Growth models on the Bethe lattice

TL;DR

This paper investigates various growth models on the Bethe lattice, revealing unique logarithmic scaling behaviors and suggesting the absence of a finite upper critical dimension for the KPZ equation.

Contribution

It provides the first extensive numerical analysis of growth models on the Bethe lattice, classifying their scaling behaviors and implications for the KPZ universality class.

Findings

Logarithmic scaling observed in nonequilibrium models

Scaling exponents depend on model and lattice coordination

No finite upper critical dimension for KPZ suggested

Abstract

I report on an extensive numerical investigation of various discrete growth models describing equilibrium and nonequilibrium interfaces on a substrate of a finite Bethe lattice. An unusual logarithmic scaling behavior is observed for the nonequilibrium models describing the scaling structure of the infinite dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This gives rise to the classification of different growing processes on the Bethe lattice in terms of logarithmic scaling exponents which depend on both the model and the coordination number of the underlying lattice. The equilibrium growth model also exhibits a logarithmic temporal scaling but with an ordinary power law scaling behavior with respect to the appropriately defined lattice size. The results may imply that no finite upper critical dimension exists for the KPZ equation.