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The single ring theorem
, 2009
"... We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical c ..."
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We study the empirical measure LAn of the eigenvalues of nonnormal square matrices of the form An = UnDnVn with Un,Vn independent Haar distributed on the unitary group and Dn real diagonal. We show that when the empirical measure of the eigenvalues of Dn converges, and Dn satisfies some technical conditions, LAn converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of FeinbergZee single ring theorem [5]. We also consider the case where Un,Vn are independent Haar distributed on the orthogonal group. 1 The problem Horn [15] asked the question of describing the eigenvalues of a square matrix with prescribed singular values. If A is a n × n matrix with singular values s1 ≥... ≥
Combinatorics of dispersionless integrable systems and universality in random matrix theory
"... Abstract. It is wellknown that the partition function of the unitary ensembles of random matrices is given by a τfunction of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τfunctions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of t ..."
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Abstract. It is wellknown that the partition function of the unitary ensembles of random matrices is given by a τfunction of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are τfunctions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function Fnm which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and 1 Fnm are the Grunsky coefficients of the Faber polynomials. nm
MONOTONE HURWITZ NUMBERS AND THE HCIZ INTEGRAL II
"... Abstract. Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking a ..."
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Abstract. Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hurwitz numbers in genus 0 and 1, for all partitions, and an explicit rational form for the generating series in arbitrary genus. This rational form implies that, up to an explicit combinatorial scaling,
Asymptotics of unitary multimatrix models: The SchwingerDyson lattice and topological recursion. ArXiv eprints
, 2014
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Key words and phrases: asymptotic expansion; partition function; Laguerretype; Riemann
"... of the partition function of a Laguerretype random matrix model ..."
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FREE ENERGIES AND FLUCTUATIONS FOR THE UNITARY BROWNIAN MOTION
"... Abstract. We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary YangMills measure on the plane. The limiting objects obtai ..."
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Abstract. We show that the Laplace transforms of traces of words in independant unitary Brownian motions converge towards an analytic function on a non trivial disc. This results allow to study asymptotics of Wilson loops under the unitary YangMills measure on the plane. The limiting objects obtained are shown to be characterized by equations analog to SchwingerDyson’s ones, named here after Makeenko and Migdal. 1.