Ergodic limits of random matrices
Spectral distributions for large non-Hermitian ensembles under multiplicative noise; we prove almost-sure convergence to the circular law and quantify the rate of convergence.
1 · The ensembles
We study products of independent random matrices with i.i.d. entries of mean zero and variance one, normalized to have bounded operator norm. Products introduce a multiplicative structure that deforms the classical semicircle picture into something genuinely two-dimensional.
2 · The circular law
The main theorem establishes almost-sure convergence of the empirical spectral measure to the uniform measure on the unit disc, for every fixed number of factors. The proof combines a logarithmic potential estimate with a delicate control of small singular values.
3 · Rates of convergence
We obtain a Kolmogorov distance bound of order N^{−1/2} up to logarithmic losses, matching the Hermitian theory. The rate deteriorates with the number of factors k, and quantifying the deterioration precisely is left open.