Communications in Mathematical Physics

, Volume 359, Issue 1, pp 223–263 | Cite as

Large Deformations of the Tracy–Widom Distribution I: Non-oscillatory Asymptotics

Article

Abstract

We analyze the left-tail 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 \({1-\gamma\in[0,1]}\). As \({\gamma}\) varies, a transition from Tracy–Widom statistics (\({\gamma=1}\)) to classical Weibull statistics (\({\gamma=0}\)) 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 leading-order left-tail asymptotics of the thinned GOE, GUE, and GSE Tracy–Widom distributions. In this paper, we obtain the asymptotic behavior in the non-oscillatory region with \({\gamma\in[0,1)}\) fixed (for the GOE, GUE, and GSE distributions) and \({\gamma\uparrow 1}\) 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 total-integral formula involving the Ablowitz–Segur solution to the second Painlevé equation.

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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