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

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 − 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 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 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 total-integral formula involving the Ablowitz–Segur solution to the second Painlevé equation.