Upper and lower bounds for the tails of the distribution of the condition number of a Gaussian matrix

Let A be an m x m real random matrix with independently and identically distributed standard Gaussian entries. We prove that there exist universal positive constants c and C such that the tail of the probability distribution of the condition number kappa(A) satisfies the inequalities c/x < P{kappa(A) > mx} < C/x for every x > 1. The proof requires a new estimation of the joint density of the largest and the smallest eigenvalues of A(T)A which follows from a formula for the expectation of the number of zeros of a certain random field defined on a smooth manifold.