q-deformed Loewner evolution

TL;DR

This paper introduces a q-deformed version of the Loewner evolution, breaking the Markov property while maintaining scale invariance, and explores its solutions, simulations, and relation to classical models.

Contribution

It proposes a novel q-calculus based generalization of Loewner evolution, extending the framework to include non-Markovian, scale-invariant stochastic processes.

Findings

Exact solutions for the deterministic q-Loewner equation

A simulation method for q-deformed Loewner evolution

Convergence analysis to classical Loewner evolution as q approaches 1

Abstract

The Loewner equation, in its stochastic incarnation introduced by Schramm, is an insightful method for the description of critical random curves and interfaces in two-dimensional statistical mechanics. Two features are crucial, namely conformal invariance and a conformal version of the Markov property. Extensions of the equation have been explored in various directions, in order to expand the reach of such a powerful method. We propose a new generalization based on q-calculus, a concept rooted in quantum geometry and non-extensive thermodynamics; the main motivation is the explicit breaking of the Markov property, while retaining scale invariance in the stochastic version. We focus on the deterministic equation and give some exact solutions; the formalism naturally gives rise to multiple mutually-intersecting curves. A general method of simulation is constructed - which can be easily extended to other q-deformed equations - and is applied to both the deterministic and the stochastic realms. The way the $q\neq 1$ picture converges to the classical one is explored as well.