## Large deformations of the Tracy-Widom distribution. I: Non-oscillatory asymptotics.(English)Zbl 1407.60005

Summary: 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 O. Bohigas et al. [“Deformations of the Tracy-Widom distribution”, Phys. Rev. E 79, No. 3, Article ID 031117, 6 p. (2009; doi:10.1103/physreve.79.031117)]. 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.

### MSC:

 60B20 Random matrices (probabilistic aspects) 15B52 Random matrices (algebraic aspects) 60F10 Large deviations

DLMF
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### References:

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