# Eigenvalues of random integer matrices

Wong, Erick

### 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

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