Quenched Growth of Normal Matrices and Harmonic Measure
Abstract
The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an N-eigenvalue configuration and sample M further eigenvalues using the projection onto polynomials of degree less than N+M that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when M/log N→∞ and M/N→0. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees N,…,N+M−1 has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree N+M and above.
Related
- Conditioned random walks on linear groups I: construction of the target harmonic measure
- Singularity of harmonic measure for finitely supported random walks
- Beyond normal fluctuations in local laws for Wigner matrices
- Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
- Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices
BibTeX
@misc{alekseev2026quenched,
title = {{Quenched Growth of Normal Matrices and Harmonic Measure}},
author = {Oleg Alekseev},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03912},
eprint = {2610.03912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03912},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quenched Growth of Normal Matrices and Harmonic Measure
AU - Oleg Alekseev
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03912
UR - https://arxiv.org/abs/2610.03912
AB - The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an N-eigenvalue configuration and sample M further eigenvalues using the projection onto polynomials of degree less than N+M that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when M/log N→∞ and M/N→0. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees N,…,N+M−1 has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree N+M and above.
ER -