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Spacing distributions in random matrix ensembles, (in) Recent perspectives in random matrix theory and number theory
 London Math. Soc. Lecture
"... 1.1 Motivation and definitions The topic of spacing distributions in random matrix ensembles is almost as old as the introduction of random matrix theory into nuclear physics. Both events can be traced back to Wigner in the mid 1950’s [37, 38]. Thus Wigner introduced the model of a large real symmet ..."
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1.1 Motivation and definitions The topic of spacing distributions in random matrix ensembles is almost as old as the introduction of random matrix theory into nuclear physics. Both events can be traced back to Wigner in the mid 1950’s [37, 38]. Thus Wigner introduced the model of a large real symmetric random
Scaled Limit and Rate of Convergence for the Largest Eigenvalue from the Generalized Cauchy Random Matrix Ensemble
, 2009
"... Abstract. In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble GCy, whose eigenvalues PDF is given by const · (xj − xk) 2 ..."
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Abstract. In this paper, we are interested in the asymptotic properties for the largest eigenvalue of the Hermitian random matrix ensemble, called the Generalized Cauchy ensemble GCy, whose eigenvalues PDF is given by const · (xj − xk) 2
Universality conjecture and results for a model of several coupled positivedefinite matrices. arXiv:1407.2597 [mathph
, 2014
"... Abstract. The paper contains two main parts: in the first part, we analyze the general case of p ≥ 2 matrices coupled in a chain subject to Cauchy interaction. Similarly to the ItzyksonZuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We fi ..."
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Abstract. The paper contains two main parts: in the first part, we analyze the general case of p ≥ 2 matrices coupled in a chain subject to Cauchy interaction. Similarly to the ItzyksonZuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a (p+1)×(p+1) matrix valued solution of a RiemannHilbert problem. In the second part, we fix the external potentials as classical Laguerre weights. We then derive strong asymptotics for the Cauchy biorthogonal polynomials when the support of the equilibrium measures contains the origin. As a result, we obtain a new family of universality classes for multilevel random determinantal point fields which include the Besselν universality for 1level and the MeijerG universality for 2level. Our analysis uses the DeiftZhou nonlinear steepest descent method and the explicit construction of a (p + 1) × (p + 1) origin parametrix in terms of Meijer Gfunctions. The solution of the full RiemannHilbert problem is derived rigorously only for p = 3 but the general framework of the proof can be extended to the Cauchy chain of arbitrary length p. 1.
Asymptotics of a Class of Fredholm Determinants
, 1998
"... In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptoti ..."
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In this expository article we describe the asymptotics of certain Fredholm determinants which provide solutions to the cylindrical Toda equations, and we explain how these asymptotics are derived. The connection with Fredholm determinants arising in the theory of random matrices, and their asymptotics, are also discussed. 1.
ensembles with orthogonal and
, 2006
"... and soft edge spacing distributions for random matrix ..."
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Painlevé transcendent evaluation of the scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles
, 2000
"... The scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles is evaluated in terms of a Painlevé V transcendent. This same Painlevé V transcendent is known from the work of Tracy and Widom, where it has been shown to specify the scaled distribution of the sm ..."
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The scaled distribution of the smallest eigenvalue in the Laguerre orthogonal and symplectic ensembles is evaluated in terms of a Painlevé V transcendent. This same Painlevé V transcendent is known from the work of Tracy and Widom, where it has been shown to specify the scaled distribution of the smallest eigenvalue in the Laguerre unitary ensemble. The starting point for our calculation is the scaled kpoint distribution of every odd labelled eigenvalue in two superimposed Laguerre orthogonal ensembles.
Dyson’s constants in the asymptotics of the determinants of WienerHopfHankel operators with the sine kernel
, 2006
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