Symmetric Polynomials Method for the Multiple Loewner Chains

We introduce the technique of Symmetric Polynomials in the study of Loewner chains driven by a family of stochastic particle systems including but not limited to Dyson Brownian motion. As a first application of this technique we obtain the phase transitions at the critical parameter $\kappa =4$ in terms of the behavior of a class of symmetric polynomials evolving in time. We obtain new proofs of the phase transitions, we also obtain new closed-form expressions relevant for the models. In addition, we apply this method for some initial deterministic multiple Loewner evolutions. Finally, as this is, to our knowledge, the first paper analyzing this method in the context of multiple Loewner chains driven by root-system based multivariate processes, we also give an overview of potential future applications of this technique in the study of these models.