# Modulus of continuity of controlled Loewner–Kufarev equations and random matrices

Modulus of continuity of controlled Loewner–Kufarev equations and random matrices First we introduce the two tau-functions which appeared either as the τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau$$\end{document}-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 of such an equation when embedded into the Segal–Wilson Grassmannian. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Modulus of continuity of controlled Loewner–Kufarev equations and random matrices

, Volume 10 (2) – May 8, 2020
29 pages

/lp/springer-journals/modulus-of-continuity-of-controlled-loewner-kufarev-equations-and-7ePMR7ZhQ4
Publisher
Springer Journals
Copyright © Springer Nature Switzerland AG 2020
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-020-00366-3
Publisher site
See Article on Publisher Site

### Abstract

First we introduce the two tau-functions which appeared either as the τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau$$\end{document}-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 of such an equation when embedded into the Segal–Wilson Grassmannian.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: May 8, 2020