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Quenched Growth of Normal Matrices and Harmonic Measure

This paper establishes that when sampling additional eigenvalues from a large normal random matrix conditioned on a fixed configuration, their spatial distribution converges to the harmonic measure on the boundary of the existing spectral droplet in the limit where the number of new eigenvalues grows faster than the logarithm of the original size but remains negligible compared to it.

Original authors: Oleg Alekseev

Published 2026-10-06

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Oleg Alekseev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how random systems settle into order. Imagine a collection of points scattered across a flat surface, where each point pushes away from the others, yet all are gently pulled toward a central region by an invisible force. This is the world of random matrices, specifically those where the points represent eigenvalues—special numbers that describe the behavior of complex systems. When these points are numerous, they do not scatter randomly; instead, they crowd together to form a solid, compact shape, often called a droplet. The way this droplet grows as more points are added is a subject of intense study, linking pure mathematics to physical phenomena like the flow of electricity or the growth of crystals. A key concept in this field is the idea of "harmonic measure," which describes how a new point is most likely to attach to the edge of an existing shape. It is a rule of probability that depends entirely on the geometry of the boundary, acting like a map that tells us where new arrivals are most welcome.

A recent study by Oleg Alekseev explores a specific and challenging question about this growth: what happens if we freeze the existing points in place and ask how new ones will arrange themselves around them? In the standard model of growth, scientists compare a system with a certain number of points to a slightly larger system where every single point is re-sampled from scratch. This is like watching a crowd grow by replacing everyone with new people and seeing how the new crowd settles. Alekseev, however, investigates a different scenario known as "quenched growth." Here, the original points are locked in their positions, acting as a fixed, unchanging landscape of obstacles. New points are then added to this frozen field, forced to navigate the repulsion from the old points while also pushing away from each other. The central question is whether this rigid, frozen environment changes the final shape of the droplet's edge, or if the new points still find their way to the same distribution as they would in the fluid, re-sampled world.

The research confirms that, under specific conditions, the answer is surprisingly consistent. When the number of new points is large enough to be statistically significant but small compared to the massive number of frozen points, the new arrivals distribute themselves along the boundary in a way that matches the harmonic measure perfectly. This means that even though the new points are reacting to a specific, frozen arrangement of old points, the randomness of that arrangement averages out. The new points do not get stuck in the gaps or cluster in the valleys of the frozen landscape; instead, they spread out along the edge exactly as if the old points were not frozen at all. This result holds true for almost every possible configuration of the frozen points, with only a vanishingly small chance of an unusual arrangement causing a deviation. The study proves that the "frozen disorder" does not alter the fundamental law of growth for the boundary.

To reach this conclusion, the author had to navigate a complex mathematical landscape involving the interaction of thousands of points. The new points are governed by a set of rules that make them repel one another, similar to how magnets with the same pole push apart. They also repel the frozen points, creating a complex energy landscape. The study shows that for a droplet with a smooth, analytic boundary, the new points are drawn to the edge with a density that depends only on the shape of the boundary, not the specific details of the frozen points inside. This was demonstrated by comparing two methods: one where the entire system is re-sampled, and the "quenched" method where the old points are fixed. While the two methods produce different results when the number of points is small, they converge to the same limit as the system grows. The new points in the frozen scenario eventually mimic the distribution of the re-sampled scenario, proving that the microscopic details of the frozen field wash out at the macroscopic scale.

The paper also delves into the differences that remain at finite scales, where the number of points is large but not infinite. In these simulations, the frozen scenario shows slightly different behavior than the re-sampled one. The new points in the frozen case exhibit weaker repulsion from each other in terms of their angles around the circle, and they show larger fluctuations in how many points land in specific sections of the boundary. This suggests that while the average distribution is the same, the "texture" of the growth is different. The frozen points create a specific, unchanging terrain that the new points must navigate, leading to a different kind of local order. However, as the number of new points increases, these differences diminish, and the system settles into the universal pattern predicted by the harmonic measure.

The study uses a specific shape with three-fold symmetry to illustrate these findings, allowing for precise numerical comparisons. In these simulations, the new points were observed to cluster near the boundary, with their density matching the theoretical prediction. The researchers measured how far the new points strayed from the boundary and found that they stayed close, with the fraction of points far from the edge decreasing as the batch size grew. This confirms that the new points are indeed confined to the edge of the droplet, forming a thin layer that follows the harmonic measure. The work also connects these findings to broader concepts in physics, such as diffusion-limited aggregation, where particles stick to a growing cluster. The results suggest that even in a system with fixed obstacles, the growth of the boundary follows a predictable, universal law.

Ultimately, this research provides a rigorous proof that the "quenched" growth of these random systems is governed by the same harmonic measure as the standard equilibrium growth. It shows that the specific arrangement of the frozen points, no matter how complex or irregular, does not prevent the new points from finding their way to the correct distribution along the boundary. The study bridges the gap between the microscopic interactions of individual points and the macroscopic laws that govern the shape of the entire system. It confirms that in the limit of large numbers, the memory of the specific frozen configuration is lost, and the system obeys a simple, elegant rule determined solely by the geometry of the boundary. This insight deepens our understanding of how order emerges from randomness in complex systems, showing that even when the past is frozen in place, the future still follows a predictable path.

Technical Summary: Quenched Growth of Normal Matrices and Harmonic Measure

Problem Statement
The paper investigates the asymptotic behavior of eigenvalues in large normal random matrices, which form a two-dimensional Coulomb gas. Specifically, it addresses the "quenched growth" scenario: a configuration of $N$ eigenvalues (the "frozen" spectrum) is fixed, and $M$ additional eigenvalues are sampled conditional on this configuration. The central question is whether the spatial distribution of these new eigenvalues converges to the exterior harmonic measure on the boundary of the original spectral droplet, analogous to the known result for "equilibrium growth" where all eigenvalues are resampled.

The study focuses on the regime where the matrix rank $N$ tends to infinity, and the batch size $M$ satisfies $\log N \ll M \ll N$ . The work compares two growth rules:

Equilibrium Growth: Comparing rank- $N$ and rank- $(N+M)$ ensembles at the same weight, where all eigenvalues are sampled anew. The increase in density is driven by the "orbital band" of orthogonal polynomials of degrees $N, \dots, N+M-1$ .
Quenched Growth: Holding the rank- $N$ spectrum $Y = \{y_1, \dots, y_N\}$ fixed and sampling $M$ new points $Z = \{z_1, \dots, z_M\}$ from the conditional law defined by projecting onto polynomials of degree $< N+M$ that vanish at the points in $Y$ .

Methodology
The author employs tools from random matrix theory, potential theory, and orthogonal polynomials.

Mathematical Formulation: The eigenvalue laws are described as determinantal point processes (DPPs). For equilibrium growth, the process is defined by the projection $\Pi_{\text{band}} = \Pi_{N+M} - \Pi_N$ . For quenched growth, the conditional law is a DPP defined by the projection $\Pi^\perp_Y = \Pi_{N+M} - P_Y$ , where $P_Y$ projects onto the span of evaluation functionals at the frozen points $Y$ .
Conditional Density: Using Proposition 2.2, the joint density of the new points $Z$ given $Y$ is shown to be an exact polynomial ensemble:
$P(dZ|Y) \propto |V(Z)|^2 \prod_{a=1}^M |q_Y(z_a)|^2 e^{-2NQ(z_a)} dA(z_a)$
where $q_Y(z) = \prod_{i=1}^N (z-y_i)$ and $V$ is the Vandermonde determinant. This represents a Coulomb gas in the external potential of the frozen charges.
Proof Strategy (Theorem 3.1): The proof of convergence to harmonic measure relies on an energy comparison argument:
1. Normalization Bounds: Lower bounds are established for the Gram determinant of the conditional basis using asymptotics of orthogonal polynomial norms (Section 4.1).
2. Frozen Field Control: For typical frozen spectra, the mean squared modulus of the frozen polynomial $q_Y$ is bounded near the droplet (Section 4.2).
3. Logarithmic Energy Gap: It is shown that configurations deviating from harmonic measure have a strictly lower logarithmic interaction energy, leading to exponential suppression on the scale of $M^2$ (Section 4.3).
4. Confinement: Estimates ensure that new eigenvalues remain close to the droplet boundary with high probability (Section 4.4).
5. Combination: These estimates are combined to show that the conditional probability of deviation from harmonic measure vanishes as $N \to \infty$ , provided $M/\log N \to \infty$ and $M/N \to 0$ .

Key Contributions and Results

Quenched Harmonic Measure Theorem (Theorem 3.1): The primary result establishes that for a regular analytic droplet, the empirical measure of a quenched batch of $M$ eigenvalues converges weakly to the exterior harmonic measure $\omega_D$ on the boundary of the frozen spectrum. This convergence holds conditionally on the frozen spectrum, except for a set of exceptional configurations whose probability tends to zero.
Equivalence of Growth Limits (Corollary 3.3): The paper proves that both the equilibrium orbital band and the quenched batch share the same limiting spatial distribution (harmonic measure) in the regime $\log N \ll M \ll N$ . This confirms that the "frozen disorder" of the old spectrum does not alter the macroscopic boundary distribution of the new eigenvalues.
Finite- $N$ Differences: While the limits are identical, the paper demonstrates that the laws differ at finite $N$ $N$ .
- Proposition 5.2: An exact finite-rank identity relates the difference between the conditional mean moments of the quenched batch and the equilibrium band to the fluctuations of the frozen spectrum and a remainder term involving polynomial modes of degree $\ge N+M$ .
- Numerical Analysis (Section 5): Using a three-fold symmetric droplet, the author shows that quenched batches exhibit weaker angular repulsion and larger fluctuations in arc counts and low angular Fourier modes compared to the equilibrium band. The structure factor decomposition reveals that these excess fluctuations arise from both the conditional mean and the variance of batches sampled at a fixed spectrum.
CUE Limit for Equilibrium Growth (Proposition 2.1): The paper provides a proof that the joint law of conformal angles for the equilibrium orbital band converges to the Circular Unitary Ensemble (CUE) eigenangle density as $N \to \infty$ with fixed $M$ .

Significance
The paper provides a conditional counterpart to the established equilibrium law for Laplacian growth. While previous work (e.g., Agam et al., Hedenmalm and Makarov) described the harmonic measure growth for equilibrium ensembles, this work extends the result to the case where the observed spectrum is fixed. This connects the growth of normal matrices to conditional point processes and clarifies the role of "frozen disorder" in Coulomb gas systems.

The results are significant for understanding the microscopic structure of random matrix ensembles. The numerical findings highlight that while macroscopic distributions converge to the same limit, the microscopic correlations (angular repulsion and number variance) retain memory of the growth rule (quenched vs. equilibrium) at finite $N$ . The paper modestly suggests that establishing the limiting law of angular fluctuations and extending these results to evolving configurations (stochastic Laplacian growth) remain open questions requiring further quantitative control.

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