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 of such an equation when embedded into the Segal–Wilson Grassmannian.
Highlights
The class of univalent functions is an extraordinarily rich mathematical object within the field of complex variables, with deep and surprising connections with, e.g. conformal field theory (CFT), random matrix theory and integrable systems, cf. [6,7,8,9,15,19,21,23], just to name the most important ones in our context
This suggests us to make a dictionary translating the language from integrable systems and free probability
In the previous paper [1], the authors introduced the notion of a solution to the controlled Loewner–Kufarev equation
Summary
The class of univalent functions is an extraordinarily rich mathematical object within the field of complex variables, with deep and surprising connections with, e.g. conformal field theory (CFT), random matrix theory and integrable systems, cf. [6,7,8,9,15,19,21,23], just to name the most important ones in our context. The set C of all such Jordan contours encircling the origin forms an infinite dimensional manifold [10,19]. It has been shown by Kirillov and Juriev [9] that there exists a canonical bijection. And ∂n is the normal derivative on the boundary ∂ Dc with respect to ξ ∈ ∂ Dc, and G0(x, ξ ) is the Dirichlet Green function associated to the Dirichlet problem in Dc. Krichever, Marshakov, Mineev-Weinstein, Wiegmann and Zabrodin [8,14,23], in different constellations, defined the logarithm of a τ -function which for a contour C = ∂ D is given by [8,19]. The τ -function connects complex analysis with the dispersionless hierarchies and integrable systems [8]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
