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Transitions for exceptional times in dynamical first-passage percolation

Abstract

In first-passage percolation (FPP), we let \((\tau _v)\) be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \(\tau _v\), there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. We study a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We prove that if \(\sum F^{-1}(1/2+1/2^k) = \infty \), then a.s. there are exceptional times at which the weight grows atypically, but if \(\sum k^{7/8} F^{-1}(1/2+1/2^k) <\infty \), then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

J. H. thanks Georgia Tech for hospitality during visits related to this work. The authors thank an anonymous referee for their comments that led to an improved presentation.

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Correspondence to Michael Damron.

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M. Damron: The research of M. D. is supported by an NSF Grant DMS-2054559 and an NSF CAREER award. J. Hanson: The research of J. H. is supported by NSF Grants DMS-1612921, DMS-1954257, a PSC-CUNY grant and a CUNY JFRASE award via the CUNY Office of Research and the Sloan Foundation. D. Harper: The research of D. H. is partially supported by NSF Grant DMS-2054559.

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Damron, M., Hanson, J., Harper , D. et al. Transitions for exceptional times in dynamical first-passage percolation. Probab. Theory Relat. Fields 185, 1039–1085 (2023). https://doi.org/10.1007/s00440-022-01178-1

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Mathematics Subject Classification

  • Primary 60K35
  • Secondary 82B43