On the heat flow conjecture for random matrices

Technical Summary: On the Heat Flow Conjecture for Random Matrices

Problem Statement
This paper addresses the "heat flow conjecture" proposed by Hall and Ho [7], which posits a deep connection between the evolution of random polynomials under the holomorphic heat equation and the deformation of random matrix models. Specifically, the conjecture asserts that applying a heat operator to the characteristic polynomial of a random matrix (specifically an elliptic Gaussian matrix) reproduces the limiting empirical spectral measure of a deformed version of that matrix.

While Hall and Ho established this correspondence at the level of holomorphic moments, a critical gap remained: holomorphic moments do not uniquely determine a probability measure on the complex plane (e.g., rotationally invariant measures centered at the origin have vanishing non-constant holomorphic moments). Consequently, proving the conjecture requires establishing the weak convergence of the empirical measure of the zeros of the heat-evolved polynomial to the limiting spectral measure of the deformed matrix, rather than merely matching moments.

Methodology
The author employs a synthesis of exact algebraic identities, logarithmic potential theory, and free probability to bridge the gap between moment information and weak convergence. The methodology is divided into two primary regimes: the planar (elliptic) case and the boundary (semicircle) case.

1. Exact Shifted Gaussian Norm Identities:
   The core algebraic tool is Proposition 2.1, which establishes an exact transfer identity for shifted $L^2$-norms. It demonstrates that applying the heat operator to a polynomial $p$ to obtain $q$ preserves the $L^2$-norm of the polynomial when the underlying Gaussian measure's covariance parameter is shifted from $c_0$ to $c_1$. This identity allows the author to relate the growth of the heat-evolved polynomial to the geometry of the target elliptic law.

2. Logarithmic Potential Theory and Confinement:
   To establish weak convergence, the paper utilizes the theory of logarithmic potentials.

   • Confinement: Using the norm identities and global bounds (Lemma 4.3), the author proves that the empirical zero measures are tight and confined to the support of the target measure (an ellipse in the planar case, a real interval in the boundary case).
   • Variational Identification: The proof identifies the limiting measure by comparing the exponential growth rate of the shifted norms (Proposition 4.2) with the upper bound derived from the logarithmic potential of the candidate limit (Lemma 5.1). By showing that the limiting logarithmic potential must agree with the "obstacle" potential of the target measure at its unique maximizers, the author forces the limiting measure to coincide with the target.

3. Free Probability and Determinantal Processes:
   For the boundary case (transitioning to the semicircle law), the paper relies on the additive free convolution. Lemma 8.2 establishes the global law for deformed Gaussian Unitary Ensemble (GUE) matrices using the Pastur equation and Gaussian concentration inequalities. The boundary case further utilizes Chebyshev polynomials and equilibrium measures (Lemma 11.1) to identify limits via positive perturbations.

Key Contributions and Results

• Theorem 1.6 (Planar Case): The paper proves the conjecture for a general class of elliptic Gaussian matrices. It shows that if $p_N$ is the characteristic polynomial of a sum of a deterministic source and an elliptic Gaussian matrix $Z_N(s, \tau_0)$, and $q_N$ is the heat-flow evolution of $p_N$ corresponding to a shift to parameter $\tau_1$, then the empirical zero measure of $q_N$ converges almost surely to the elliptic law $\mu_{S, c_1}$. This matches the limiting spectral measure of the matrix with parameter $\tau_1$.
• Theorem 1.9 (Boundary Case): The result is extended to the boundary of the covariance disk, where the limiting measure becomes the semicircle law. This covers the transition from the complex Ginibre ensemble to the Gaussian Unitary Ensemble (GUE).
• Corollary 1.10 (Ginibre to Semicircle): A specific, distinguished case is proven: the heat flow of the characteristic polynomial of a complex Ginibre matrix converges almost surely to the standard semicircle law. This confirms the specific endpoint described in Hall and Ho's work.
• Generalization to External Sources: The theorems accommodate deterministic external sources (scalar or matrix-valued), proving convergence for a broad class of initial conditions, provided the source converges in $\ast$-distribution.

Significance and Claims
The paper claims to resolve the characteristic-polynomial problem for the Hall–Ho conjecture in general cases, moving beyond the limitations of moment-based proofs. By combining exact shifted Gaussian norm identities with logarithmic potential theory and global confinement arguments, the author establishes that the differential operation of the heat flow on a random polynomial indeed reproduces the deformation of the underlying matrix model at the level of weak convergence.

The work demonstrates the robustness of the heat-flow picture in random matrix theory, specifically handling the strong matrix-induced dependence between coefficients and zeros that prevents the application of simpler independent-coefficient models. The proof relies on the specific structure of elliptic Gaussian matrices and their connection to determinantal point processes and free probability, providing a rigorous foundation for the "model-deformation" phenomenon.

Note: The author explicitly discloses the use of AI tools (ChatGPT Pro 5.5 and ChatGPT 5.6 Sol Ultra) for ideation, technical assistance, and proof checking, while retaining full responsibility for the content and any errors.