## Universality of general β-ensembles (2011)

Citations: | 4 - 3 self |

### BibTeX

@TECHREPORT{Bourgade11universalityof,

author = {Paul Bourgade and László Erdős and Horng-tzer Yau},

title = {Universality of general β-ensembles},

institution = {},

year = {2011}

}

### OpenURL

### Abstract

We prove the universality of the β-ensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of log-gases at any inverse temperature β coincide with those of the Gaussian β-ensembles.

### Citations

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Random Matrices
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Citation Context ...V (x)/2. Thus the analysis of the correlation functions relies heavily on the asymptotic properties of the corresponding orthogonal polynomials. In the pioneering work of Gaudin, Mehta and Dyson (see =-=[23]-=- for a review), the potential V is the quadratic polynomial V (x) = x2 and the orthogonal polynomials are the Hermite polynomials for which asymptotic properties are well-known. The major input of the... |

295 |
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
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(Show Context)
Citation Context ...the interior of the spectrum, although similar questions regarding the edge distribution are also important. Over the past two decades, spectacular progress (see, e.g., [5, 10, 11, 24, 25, 9, 22] and =-=[2, 8, 9]-=- for a review) on bulk universality was made for classical invariant ensembles, i.e., matrix models with probability measure given by e −NβTrV (H)/2 /Z where N is the size of the matrix H, V is a real... |

201 |
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
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(Show Context)
Citation Context ... on eigenvalue distribution in the interior of the spectrum, although similar questions regarding the edge distribution are also important. Over the past two decades, spectacular progress (see, e.g., =-=[5, 10, 11, 24, 25, 9, 22]-=- and [2, 8, 9] for a review) on bulk universality was made for classical invariant ensembles, i.e., matrix models with probability measure given by e −NβTrV (H)/2 /Z where N is the size of the matrix ... |

139 |
Strong asymptotics of orthogonal polynomials with respect to exponential weights
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(Show Context)
Citation Context ... on eigenvalue distribution in the interior of the spectrum, although similar questions regarding the edge distribution are also important. Over the past two decades, spectacular progress (see, e.g., =-=[5, 10, 11, 24, 25, 9, 22]-=- and [2, 8, 9] for a review) on bulk universality was made for classical invariant ensembles, i.e., matrix models with probability measure given by e −NβTrV (H)/2 /Z where N is the size of the matrix ... |

95 |
A Brownian-motion model for the eigenvalues of a random matrix
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Citation Context ...ant C uniformly in N. Here µW is the law given by the Wigner ensemble. Under this assumption, a strong estimate on the local ergodicity of Dyson Brownian motion (DBM) was established in [14, 15]. DBM =-=[13]-=- establishes a dynamical interpolation between Wigner matrices and the invariant equilibrium measure µ. This estimate then implies the universality of Wigner matrices. Thus the main task in proving th... |

88 | Matrix models for beta-ensembles
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Citation Context ...iori can be an arbitrary positive number. These measures are called general β-ensembles. We will often refer to the variables λj as particles or points and the system is called log-gas. It was proved =-=[12]-=- that in the Gaussian case, i.e., when V is quadratic, the measure (1.1) describes eigenvalues of tri-diagonal matrices. This observation allowed one to establish detailed properties, including the lo... |

87 |
On extensions of the Brunn-Minkowski and PrékopaLeindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation
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Citation Context ...ocal criterion for the logarithmic Sobolev inequality, will yield a strong concentration for ∑ i∈I (k,M) viλi under ω (k,M), if ∑ i vi = 0. This lemma is a consequence of the Brascamp-Lieb inequality =-=[7]-=-. Notice that the original inequality applied only to measures on RN , but a mollifying argument in Lemma 4.4 of [17] has extended it to the measures on the simplex {λ1 < λ2 < . . . < λN } considered ... |

67 |
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Citation Context ...oncentration and accuracy of the particles location at scale N −1/2 , in the bulk. Concentration is a simple consequence of the Bakry-Émery convexity criterion for the logarithmic Sobolev inequality (=-=[3]-=-, see also [2]): define H by µ(dλ) = 1 ZN e−NH(λ) dλ, and assume ∇ 2 H ≥ σ IdN (3.9) in the sense of partial order for positive definite operators. Then µ satisfies a logarithmic Sobolev inequality wi... |

62 | Large deviations for Wigner’s law and Voiculescu’s non-commutative entropy. Probab. Theory Relat
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Citation Context ...) = 1 ∫ B 2π A V ′ (z) − V ′ (t) z − t In particular, for convex V , r has no zero in R. dt √ . (3.4) (t − A)(B − t) It is known that the particles locations cannot be far from its classical location =-=[4, 26]-=-: for any ε > 0 there are positive constants C, c, such that, for all N ≥ 1, c −cN Pµ (∃k ∈ �1, N� | |λk − γk| ≥ ε) ≤ Ce . (3.5) In order to have density strictly in a compact support, for given R > 0... |

55 | Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles
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- 1997
(Show Context)
Citation Context ... on eigenvalue distribution in the interior of the spectrum, although similar questions regarding the edge distribution are also important. Over the past two decades, spectacular progress (see, e.g., =-=[5, 10, 11, 24, 25, 9, 22]-=- and [2, 8, 9] for a review) on bulk universality was made for classical invariant ensembles, i.e., matrix models with probability measure given by e −NβTrV (H)/2 /Z where N is the size of the matrix ... |

53 |
An Introduction to Random Matrices
- Anderson, Guionnet, et al.
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(Show Context)
Citation Context ...the interior of the spectrum, although similar questions regarding the edge distribution are also important. Over the past two decades, spectacular progress (see, e.g., [5, 10, 11, 24, 25, 9, 22] and =-=[2, 8, 9]-=- for a review) on bulk universality was made for classical invariant ensembles, i.e., matrix models with probability measure given by e −NβTrV (H)/2 /Z where N is the size of the matrix H, V is a real... |

32 |
Master loop equations, free energy and correlations for the chain of matrices
- Eynard
(Show Context)
Citation Context ...(X)| ≤ E(|X − E(X)| 2 ). The equation used by Johansson (which can be obtained by a change of variables in (2.1) [20] or by integration by parts [26]), is a variation of the loop equation (see, e.g., =-=[19]-=-) used in physics literatures and it takes the form ) (mN − m) 2 + s(mN − m) + bN = cN. (3.13) Equation (3.13) expresses the difference mN − m in terms of (mN − m) 2 , bN and cN . In the regime |mN − ... |

26 |
On the statistical mechanics approach to random matrix theory: integrated density of states
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- 1995
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Citation Context ...ection. We first state well-known facts about the equilibrium measure. For convex analytic potential V satisfying the asymptotic growth condition (2.2) (or even with weaker hypotheses on V , see e.g. =-=[6, 1]-=-), the equilibrium measure ρ(s)ds associated with (µ (N) )N≥0 can be defined as the unique minimizer (in the set of probability measures on R endowed with the weak topology) of the functional ∫ I(ν) =... |

24 | A new approach to universality limits involving orthogonal polynomials
- Lubinsky
(Show Context)
Citation Context |

22 |
On the 1/n expansion for some unitary invariant ensembles of random matrices
- Albeverio, Pastur, et al.
(Show Context)
Citation Context ...ection. We first state well-known facts about the equilibrium measure. For convex analytic potential V satisfying the asymptotic growth condition (2.2) (or even with weaker hypotheses on V , see e.g. =-=[6, 1]-=-), the equilibrium measure ρ(s)ds associated with (µ (N) )N≥0 can be defined as the unique minimizer (in the set of probability measures on R endowed with the weak topology) of the functional ∫ I(ν) =... |

17 |
The local relaxation flow approach to universality of the local statistics of random matrices
- Erdős, Schlein, et al.
(Show Context)
Citation Context ... gap statistics of two measures are approximately the same if τD/K → 0. More precisely, we have the following general theorem which is a slight modification of Lemma 3.4 [14] (see also Theorem 4.1 in =-=[15]-=-). Lemma 5.9 Let G : R → R be a bounded smooth function with compact support. Let I be an interval of indices with |I| = K. Consider a measure ω with relaxation time τ and let qdω be another probabili... |

15 | Yau Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation, Electron
- Erdős, Ramírez, et al.
(Show Context)
Citation Context ...N due to the extra 1/N factor. In the convex setting, the variance can be estimated by the logarithmic Sobolev inequality and we immediately obtain an estimate on mN − m. We then follow the method in =-=[16]-=- to use the HelfferSjöstrand functional calculus to have an estimate on the particle locations. Although it is tempting to use this new accuracy information on the particle locations to estimate the v... |

14 |
Random Matrix Theory: Invariant Ensembles and Universality
- Deift, Gioev
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(Show Context)
Citation Context |

14 | Bulk universality and related properties of Hermitian matrix models
- Pastur, Shcherbina
- 2008
(Show Context)
Citation Context |

12 | Yau Universality of random matrices and local relaxation flow
- Erdős, Schlein, et al.
(Show Context)
Citation Context ...ility measures of invariant ensembles. We note that the assumption of analyticity on V is needed only for using the loop equation in (1). The basic idea of our proof is to use the following tool from =-=[14]-=-: For two probability measures µ and ω define the Dirichlet form by D(µ | ω) := 1 ∫ ∣ √ ∣∣∇ dµ ∣ 2N dω 2 dω. Then the difference of the local spacing distributions of the two measures is negligible pr... |

12 |
Rigidity of eigenvalues of Generalized Wigner Matrices, Preprint (2010) arXiv
- Erdös, Yau, et al.
(Show Context)
Citation Context ...hus the main task in proving the universality of Wigner matrices is reduced to verifying Assumption III. There are several similarities between the method used for the universality of Wigner matrices =-=[14, 18]-=- and the current proof for β-ensembles: (i) Both rely on crude estimates such as (1.3) and (1.4) on the location of the eigenvalues to establish the local spacing distributions are the same as in the ... |

11 | Spectral statistics of Erdős-Rényi Graphs II: Eigenvalue spacing and the extreme eigenvalues
- Erdős, Knowles, et al.
(Show Context)
Citation Context ...ω (k,M), if ∑ i vi = 0. This lemma is a consequence of the Brascamp-Lieb inequality [7]. Notice that the original inequality applied only to measures on RN , but a mollifying argument in Lemma 4.4 of =-=[17]-=- has extended it to the measures on the simplex {λ1 < λ2 < . . . < λN } considered in this paper. Lemma 3.9 Decompose the coordinates λ = (λ1, . . . , λN) of a point in RN = Rm × RN−m as λ = (x, y), w... |

9 | Continuum limits of random matrices and the Brownian carousel - Valkó, Virág - 2009 |

3 |
M.: Fluctuations of eigenvalues of matrix models and their applications. http://arxiv.org/abs/1003.6121v1 [math-ph
- Kriecherbauer, Shcherbina
- 2010
(Show Context)
Citation Context ...d symplectic cases, i.e., β = 1, 4, are much more difficult to use than the one for the unitary case. While universality for β = 2 was proved for very general potential, the best results for β = 1, 4 =-=[9, 21, 26]-=- are still restricted to analytic V with additional conditions. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, one can still consider the measure (1.1), but there is no simple expression of the ... |

3 |
Orthogonal and Symplectic Matrix Models: Universality and Other Properties
- Shcherbina
(Show Context)
Citation Context ...d symplectic cases, i.e., β = 1, 4, are much more difficult to use than the one for the unitary case. While universality for β = 2 was proved for very general potential, the best results for β = 1, 4 =-=[9, 21, 26]-=- are still restricted to analytic V with additional conditions. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, one can still consider the measure (1.1), but there is no simple expression of the ... |