Large deformations of the Tracy-Widom distribution. I: Non-oscillatory asymptotics

DOI10.1007/S00220-017-3006-7zbMATH Open1407.60005arXiv1702.04462OpenAlexW2588635180MaRDI QIDQ1749263

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Publication date: 16 May 2018

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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 $1-gammain[0,1]$. 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 $gammain[0,1)$ fixed (for the GOE, GUE, and GSE distributions) and $gammauparrow1$ 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.

Full work available at URL: https://arxiv.org/abs/1702.04462

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