This thesis focuses on various aspects of non-Gaussian distributions and processes sharing scaling properties where the exponent 2/3 appears. The two probabilistic objects that we will introduce are: 1) Tracy-Widom distribution: This is the large dimensional limit of the top eigenvalue of random matrices in beta-ensembles. In a joint work with Balint Virag, we studied the asymptotic behavior of its right tail for all positive beta, using tools coming from diffusion analysis, such as the Girsanov formula. 2) The “true self repelling motion” (TSRM): This is a self-interacting process which was introduced by Balint Toth and Wendelin Werner. We have been interested in properties related to trajectories of this motion (large deviations, law of the iterated logarithm) and explicit distribution computations (joint work with Balint Toth). During this study, we have also dealt with questions related to game theory.
