Brian Hall, Notre Dame

Abstract. Several recent results have demonstrated a “model deformation phenomenon” in random matrix theory, in which the limiting eigenvalue distributions of two different random matrix models are, in certain cases, related by push-forward under an explicit, canonical map of the plane to itself. The prototype example is the case of the circular and semicircular laws, which are related by push-forward under the map z —> 2Re(z). There are by now several broad families of examples extending this simple case.

I will discuss a conjecture, developed with Ching Wei Ho, that provides a finite-N version of the model deformation phenomenon, at the level of characteristic polynomials. Specifically, the conjecture says that applying the heat operator to the characteristic polynomial of one random matrix gives a polynomial whose bulk distribution of zeros resembles that of a different random matrix. As an example, consider applying the heat operator for time 1/N to the characteristic polynomial of an NxN GUE matrix. We believe that the zeros of the resulting polynomial will be almost surely asymptotically uniform over the unit disk. Thus, the heat operator can turn the semicircular law into the circular law. I will explain the conjecture and describe some rigorous results in this direction.