## Bulk Universality of General β-Ensembles with Non-convex Potential (2012)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Bourgade12bulkuniversality,

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

title = {Bulk Universality of General β-Ensembles with Non-convex Potential},

year = {2012}

}

### OpenURL

### Abstract

We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β> 0. This removes the convexity assumption appeared in the earlier work [6]. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure ” so that the local statistics are preserved under this convexification.

### Citations

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Citation Context ... universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved for analytic V with some additional conditions. A summary of recent developments can be found in =-=[2, 9, 10, 24]-=-. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, there is no simple expression of the correlation functions in terms of orthogonal polynomials. In [6], we initiated a new approach to prove bulk ... |

200 |
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Citation Context ...ew) was the starting point for all results on classical universality. Precise analysis on orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach =-=[5,11,12]-=-. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved f... |

139 |
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Citation Context ...x)/2 . Thus the analysis of the correlation functions relies heavily on the asymptotic properties of the corresponding orthogonal polynomials. This approach, initiated by Dyson, Gaudin and Mehta (see =-=[21]-=- for a review) was the starting point for all results on classical universality. Precise analysis on orthogonal polynomials for general class of weight functions was made possible by the RiemannHilber... |

114 | Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model
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Citation Context ...ew) was the starting point for all results on classical universality. Precise analysis on orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach =-=[5,11,12]-=-. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved f... |

86 | Matrix models for beta ensembles
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Citation Context ...r the measure (1.1) on ΣN , but in general there is no simple natural matrix model producing this measure except for the Gaussian case, V (x) = x 2 , which corresponds to a tri-diagonal random matrix =-=[13, 26]-=-. We will view µ = µ (N) as a Gibbs measure of particles in R with a logarithmic interaction, where the parameter β > 0 is interpreted as the inverse temperature. We will refer to the variables λj as ... |

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Citation Context ..., c1, c2 > 0 such that for any N ≥ 1 and k ∈ �αN, (1 − α)N�, Pµ ( 1 − |λk − γk| > N 2 +ε) δ −c2N ≤ c1e . Proof. From Lemma 3.5 about the convexity of Hν, we get by the classical Bakry-Émery criterion =-=[3]-=- that ν satisfies a logarithmic Sobolev inequality with constant of order 1/N, so by Herbst’s lemma concentration at scale N −1/2 holds for individual particles for ν: there is a constant c > 0 such t... |

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Citation Context ...V ) − Gk(W )| 1 + |Gk(V ) − Gk(W )| , where Gk(V ) = inf |x|>k V (x)/ log |x|. 2 For eigenvalues in the bulk, (1.7) follows from the large deviations for the empirical spectral measure with speed N 2 =-=[2, 4]-=-, and for the extreme eigenvalues the large deviations principle with speed N was proved in [2], Theorem 2.6.6, up to a condition on the partition function that follows from Theorem 1 (iii) in [25]. 3... |

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Citation Context ...orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., =-=[20,22,23]-=-. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved for analytic V with some additional conditions. A summary of recent developments can be foun... |

50 |
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Citation Context ... universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved for analytic V with some additional conditions. A summary of recent developments can be found in =-=[2, 9, 10, 24]-=-. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, there is no simple expression of the correlation functions in terms of orthogonal polynomials. In [6], we initiated a new approach to prove bulk ... |

36 | Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields
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Citation Context ...rges weakly to a continuous function ρ, the equilibrium density, with compact support. We additionally assume that ρ(s) is supported on a single interval [A, B], and that V is regular in the sense of =-=[19]-=-. We recall that V is regular if its equilibrium density ρ is positive on (A, B) and vanishes like a square root1 at each of the endpoints of [A, B], that is √ + ρ(t) = sA t − A (1 + O (t − A)) , t → ... |

32 |
Master loop equations, free energy and correlations for the chain of matrices
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Citation Context ... dt; Here the var of a complex random variable denotes var(X) = E(X 2 ) − E(X) 2 , i.e. without absolute value unlike the usual variance. Note that |var(X)| ≤ E(|X − E(X)| 2 ). The loop equation (see =-=[16, 17, 25]-=- for various proofs) is ) (mN − m) 2 + s(mN − m) + bN = cN. (2.10) In the regime where |mN − m| is small, we can neglect the quadratic term. The term bN is the same order as |mN − m| and is difficult ... |

25 |
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Citation Context ...w that r is nonzero in the interval [A, B]. By continuity, it is also nonzero in [A − κ, B + κ] for some small κ. Let ρ (N,κ) k denote the correlation functions of the measure µ(N,κ). Then Lemma 1 in =-=[7]-=- states that under condition (1.3), for some large enough κ there exists some c > 0, depending only on V , such that for any x1, . . . , xk ∈ [A − κ, B + κ], we have ∣ ∣ρ (N,κ) k (x1, . . . , xk) − ρ ... |

24 | A new approach to universality limits involving orthogonal polynomials
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Citation Context ...orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., =-=[20,22,23]-=-. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved for analytic V with some additional conditions. A summary of recent developments can be foun... |

22 |
On the 1/n expansion for some unitary invariant ensembles of random matrices
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Citation Context ... (mN − m) 2 + s(mN − m) + bN = cN. (2.10) In the regime where |mN − m| is small, we can neglect the quadratic term. The term bN is the same order as |mN − m| and is difficult to treat. As observed in =-=[1, 25]-=-, for analytic V (hence analytic bN), this term vanishes when we perform a contour integration. So we have roughly the relation (mN − m) ∼ 1 ( N∑ ) 1 varµ , (2.11) N 2 z − λk k=1 where we dropped the ... |

14 |
Random Matrix Theory: Invariant Ensembles and Universality
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Citation Context ...nnHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 =-=[10, 18, 25]-=- it was proved for analytic V with some additional conditions. A summary of recent developments can be found in [2, 9, 10, 24]. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, there is no simple ... |

14 | Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation
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Citation Context ...this implies that (2.7) actually holds when replacing g by any smooth compactly supported φ, for example by a Helffer-Sjöstrand type argument, similar to Lemma 2.3. We can now apply formula (B.13) in =-=[15]-=- for the signed measure ˜ρ = ρ (N,µh) 1 − ρ, with Stieljes transform S, where ρ (N,µh) 1 is the one-point correlation function of µh. We obtain ∫ ∞ ∣ φ(λ)˜ρ(λ)dλ ∣ −∞ ≤C ∫∫ ∣ yφ ′′ (x)χ(y) Im S(x + iy... |

10 | Spectral statistics of Erdös-Rényi graphs II: eigenvalue spacing and the extreme eigenvalues - Erdös, Knowles, et al. - 2012 |

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

4 | Universality of general β-ensembles
- Bourgade, Erdös, et al.
- 2009
(Show Context)
Citation Context ...2 Jan 11, 2012 Abstract We prove the bulk universality of the β-ensembles with non-convex regular analytic potentials for any β > 0. This removes the convexity assumption appeared in the earlier work =-=[6]-=-. The convexity condition enabled us to use the logarithmic Sobolev inequality to estimate events with small probability. The new idea is to introduce a “convexified measure” so that the local statist... |

3 |
M.: Fluctuations of eigenvalues of matrix models and their applications. http://arxiv.org/abs/1003.6121v1 [math-ph
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(Show Context)
Citation Context ...nnHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 =-=[10, 18, 25]-=- it was proved for analytic V with some additional conditions. A summary of recent developments can be found in [2, 9, 10, 24]. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, there is no simple ... |

3 |
Orthogonal and Symplectic Matrix Models: Universality and Other Properties
- Shcherbina
(Show Context)
Citation Context ...nnHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 =-=[10, 18, 25]-=- it was proved for analytic V with some additional conditions. A summary of recent developments can be found in [2, 9, 10, 24]. For non-classical values of β, i.e., β ̸∈ {1, 2, 4}, there is no simple ... |

2 |
asymptotics of orthogonal polynomials with respect to exponential weights
- Strong
- 1999
(Show Context)
Citation Context ...ew) was the starting point for all results on classical universality. Precise analysis on orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach =-=[5,11,12]-=-. There are also methods independent of the Riemann-Hilbert approach, see, e.g., [20,22,23]. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved f... |

2 |
universality and related properties of Hermitian matrix models
- Bulk
(Show Context)
Citation Context ...orthogonal polynomials for general class of weight functions was made possible by the RiemannHilbert approach [5,11,12]. There are also methods independent of the Riemann-Hilbert approach, see, e.g., =-=[20,22,23]-=-. The universality for β = 2 was proved for very general potential. For β = 1, 4 [10, 18, 25] it was proved for analytic V with some additional conditions. A summary of recent developments can be foun... |