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 $1-\gamma\in[0,1]$. 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, de Carvalho, and Pato \cite{BohigasCP: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 Painlev\'e equation.