Averaging Principle and Shape Theorem for a Growth Model with Memory
Abstract
We present a general approach to study a class of random growth models in -dimensional Euclidean space. These models are designed to capture basic growth features which are expected to manifest at the mesoscopic level for several classical self-interacting processes originally defined at the microscopic scale. It includes once-reinforced random walk with strong reinforcement, origin-excited random walk, and few others, for which the set of visited vertices is expected to form a "limiting shape". We prove an averaging principle that leads to such shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain.
Cite
@article{arxiv.1812.00726,
title = {Averaging Principle and Shape Theorem for a Growth Model with Memory},
author = {Amir Dembo and Pablo Groisman and Ruojun Huang and Vladas Sidoravicius},
journal= {arXiv preprint arXiv:1812.00726},
year = {2020}
}
Comments
To appear in Comm. Pure Appl. Math.; Fixed earlier error in proof of Prop. 1.6(a) by adding to Assumption (L) the boundedness of F and the case p=infty in (1.15); In Prop. 1.6, parts (a) and (b) revised and new results added as parts (d) and (e). Example from Sec. 4.2 now included in Thm. 1.10, where the case H=0 revised as part (c). Proofs in Sec. 2 and Sec. 4 have been reorganized