Modulus of Continuity of Controlled Loewner-Kufarev Equations and Random Matrices

TL;DR

This paper explores the interplay between conformal maps, free probability, and integrable systems, establishing new connections and providing continuity estimates for solutions to controlled Loewner-Kufarev equations within the context of random matrices.

Contribution

It introduces new tau-functions related to conformal maps and free probability, and establishes a novel connection between these areas and integrable systems, including continuity estimates for Loewner-Kufarev equations.

Findings

Established a connection between free probability, growth models, and integrable systems.

Determined a class of driving functions for controlled Loewner-Kufarev equations.

Provided a continuity estimate for solutions embedded in the Segal-Wilson Grassmannian.

Abstract

First we introduce the two tau-functions which appeared either as the $\tau$-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large $N$-matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner-Kufarev equations, which enables us to give a continuity estimate for the solution to such equations when embedded into the Segal-Wilson Grassmannian.