Results 11  20
of
23
Dyson's model of interacting Brownian motions at arbitrary coupling strength
 MARKOV PROC. REL. FIELDS
, 1998
"... For Dyson's model of Brownian motions we prove that the fluctuations are of order one and, in a scaling limit, are governed by an infinite dimensional OrnsteinUhlenbeck process. This extends a previous result valid only at the free Fermion point fi = 2. Dyson's model can also be interpreted as a ra ..."
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Cited by 5 (1 self)
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For Dyson's model of Brownian motions we prove that the fluctuations are of order one and, in a scaling limit, are governed by an infinite dimensional OrnsteinUhlenbeck process. This extends a previous result valid only at the free Fermion point fi = 2. Dyson's model can also be interpreted as a random surface. Our result implies that the surface statistics is governed by a massless Gaussian field in the scaling limit.
A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.
Universality of general βensembles
, 2011
"... We prove the universality of the βensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of loggases at any inverse temperature β coincide with those of the Gaussian βensembles. ..."
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Cited by 4 (3 self)
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We prove the universality of the βensembles with convex analytic potentials and for any β> 0, i.e. we show that the spacing distributions of loggases at any inverse temperature β coincide with those of the Gaussian βensembles.
Lectures on random matrix models. The RiemannHilbert approach
, 2008
"... This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its appli ..."
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Cited by 2 (0 self)
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This is a review of the RiemannHilbert approach to the large N asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the RiemannHilbert approach to the large N asymptotics of orthogonal polynomials and its applications to the problem of universality in random matrix models, the double scaling limits, the large N asymptotics of the partition function, and random matrix models with external source.
Asymptotic Expansion of β Matrix Models in the Onecut Regime
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2012
"... We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov ..."
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We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion ” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of SchwingerDyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and MaurelSegala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.
Bulk Universality of General βEnsembles with Nonconvex Potential
, 2012
"... We prove the bulk universality of the βensembles with nonconvex 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. ..."
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We prove the bulk universality of the βensembles with nonconvex 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.
ON FLUCTUATIONS OF EIGENVALUES OF RANDOM PERMUTATION MATRICES
, 2013
"... Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting nonuniversality phenomenon. Though they have bounded variance, their fluctuations are asymptotically nonGaussian but infinitely divisible. The fluctuations are asymptoticall ..."
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Smooth linear statistics of random permutation matrices, sampled under a general Ewens distribution, exhibit an interesting nonuniversality phenomenon. Though they have bounded variance, their fluctuations are asymptotically nonGaussian but infinitely divisible. The fluctuations are asymptotically Gaussian for less smooth linear statistics for which the variance diverges. The degree of smoothness is measured in terms of the quality of the trapezoidal approximations of the integral of the observable.
Asymptotic Analysis of the Density of States in Random Matrix Models Associated With a Slowly Decaying Weight
, 2001
"... The asymptotic behavior of polynomials that are orthogonal with respect to a slowly decaying weight is very different from the asymptotic behavior of polynomials that are orthogonal with respect to a Freudtype weight. While the latter has been extensively studied, much less is known about the forme ..."
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The asymptotic behavior of polynomials that are orthogonal with respect to a slowly decaying weight is very different from the asymptotic behavior of polynomials that are orthogonal with respect to a Freudtype weight. While the latter has been extensively studied, much less is known about the former. Following an earlier investigation into the zero behavior, we study here the asymptotics of the density of states in a unitary ensemble of random matrix with a slowly decaying weight. This measure is also naturally connected with the orthogonal polynomials. It is shown that, after suitable rescaling, the weak limit is the same as the weak limit of the rescaled zeros. 1
On Asymptotic Behavior of Multilinear Eigenvalue Statistics of Random Matrices
"... We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U and V(von Mises) statistics of eigenvalues of random matrices as their size tends to in nity. We show rst that for a certain class of test functions (kernels), determining the statistics, the validity of these limitin ..."
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We prove the Law of Large Numbers and the Central Limit Theorem for analogs of U and V(von Mises) statistics of eigenvalues of random matrices as their size tends to in nity. We show rst that for a certain class of test functions (kernels), determining the statistics, the validity of these limiting laws reduces to the validity of analogous facts for certain linear eigenvalue statistics. We then check the conditions of the reduction statements for several most known ensembles of random matrices, The reduction phenomenon is well known in statistics, dealing with i.i.d. random variables. It is of interest that an analogous phenomenon is also the case for random matrices, whose eigenvalues are strongly dependent even if the entries of matrices are independent. 1
Universal scaling limits . . .
, 2009
"... We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ xp/q, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)minimal models kernels. Those (p, q) kernels a ..."
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We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ∼ xp/q, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the universal (p, q)minimal models kernels. Those (p, q) kernels are written in terms of functions solutions of a linear equation of order q, with polynomial coefficients of degree ≤ p. For example, near a regular edge y ∼ x1/2, the (1, 2) kernel is the Airy kernel. Those kernels are associated to the (p, q) minimal model, i.e. the (p, q) reduction of the KP hierarchy solution of the string equation. Here we consider only the 1matrix model, for which q = 2.