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Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

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Abstract

In this paper, we study the singularly perturbed Laguerre unitary ensemble

$$\frac{1}{Z_n} ({\rm det}\,\, M)^\alpha e^{- {\rm tr}\, V_t(M)}dM, \qquad \alpha > 0,$$

with \({V_t(x) = x + t/x,\,\, x \in (0,+\infty)}\) and t >  0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves \({\psi}\)-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.

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Authors and Affiliations

  1. Institut Franco-Chinois de l’Energie Nucléaire, Sun Yat-sen University, GuangZhou, 510275, China

    Shuai-Xia Xu

  2. Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

    Dan Dai

  3. Department of Mathematics, Sun Yat-sen University, GuangZhou, 510275, China

    Yu-Qiu Zhao

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  1. Shuai-Xia Xu
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  2. Dan Dai
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Correspondence to Yu-Qiu Zhao.

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Communicated by P. Deift

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Xu, SX., Dai, D. & Zhao, YQ. Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble. Commun. Math. Phys. 332, 1257–1296 (2014). https://doi.org/10.1007/s00220-014-2131-9

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