Year: 2026
Type: preprint
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 $M/\log N\to\infty$ and $M/N\to0$. 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 $N,\ldots,N+M-1$ 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.
Source: arXiv (Cornell University)
Author: Alekseev, Oleg
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220742659