Tracy–Widom at High Temperature

Abstract

We investigate the marginal distribution of the bottom eigenvalues of the stochastic Airy operator when the inverse temperature \(\beta \) tends to \(0\). We prove that the minimal eigenvalue, whose fluctuations are governed by the Tracy–Widom \(\beta \) law, converges weakly, when properly centered and scaled, to the Gumbel distribution. More generally we obtain the convergence in law of the marginal distribution of any eigenvalue with given index \(k\). Those convergences are obtained after a careful analysis of the explosion times process of the Riccati diffusion associated to the stochastic Airy operator. We show that the empirical measure of the explosion times converges weakly to a Poisson point process using estimates proved in Dumaz and Virág (Ann Inst H Poincaré Probab Statist 49(4):915–933, 2013). We further compute the empirical eigenvalue density of the stochastic Airy ensemble on the macroscopic scale when \(\beta \rightarrow 0\). As an application, we investigate the maximal eigenvalues statistics of \(\beta _N\)-ensembles when the repulsion parameter \(\beta _N\rightarrow 0\) when \(N\rightarrow +\infty \). We study the double scaling limit \(N\rightarrow +\infty , \beta _N \rightarrow 0\) and argue with heuristic and numerical arguments that the statistics of the marginal distributions can be deduced following the ideas of Edelman and Sutton (J Stat Phys 127(6):1121–1165, 2007) and Ramírez et al. (J Am Math Soc 24:919–944, 2011) from our later study of the stochastic Airy operator.