Eigenvalues of random integer matrices

Abstract

The probability that a real TeX Embedding failed! matrix with entries in TeX Embedding failed! has both eigenvalues real is precisely TeX Embedding failed!. The same also holds (asymptotically) if we draw entries from the discrete set TeX Embedding failed!. The probability of the latter matrix having rational eigenvalues is TeX Embedding failed!. In joint work with Greg Martin, we determine the precise limiting distribution of eigenvalues in this subset, which is notably different from that of the real eigenvalues. I'll survey a medley of related results, and possibly mention applications to the analysis of random integer programs (joint with Gabor Pataki and Mustafa Tural).

Details