The heat flow conjecture for random matrices

Brian C. Hall, Ching-Wei Ho

Published 2022-02-19, updated 2022-04-22Version 2

Recent results by various authors have established a "model deformation phenomenon" in random matrix theory. Specifically, it is possible to construct pairs of random matrix models such that the limiting eigenvalue distributions are connected by push-forward under an explicitly constructible map of the plane to itself. In this paper, we argue that the analogous transformation at the finite-$N$ level can be accomplished by applying an appropriate heat flow to the characteristic polynomial of the first model. Let the "second moment" of a random polynomial $p$ denote the expectation value of the square of the absolute value of $p.$ We find certain pairs of random matrix models and we apply a certain heat-type operator to the characteristic polynomial $p_{1}$ of the first model, giving a new polynomial $q.$ We prove that the second moment of $q$ is equal to the second moment of the characteristic polynomial $p_{2}$ of the second model. This result leads to several conjectures of the following sort: when $N$ is large, the zeros of $q$ have the same bulk distribution as the zeros of $p_{2},$ namely the eigenvalues of the second random matrix model. At a more refined level, we conjecture that, as the characteristic polynomial of the first model evolves under the appropriate heat flow, its zeros will evolve close to the characteristic curves of a certain PDE. All conjectures are formulated in "additive" and "multiplicative" forms. As a special case, suppose we apply the standard heat operator for time $1/N$ to the characteristic polynomial $p$ of an $N\times N$ GUE matrix, giving a new polynomial $q.$ We conjecture that the zeros of $q$ will be asymptotically uniformly distributed over the unit disk. That is, the heat operator converts the distribution of zeros from semicircular to circular.