Hello everybody, here is my question:
Assume A is a random symmetric $nxn$ matrix whose entries are independent, normally distributed with mean zero and variance 2 on the diagonal and 1 off diagonal (to my knowledge such a random matrix is said to belong to the 'Gaussian orthogonal ensemble'). The joint pdf for the eigenvalues of A is well known, but i was wondering:
does there exist a precise formula for the probability that all the eigenvalues of A have norm greater then epsilon?
Equivalently which is the probability that the least singular value of A is smaller than epsilon?
I am computing the intrinsic volume of singular symmetric matrices of (Frobenius or trace-square) norm one and I need this precise formula to perform the epsilon limit using tubes. I am sorry if this is a well known result, but I was not able to find it in the literature. In case I would really appreciate a reference for this.