Properties of first passage percolation above the (hypothetical) critical dimension

2023/04/18 by Kenneth S. Alexander, Alexander, Kenneth S.
Mathematics · Physics and Astronomy · #60K35 (Primary) 92B43 (Secondary) #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2304.08669

openalex publication_date 2023/04/18 · openalex created_date 2023/04/22 · openalex updated_date 2026/07/28

Abstract

It is not known (and even physicists disagree) whether first passage percolation (FPP) on ℤ^d has an upper critical dimension d_c, such that the fluctuation exponent χ=0 in dimensions d>d_c. In part to facilitate study of this question, we may nonetheless try to understand properties of FPP in such dimensions should they exist, in particular how they should differ from d0 must be false if χ=0. A particular one of the three is most plausible to fail, and we explore the consequences if it is indeed false. These consequences support the idea that when χ=0, passage times are ``local'' in the sense that the passage time from x to y is primarily determined by the configuration near x and y. Such locality is manifested by certain ``disc--to--disc'' passage times, between discs in parallel hyperplanes, being typically much faster than the fastest mean passage time between points in the two discs.

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