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Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble

Shulin Lyu, ,
Journal of Mathematical Physics , Volume 62; doi:10.1063/5.0010029

Abstract: We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight xγe−4nx,x∈0,∞,γ>−1, lie in the interval [0, α]. By using previous results for finite n obtained by the ladder operator approach of orthogonal polynomials, we derive the large n asymptotics of the largest eigenvalue distribution function with α ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures.
Keywords: unitary / ensemble / Laguerre / eigenvalue / function / soft / Math / Deift / Phys / edge

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