Skip to main content
We gratefully acknowledge support from the Simons Foundation, member institutions, and all contributors. Donate
arXiv logo
Cornell University Logo

Mathematics > Probability

arXiv:1812.00726 (math)
[Submitted on 3 Dec 2018 (v1), last revised 19 Aug 2020 (this version, v2)]

Title:Averaging Principle and Shape Theorem for a Growth Model with Memory

Authors:Amir Dembo, Pablo Groisman, Ruojun Huang, Vladas Sidoravicius
View PDF
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.
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
Subjects: Probability (math.PR)
MSC classes: 60K35, 60K37, 82C22, 82C24
Cite as: arXiv:1812.00726 [math.PR]
  (or arXiv:1812.00726v2 [math.PR] for this version)
  https://doi.org/10.48550/arXiv.1812.00726

Submission history

From: Pablo Groisman [view email]
[v1] Mon, 3 Dec 2018 13:33:14 UTC (2,524 KB)
[v2] Wed, 19 Aug 2020 03:33:13 UTC (2,527 KB)
Full-text links:

Access Paper:

  • View PDF
  • TeX Source
  • Other Formats
view license
Current browse context:
math.PR
< prev   |   next >
new | recent | 2018-12
Change to browse by:
math

References & Citations

export BibTeX citation

Bookmark

BibSonomy logo Reddit logo

Bibliographic and Citation Tools

Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)

Code, Data and Media Associated with this Article

alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)

Demos

Replicate (What is Replicate?)
Hugging Face Spaces (What is Spaces?)
TXYZ.AI (What is TXYZ.AI?)

Recommenders and Search Tools

Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.

Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)