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Title:
Deformations of the Tracy-Widom distribution and transition asymptotics for Painleve II
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Abstract:
The distribution functions of the largest eigenvalue of random matrices drawn from the three classical Gaussian ensembles were derived in the early 1990s by Tracy and Widom. For the famous GUE ensemble this result first expresses the distribution function as Fredholm determinant of the Airy integral operator and secondly identifies the underlying integrable system as a distinguished solution of the second Painlev ́e equation. During the following 25 years many links of Tracy-Widom distribution functions to other probabilistic models were discovered, here we will discuss a thinning model in which the Tracy-Widom distribution undergoes a deformation: drop a certain fraction of edge scaled eigenvalues in the GUE. As shown by Bohigas, Carvalho and Pato the Fredholm determinant formalism naturally extends to such incomplete spectra and we have now a one parametric family of Fredholm determinants for the distribution function of the largest eigenvalue in the new particle system. The underlying integrable system in this new system is different from the standard one, still the model offers the possibility to interpolate between random matrix theory statistics and classical Weilbull statistics. We will discuss this interpolation process on the level of the large negative transition asymptotics of the Painlev ́e II
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