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arXiv · 2001.00171

Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble

Shulin Lyu·Chao Min·Yang Chen
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Abstract

We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^γ\mathrm{e}^{-4nx},\;x\in[0,\infty),\;γ>-1$, 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 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when $γ=0$.

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Shulin Lyu, Chao Min, Yang Chen. 2020-03-29. Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble. https://doi.org/10.1063/5.0010029

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