Universality conjecture for heat-evolved polynomial zeros

Let $\nu$ be a rotationally invariant probability measure on $\mathbb{C}\backslash\{0\}$ satisfying $\int_{0}^{\varepsilon} \nu(\{|z| < r\})r^{-1}\,\operatorname{d}r<\infty$ for some $\varepsilon>0$, and let $\nu=\nu_0$ be realized by a concave function $g:[0,1]\to\mathbb{R}$. Suppose that $Q_n$ is a sequence of degree-$n$ polynomials whose roots are asymptotically distributed according to $\nu_0$ and are not “too evenly spaced”. Universality conjecture. The conclusion of Theorem 2.9 remains in force with $P_n$ replaced by $Q_n$; in particular, the empirical zero distribution of

$$\exp\left\{-\frac{t}{2n}\partial_z^2\right\}Q_n$$

converges to the same limiting measure as in the original model. This is an extended heat-flow universality principle: the limiting behavior should depend on the asymptotic root distribution, subject to excluding overly regular spacing. The source gives examples involving i.i.d. roots, Haar orthogonal or unitary matrices, and products or powers of Ginibre matrices.