Perturbations of Multiple Schramm-Loewner Evolution with Two Non-colliding Dyson Brownian Motions

TL;DR

This paper investigates the behavior of multiple SLE processes driven by Dyson Brownian motion, proving convergence and estimating hull distances, thus advancing understanding of complex stochastic Loewner evolutions with multiple forces.

Contribution

It establishes Carathéodory convergence for perturbed multiple SLEs driven by Dyson Brownian motion with two forces, using Bessel process analysis and Loewner differential equation estimates.

Findings

Proved convergence of perturbed Loewner chains for two driving forces.

Estimated Hausdorff distance between hulls under perturbations.

Analyzed the impact of initial conditions and diffusivity on the evolution.

Abstract

In this article, we study multiple $SLE_\kappa$, for $\kappa\in(0,4]$, driven by Dyson Brownian motion. This model was introduced in the unit disk by Cardy in connection with the Calogero-Sutherland model. We prove the Carath\'eodory convergence of perturbed Loewner chains under different initial conditions and under different diffusivity $\kappa \in (0,4]$ for the case of $N=2$ driving forces. Our proofs use the analysis of Bessel processes and estimates on Loewner differential equation with multiple driving forces. In the last section, we estimate the Hausdorff distance of the hulls under perturbations of the driving forces, with assumptions on the modulus of the derivative of the multiple Loewner maps.