Stochastic Loewner Evolutions, Fuchsian Systems and Orthogonal Polynomials

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.