How many eigenvalues of a Gaussian matrix are positive?

by Piepaolo Vivo (ICTP)

Tuesday 9 Feb 2010, 13:30 → 14:30 Europe/Stockholm

122:028

The index of a random matrix (i.e. the number of positive or negative eigenvalues) is a random variable providing information about the stability of stationary points in high-dimensional potential landscapes. For a Gaussian matrix model of large size N, typically half of the eigenvalues are positive and half negative (Wigner's semicircle law), however atypical fluctuations around the semicircle are quite interesting and surprisingly not well understood until very recent times. Using a Coulomb gas technique and functional methods, we find that the distribution of the index is not strictly Gaussian around the mean due to an unusual logarithmic singularity in the rate function. The variance of the index increases logarithmically with the matrix size, and such finding is confirmed by comparing with an exact finite N result based on the Andrejeff integration formula. The combinatorics behind such formula is still to be understood.