Abstract
We analyze the lefttail asymptotics of deformed Tracy–Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 − γ ∈ [0, 1]. As γ varies, a transition from Tracy–Widom statistics (γ = 1) to classicalWeibull statistics (γ = 0) was observed in the physics literature by Bohigas et al. (Phys Rev E 79:031117, 2009). We provide a description of this transition by rigorously computing the leadingorder lefttail asymptotics of the thinned GOE, GUE, and GSE Tracy–Widom distributions. In this paper, we obtain the asymptotic behavior in the nonoscillatory region with γ ∈ [0, 1) fixed (for the GOE, GUE, and GSE distributions) and γ ↑ 1 at a controlled rate (for the GUE distribution). This is the first step in an ongoing program to completely describe the transition between Tracy–Widom and Weibull statistics. As a corollary to our results, we obtain a new totalintegral formula involving the Ablowitz–Segur solution to the second Painlevé equation.
Original language  English 

Pages (fromto)  223263 
Number of pages  41 
Journal  Communications in Mathematical Physics 
DOIs  
Publication status  Published  13 Oct 2017 
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Dr Thomas Bothner
 School of Mathematics  Associate Professor in Mathematical Physics
Person: Academic