Numerical Integration of the KPZ and Related Equations on Networks: The Case of the Cayley Tree

TL;DR

This paper develops numerical methods to simulate nonlinear stochastic surface growth equations on Cayley trees, revealing how network topology affects interface dynamics and scaling, with results aligning with prior discrete models.

Contribution

It introduces and compares numerical schemes for integrating KPZ and related equations on Cayley trees, advancing understanding of growth processes on non-Euclidean networks.

Findings

Good agreement with previous discrete model results

Network topology influences interface growth behavior

Boundary effects impact scaling properties

Abstract

The numerical integration of stochastic growth equations on non-Euclidean networks presents unique challenges due to the nonlinearities that occur in many relevant models and of the structural constraints of the networks. In this work, we integrate the KPZ, Edwards-Wilkinson, and tensionless KPZ equations on Cayley trees using different numerical schemes and compare their behavior with previous results obtained for discrete growth models. By assessing the stability and accuracy of these methods, we explore how network topology influences interface growth and how boundary effects shape the observed scaling properties. Our results show good agreement with previous studies on discrete models, reinforcing key scaling behaviors while highlighting some differences. These findings contribute to a better understanding of surface growth on networked substrates and provide a computational framework for studying nonlinear stochastic processes beyond Euclidean lattices.