Geometric Dyson Brownian Motions and the Free Log-Normal Limit for a Non-Square Product of Random Matrices

We study the squared singular value spectrum of a product of non-square random matrices, a setting that also corresponds to the feature covariance eigenvalues of a deep linear neural network at initialization.

We first take a proportional depth-width $d,n$ limit with the number of data points $m$ held fixed, and show that the resulting covariance eigenvalue process satisfies a geometric version of Dyson Brownian motion.

We then take a second, sequential mean-field limit corresponding to the scaling $dm/n\to\bar\tau$, and show that the limiting $T$-transform of the spectrum solves a Burgers equation.

In the identity-start case this equation yields the free log-normal law, and the general limit is obtained by free multiplicative convolution with the free log-normal.

We further obtain the free log-normal support formula, a fixed-point iteration for numerical evaluation, and a formal small-time Marchenko--Pastur approximation.

We also use the limiting spectral law to predict a toy random-feature regression risk, finding close agreement with a finite-dimensional simulation.