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Differentiability and monotonicity of expected passage time in Euclidean first-passage percolation

Part of: Equilibrium statistical mechanics Applications to specific types of physical systems Limit theorems Special processes

Published online by Cambridge University Press:  14 July 2016

C. Douglas Howard
C. Douglas Howard*
Affiliation:
Baruch College, The City University of New York
*
∗ Postal address: Baruch College, The City University of New York, Mathematics Department, 17 Lexington Avenue, New York, NY 10010, USA. Email address: [email protected]
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Abstract

In first-passage percolation (FPP) models, the passage time Tℓ from the origin to the point ℓeℓ satisfies f(ℓ) := ETℓ = μℓ + o(ℓ½+ε), where μ ∊ (0,∞) denotes the time constant. Yet, for lattice FPP, it is not known rigorously that f(ℓ) is eventually monotonically increasing. Here, for the Poisson-based Euclidean FPP of Howard and Newman (Prob. Theory Relat. Fields108 (1997), 153–170), we prove an explicit formula for f′(ℓ). In all dimensions, for certain values of the model's only parameter we have f′(ℓ) ≥ C > 0 for large ℓ.

Keywords

First-passage percolationpassage timemonotonicity

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82B21: Continuum models (systems of particles, etc.) 82B43: Percolation
Secondary: 60F10: Large deviations 82D30: Random media, disordered materials (including liquid crystals and spin glasses)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 815 - 827
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

Research supported in part by NSF Grant DMS-98-15226 and a Eugene Lang Research Fellowship.

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