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Averaging Principle and Shape Theorem for a Growth Model with Memory
Author(s) -
Dembo Amir,
Groisman Pablo,
Huang Ruojun,
Sidoravicius Vladas
Publication year - 2021
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.21965
Subject(s) - mathematics , limiting , random walk , markov chain , euclidean geometry , mesoscopic physics , euclidean space , class (philosophy) , statistical physics , markov process , pure mathematics , discrete mathematics , geometry , statistics , computer science , physics , quantum mechanics , artificial intelligence , engineering , mechanical engineering
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 that 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 a 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 a shape theorem. The limiting shape can be computed in terms of the invariant measure of an associated Markov chain. © 2020 Wiley Periodicals LLC