scholarly journals Quantifying Dip–Ramp–Plateau for the Laguerre Unitary Ensemble Structure Function

2021 ◽  
Author(s):  
Peter J. Forrester
Keyword(s):  
Structure Function ◽  
Ensemble Structure ◽  
Laguerre Unitary Ensemble ◽  
Unitary Ensemble

scholarly journals Painlevé V, Painlevé XXXIV and the degenerate Laguerre unitary ensemble

2019 ◽  
Vol 09 (02) ◽  
pp. 2050016 ◽  
Author(s):  
Chao Min ◽  
Yang Chen
Keyword(s):  
Asymptotic Behavior ◽  
Compatibility Condition ◽  
Logarithmic Derivative ◽  
Hankel Determinant ◽  
Ladder Operators ◽  
Coupled Riccati Equations ◽  
Coulomb Fluid ◽  
Laguerre Unitary Ensemble ◽  
Soft Edge ◽  
Unitary Ensemble

In this paper, we study the Hankel determinant associated with the degenerate Laguerre unitary ensemble (dLUE). This problem originates from the largest or smallest eigenvalue distribution of the dLUE. We derive the ladder operators and its compatibility condition with respect to a general perturbed weight function. By applying the ladder operators to our problem, we obtain two auxiliary quantities [Formula: see text] and [Formula: see text] and show that they satisfy the coupled Riccati equations, from which we find that [Formula: see text] satisfies the Painlevé V equation. Furthermore, we prove that [Formula: see text], a quantity related to the logarithmic derivative of the Hankel determinant, satisfies both the continuous and discrete Jimbo–Miwa–Okamoto [Formula: see text]-form of the Painlevé V. In the end, by using Dyson’s Coulomb fluid approach, we consider the large [Formula: see text] asymptotic behavior of our problem at the soft edge, which gives rise to the Painlevé XXXIV equation.

scholarly journals PDEs SATISFIED BY EXTREME EIGENVALUES DISTRIBUTIONS OF GUE AND LUE

2012 ◽  
Vol 01 (01) ◽  
pp. 1150003 ◽  
Author(s):  
ESTELLE BASOR ◽  
YANG CHEN ◽  
LUN ZHANG
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Gaussian Unitary Ensemble ◽  
Analytic Proof ◽  
Smallest Eigenvalue ◽  
Extreme Eigenvalues ◽  
Laguerre Unitary Ensemble ◽  
Unitary Ensemble

In this paper we study, Prob (n, a, b), the probability that all the eigenvalues of finite n unitary ensembles lie in the interval (a, b). This is identical to the probability that the largest eigenvalue is less than b and the smallest eigenvalue is greater than a. It is shown that a quantity allied to Prob (n, a, b), namely, [Formula: see text] in the Gaussian Unitary Ensemble (GUE) and [Formula: see text], in the Laguerre Unitary Ensemble (LUE) satisfy certain nonlinear partial differential equations for fixed n, interpreting Hn(a, b) as a function of a and b. These partial differential equations may be considered as two variable generalizations of a Painlevé IV and a Painlevé V system, respectively. As an application of our result, we give an analytic proof that the extreme eigenvalues of the GUE and the LUE, when suitably centered and scaled, are asymptotically independent.

scholarly journals On the ratio probability of the smallest eigenvalues in the Laguerre unitary ensemble

Nonlinearity ◽  
2018 ◽  
Vol 31 (4) ◽  
pp. 1155-1196 ◽  
Author(s):  
Max R Atkin ◽  
Christophe Charlier ◽  
Stefan Zohren
Keyword(s):  
Laguerre Unitary Ensemble ◽  
Unitary Ensemble

scholarly journals On the condition number of the critically-scaled Laguerre Unitary Ensemble

2016 ◽  
Vol 36 (8) ◽  
pp. 4287-4347 ◽  
Author(s):  
Govind Menon ◽  
Thomas Trogdon ◽  
Percy A. Deift
Keyword(s):  
Condition Number ◽  
Laguerre Unitary Ensemble ◽  
Unitary Ensemble
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