The heat flow conjecture for polynomials and random matrices

We study the evolution of the roots of a polynomial of degree $N$, when the polynomial itself is evolving according to the heat flow. We propose a general conjecture for the large-$N$ limit of this evolution. Specifically, we propose (1) that the log potential of the limiting root distribution should evolve according to a certain first-order, nonlinear PDE, and (2) that the limiting root distribution at a general time should be the push-forward of the initial distribution under a certain explicit transport map. These results should hold for sufficiently small times, that is, until singularities begin to form. We offer three lines of reasoning in support of our conjecture. First, from a random matrix perspective, the conjecture is supported by a deformation theorem for the second moment of the characteristic polynomial of certain random matrix models. Second, from a dynamical systems perspective, the conjecture is supported by the computation of the second derivative of the roots with respect to time, which is formally small before singularities form. Third, from a PDE perspective, the conjecture is supported by the exact PDE\ satisfied by the log potential of the empirical root distribution of the polynomial, which formally converges to the desired PDE as $N\rightarrow \infty.$ We also present a "multiplicative" version of the the conjecture, supported by similar arguments. Finally, we verify rigorously that the conjectures hold at the level of the holomorphic moments.