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2021 | OriginalPaper | Chapter

Existence and Coexistence in First-Passage Percolation

Author : Daniel Ahlberg

Published in: In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius

Publisher: Springer International Publishing

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Abstract

We consider first-passage percolation with i.i.d. non-negative weights coming from some continuous distribution under a moment condition. We review recent results in the study of geodesics in first-passage percolation and study their implications for the multi-type Richardson model. In two dimensions this establishes a dual relation between the existence of infinite geodesics and coexistence among competing types. The argument amounts to making precise the heuristic that infinite geodesics can be thought of as ‘highways to infinity’. We explain the limitations of the current techniques by presenting a partial result in dimensions d > 2.

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next chapter Ground State Stability in Two Spin Glass Models

Consider the sequence of finite geodesics between the origin and n e ₁, where e ₁ denotes the first coordinate vector. Since the number of edges that connect to the origin is finite, one of them must be traversed for infinitely many n. Repeating the argument results in an infinite path which by construction is a geodesic.

This will be referred to as having unique passage times.

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Title: Existence and Coexistence in First-Passage Percolation
Author: Daniel Ahlberg
Publisher: Springer International Publishing
Book: In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius
Print ISBN: 978-3-030-60753-1

Electronic ISBN: 978-3-030-60754-8

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-60754-8_1