arXiv subjects
Publications and source records attributed to Bálint Virág.
The Tracy-Widom beta distribution is the large dimensional limit of the top eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator representation to show that as a tends to infinity the tail of the Tracy Widom distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2)).
We show that in the point process limit of the bulk eigenvalues of $β$-ensembles of random matrices, the probability of having no eigenvalue in a fixed interval of size $λ$ is given by \[\bigl(\ kappa_β+o(1)\bigr)λ^{γ_β}\exp\biggl(-{\bet a}{64}λ^2+\biggl(β{8}-{1}{4}\biggr)λ\biggr)\] as $λ\to\infty$, where \[γ_β={1}{4}\biggl(β{2}+{2}β-3\biggr)\] and $κ_β$ is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson [J. Math. Phys. 3 (1962) 157--165]. Our proof uses the new Brownian carousel representation of the limit process, as well as the Cameron--Martin--Girsanov transformation in stochastic calculus.
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees). They have the striking property that the number of points in a region $D$ is a sum of independent Bernoulli random variables, with parameters which are eigenvalues of the relevant operator on $L^2(D)$. Moreover, any determinantal process can be represented as a mixture of determinantal projection processes. We give a simple explanation for these known facts, and establish analogous representations for permanental processes, with geometric variables replacing the Bernoulli variables. These representations lead to simple proofs of existence criteria and central limit theorems, and unify known results on the distribution of absolute values in certain processes with radially symmetric distributions.