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 . As varies, a transition from Tracy-Widom statistics () to classical Weibull statistics () 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 fixed (for the GOE, GUE, and GSE distributions) and 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 Painlev, equation.