Modulus of continuity of controlled Loewner-Kufarev equations and random matrices

Area of Science:

• Integrable systems
• Conformal field theory
• Free probability theory

Background:

• The study of tau-functions is crucial in integrable hierarchies and conformal field theory.
• Existing research links conformal maps with large N-matrix integrals.

Purpose of the Study:

• Introduce two specific tau-functions and explore their interrelations.
• Establish a novel connection between free probability, growth models, and integrable systems.
• Extend existing links to higher-order free probability.

Main Methods:

• Analysis of tau-functions in the context of integrable hierarchies and conformal field theory.
• Development of a dictionary to summarize connections between free probability, growth models, and integrable systems.
• Application of controlled Loewner-Kufarev equations within the Segal-Wilson Grassmannian framework.

Main Results:

• Introduction of two tau-functions relevant to Jordan curve Riemann mapping and conformal field theory.
• Establishment of a novel dictionary-summarized link between free probability (specifically second-order freeness), growth models, and integrable systems.
• Extension of the connection between conformal maps and large N-matrix integrals to higher-order free probability.
• Determination of driving functions for controlled Loewner-Kufarev equations, yielding continuity estimates for solutions in the Segal-Wilson Grassmannian.

Conclusions:

• The interrelation of tau-functions, free probability, growth models, and integrable systems offers new insights.
• The findings extend the applicability of free probability concepts to complex systems.
• The study provides a framework for analyzing dynamically evolving contours in mathematical physics.