Free Convolution and Generalized Dyson Brownian Motion
Pierre Bousseyroux, Jean-Philippe Bouchaud
TL;DR
The paper addresses how freeness and the additivity of $R$-transforms for large free matrices translate into a concrete dynamical description of eigenvalues. By deriving a generalized Dyson Brownian motion with higher-body interactions and leveraging the Stieltjes transform and subordination, it produces a hydrodynamic evolution for the eigenvalue density: $ rac{ d}{ d t} ho(\xi,t) = - rac{ d}{ d \xi}ig[ ho(\xi,t)F(\xi,t)ig]$, where $F$ encodes infinite cumulant contributions. The main contributions include an explicit velocity formula $F_i(\lambda_i,t) = rac{ d}{ d t} oldsymbol{ m B}(t) ext{ terms} imes rak{h}_{n-1}(\lambda_i,t)$ with generalized Hilbert transforms $ rak{h}_n$, a demonstration that this dynamics recovers the additive $R$-transform law in the large-$N$ limit, and a perturbative correspondence up to order $t^4$ that connects with standard quantum perturbation theory. The results offer a physically intuitive, dynamical interpretation of free additive and multiplicative calculus, with potential implications for fields where large random matrices and outlier formation are relevant, such as physics, finance, and complex systems.
Abstract
The eigenvalue spectrum of the sum of large random matrices that are mutually "free", i.e., randomly rotated, can be obtained using the formalism of R-transforms, with many applications in different fields. We provide a direct interpretation of the otherwise abstract additivity property of R-transforms for the sum in terms of a dynamical evolution of "particles" (the eigenvalues), interacting through two-body and higher-body forces and subject to a Gaussian noise, generalizing the usual Dyson Brownian motion with Coulomb interaction. Interestingly, the appearance of an outlier outside of the bulk of the spectrum is signalled by a divergence of the "velocity" of the generalized Dyson motion. We extend our result to products of free matrices.