-gas ensembles on (shifted) quadratic lattices. We prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a q-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field.Our approach is based on a q-analogue of the Schwinger-Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to quadratic lattices.

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Section: )mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Dimitrov

Knizel

2019

Journal of Functional Analysis

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“…In different directions of generalization, sparse matrices [ 1 , 32 , 47 , 56 ], adjacency matrices of regular graphs [ 14 ] and band matrices [ 19 , 20 , 66 ] have also been considered. In parallel developments bulk and edge universal statistics have been proven for invariant -ensembles [ 12 , 15 , 17 , 18 , 29 , 30 , 52 , 61 , 62 , 64 , 65 , 73 ] and even for their discrete analogues [ 13 , 16 , 41 , 48 ] but often with very different methods.…”

Section: Introductionmentioning

confidence: 99%

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

Erdős

Krüger

Schröder

2020

Commun. Math. Phys.

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For complex Wigner-type matrices, i.e. Hermitian random matrices with independent, not necessarily identically distributed entries above the diagonal, we show that at any cusp singularity of the limiting eigenvalue distribution the local eigenvalue statistics are universal and form a Pearcey process. Since the density of states typically exhibits only square root or cubic root cusp singularities, our work complements previous results on the bulk and edge universality and it thus completes the resolution of the Wigner-Dyson-Mehta universality conjecture for the last remaining universality type in the complex Hermitian class. Our analysis holds not only for exact cusps, but approximate cusps as well, where an extended Pearcey process emerges. As a main technical ingredient we prove an optimal local law at the cusp for both symmetry classes. This result is also the key input in the companion paper [ ] where the cusp universality for real symmetric Wigner-type matrices is proven. C

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“…Since their introduction loop equations have also been used to prove local universality for random matrices [BEY14,BFG15]. In the discrete setup loop equations have been used to study global fluctuations in [BGG17] and edge fluctuations in [GH19] for discrete β-ensembles. It is our strong hope that the multi-level loop equations we derive in the present paper can also be used to study the global fluctuations of continuous and discrete β-corners processes.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Multi-level loop equations for $β$-corners processes

Dimitrov¹,

Knizel²

2021

Preprint

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The goal of the paper is to introduce a new set of tools for the study of discrete and continuous β-corners processes. In the continuous setting, our work provides a multi-level extension of the loop equations (also called Schwinger-Dyson equations) for β-log gases obtained by Borot and Guionnet in (Commun. Math. Phys. 317, 447-483, 2013). In the discrete setting, our work provides a multi-level extension of the loop equations (also called Nekrasov equations) for discrete β-ensembles obtained by Borodin, Gorin and Guionnet in (Publications mathématiques de l 'IHÉS 125, 1-78, 2017). Contents 1. Introduction and main result 1 2. General setup and the equations 6 3. Application of Nekrasov equations 28 4. Continuous limits 33 References 46

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Rigidity and Edge Universality of Discrete β‐Ensembles

Cited by 21 publications

References 102 publications

Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Log-gases on quadratic lattices via discrete loop equations and q-boxed plane partitions

Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case

Multi-level loop equations for $β$-corners processes

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