Averaging Principle and Shape Theorem for a Growth Model with Memory

2018/12/03 by Dembo, Amir, Groisman, Pablo, Huang, Ruojun +1 · 1 citation
#60K35 #60K37 #82C22 #82C24 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1812.00726

Abstract

We present a general approach to study a class of random growth models in n-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.

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