arXiv:2111.08522v1 [math.PR] 16 Nov 2021 Perturbations of Simultaneously Growing Multiple Schramm-Loewner Evolutions Jiaming Chen * and Vlad Margarint † November 2021 Abstract In this article we study multiple SLEκ, for κ ∈ (0, 4], driven by Dyson Brownian motion. This model was introduced in the unit disk by Cardy [18] 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 κ ∈ (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. 1 Introduction The forward multiple Loewner chain encodes the dynamics of a family of con- formal maps g t (z ) defined on simply connected domains H\K t of the upper-half plane H, where K t are growing hulls ([5] Sec. 4.1.2) in the sense that K s ⊂ K t for all 0 ≤ s ≤ t. In this work we study a Loewner chain generated by N ∈ N continuous driving forces {λ 1 (t),λ 2 (t),...,λ N (t)} from R to R. We denote these driving functions by λ j : [0,T ] → R, j =1,...,N . We have ∂ t g t (z )= 1 N N  j=1 2 g t (z ) − λ j (t) , (1.1) * ETH Z¨ urich; jiamchen@student.ethz.ch † NYU Shanghai; vdm2@nyu.edu 1