Baire categorical aspects of first passage percolation. II

DOI10.1007/S10474-019-00943-4zbMATH Open1449.54042arXiv1811.06483OpenAlexW4253815678WikidataQ127944689 ScholiaQ127944689MaRDI QIDQ2338578

Balázs Maga

Publication date: 21 November 2019

Published in: Acta Mathematica Hungarica (Search for Journal in Brave)

Abstract: In this paper we continue our earlier work about topological first passage percolation and answer certain questions asked in our previous paper. Notably, we prove that apart from trivialities, in the generic configuration there exists exactly one geodesic ray, in the sense that we consider two geodesic rays distinct if they only share a finite number of edges. Moreover, we show that in the generic configuration any not too small and not too large convex set arises as the limit of a sequence $fracB(t_n)t_n$ for some $(t_n)oinfty$. Finally, we define topological Hilbert first passage percolation, and amongst others we prove that certain geometric properties of the percolation in the generic configuration guarantee that we consider a setting linearly isomorphic to the ordinary topological first passage percolation.

Full work available at URL: https://arxiv.org/abs/1811.06483

zbMATH Keywords

Hilbert spacemetric spacefirst passage percolationresiduality

Mathematics Subject Classification ID

Metric spaces, metrizability (54E35) Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) (46C05) Baire category, Baire spaces (54E52) Interacting random processes; statistical mechanics type models; percolation theory (60K35)

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