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

Abstract

First we introduce the two tau-functions which appeared either as the τ -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.

Univalence and holomorphic extension of the solution to $ω$-controlled Loewner-Kufarev equations

“…The assumptions on the driving functions are mentioned in Section 2.1, and then the definition of a solution to controlled Loewner-Kufarev equation is given in Section 2.2. Although these concepts appear in the previous papers [1,2], we summarize them to confirm the terminology. In Section 3, we prove Theorem 1.1 through four steps, namely, the uniqueness (Theorem 3.1), existence (Theorem 3.2), holomorphic extension (Corollary 3.…”

“…It should be noted that, in the paper [2], they assume a priori that f t is univalent on D and extends to a holomorphic function on an open neighbourhood of D. However, we expect that this property is intrinsic to a large class of controlled Loewner-Kufarev equations. The purpose of the present article is to confirm this belief in the ω-controlled case.…”

“…As is described in [1, Section 3.2], the solution (f t ) 0≤t≤T is embedded into the Segal-Wilson Grassmannian through Krichever's construction if it is univalent on the unit disk D and extends holomorphically across ∂D. Moreover, the first two authors gave a sufficient condition [2,Theorem 3.3] for the embedded solution to be continuous as a curve in the Grassmannian. In this theorem, they consider the case in which the driving functions x 0 and x are controlled by a control function ω :…”