Growth models on the Bethe lattice

¹ Department of Physics, University of Tehran - Post Office Box 14395-547, Tehran, Iran
² Institut für Theoretische Physik, Universität zu Köln - Zülpicher Str. 77, 50937 Köln, Germany, EU
³ Institute for Research in Fundamental Sciences (IPM), School of Particles and Accelerators - P.O. Box 19395-5531, Tehran, Iran

^(a) ab.saberi@ut.ac.ir

Received: 14 April 2013
Accepted: 24 June 2013

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.