# Stochastic Loewner Evolutions, Fuchsian Systems and Orthogonal Polynomials

## TL;DR

This paper links Levy-Loewner evolutions to eigenvalues of matrices and Fuchsian systems, providing a new way to analyze the integral means spectrum and derivatives of conformal maps.

## Contribution

It introduces a class of Levy-Loewner evolutions where the beta-spectrum at q=2 is characterized by matrix eigenvalues and connects derivatives of conformal maps to Fuchsian system solutions.

## Key findings

- Beta-spectrum at q=2 equals the maximal eigenvalue of a specific matrix.
- Second moments of derivatives are expressed via Fuchsian system solutions.
- Provides a new analytical framework for studying Levy-Loewner evolutions.

## Abstract

We find a wide class of Levy-Loewner evolutions for which the value of integral means beta-spectrum $\beta(q)$ at $q=2$ is the maximal real eigenvalue of a three-diagonal matrix. The second moments of derivatives of corresponding conformal mappings are expressed through solutions of matrix Fuchsian systems with three singular points.