Eigenvalues of GUE Minors

Eigenvalues of GUE minors

Consider an infinite random matrix H = (hij)0<i,j picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by Hi = (hrs)1≤r,s≤i and let the j:th largest eigenvalue of Hi be μ i j . We show that the configuration of all these eigenvalues (i, μ i j) form a determinantal point process on N × R. Furthermore we show that this process can be obtained as the scaling limit in random tili...

Gaussian Fluctuations of Eigenvalues in the Gue

Under certain conditions on k we calculate the limit distribution of the kth largest eigenvalue, xk, of the Gaussian Unitary Ensemble (GUE). More specifically, if n is the dimension of a random matrix from the GUE and k is such that both n − k and k tends to infinity as n → ∞ then xk is normally distributed in the limit. We also consider the joint limit distribution of xk1 < . . . < xkm where w...

Asymptotic Independence of the Extreme Eigenvalues of Gue

We give a short, operator-theoretic proof of the asymptotic independence of the minimal and maximal eigenvalue of the n× n Gaussian Unitary Ensemble in the large matrix limit n→ ∞. This is done by representing the joint probability distribution of those extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. The method is amenable to explicitl...

And the Gue Hypothesis

Generating series for GUE correlators

We extend to the Toda lattice hierarchy the approach of [3, 4] to computation of logarithmic derivatives of tau-functions in terms of the so-called matrix resolvents of the corresponding difference Lax operator. As a particular application we obtain explicit generating series for connected GUE correlators. On this basis an efficient recursive procedure for computing the correlators in full gene...