Let p_n(z)=∏_j=1^n(z-z_j), where z_1,…,z_n are independent with common law mu_0 on C, and let p_n,t denote the holomorphic heat evolution of p_n in the normalization of Hall and Ho. If nu_n,t is the empirical measure of the zeros of p_n,t, the conjecture asserts that, for macroscopic times before the first singular time, nu_n,t converges weakly as n→∞ to a deterministic measure mu_t. The limiting measure is given by the Hall–Ho transport description, i.e. mu_t=(T_t)_#mu_0 for the corresponding explicit transport map T_t; in particular, the circular law is transported to the elliptic law. The conjecture also includes convergence of the associated zero-density evolution to the proposed heat-flow PDE.

Random-polynomial root-evolution conjectures

Let $p_n(z)=\prod_{j=1}^n(z-z_j)$ , where $z_1,\ldots,z_n$ are independent with common law $\mu_0$ on $\mathbb{C}$ , and let $p_{n,t}$ denote the holomorphic heat evolution of $p_n$ in the normalization of Hall and Ho. If $\nu_{n,t}$ is the empirical measure of the zeros of $p_{n,t}$ , the conjecture asserts that, for macroscopic times before the first singular time, $\nu_{n,t}$ converges weakly as $n\to\infty$ to a deterministic measure $\mu_t$ . The limiting measure is given by the Hall–Ho transport description, i.e. $\mu_t=(T_t)_\#\mu_0$ for the corresponding explicit transport map $T_t$ ; in particular, the circular law is transported to the elliptic law. The conjecture also includes convergence of the associated zero-density evolution to the proposed heat-flow PDE.

References

Primary source

arXiv

Additional references

Heat flow and repeated differentiation of polynomials with i.i.d. roots — arXiv — Jonas Jalowy

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to prove the two main root-evolution predictions under restricted assumptions, but the result is not yet independently verified and does not cover all initial distributions.

The problem concerns the macroscopic motion of zeros of large random polynomials under heat flow and repeated differentiation. Hall and Ho formulated the heat-flow conjecture; subsequent work extended the analysis to rigorous convergence and transport descriptions under explicit hypotheses.

Known results

Hall and Ho, 2022: proposed the heat-flow PDE and transport-map description, with supporting arguments and moment-level verification.
Hall, Ho, Jalowy, and Kabluchko, 2023–2024: proved convergence in probability for broad random-polynomial classes under assumptions, including transport before $t_{\mathrm{sing}}$ .
Jalowy, 2023: proved a bulk push-forward theorem for repeated differentiation and related fractional differential operators.
The limiting measure need not be determined uniquely by the initial measure for nonreal initial support.

September 21, 2026 claimed proof

A preprint by Jonas Jalowy claims almost-sure weak convergence for heat flow and repeated differentiation, with explicit limiting transport maps under bounded-density and Stieltjes-transform assumptions. If correct, it settles the tracked conjectures in those regimes, but the claim is unverified and explicitly does not establish universality for all initial laws.

Current status (as of September 2026): heat-flow and repeated-differentiation evolution have a claimed proof under stated hypotheses, but remain unverified, while the unrestricted problem remains open.

Sources

No solutions have been posted yet.