Lee--Yang zeroes of the Curie--Weiss ferromagnet, unitary Hermite polynomials, and the backward heat flow

The backward heat flow on the real line started from the initial condition $z^n$ results in the classical $n$-th Hermite polynomial whose zeroes are distributed according to the Wigner semicircle law in the large $n$ limit. Similarly, the backward heat flow with the periodic initial condition $(\sin \frac \theta 2)^n$ leads to trigonometric or unitary analogues of the Hermite polynomials. These polynomials are closely related to the partition function of the Curie--Weiss model and appeared in the work of Mirabelli on finite free probability. We relate the $n$-th unitary Hermite polynomial to the expected characteristic polynomial of a unitary random matrix obtained by running a Brownian motion on the unitary group $U(n)$. We identify the global distribution of zeroes of the unitary Hermite polynomials as the free unitary normal distribution. We also compute the asymptotics of these polynomials or, equivalently, the free energy of the Curie--Weiss model in a complex external field. We identify the global distribution of the Lee--Yang zeroes of this model. Finally, we show that the backward heat flow applied to a high-degree real-rooted polynomial (respectively, trigonometric polynomial) induces, on the level of the asymptotic distribution of its roots, a free Brownian motion (respectively, free unitary Brownian motion).