Models of First-Passage Percolation
First-passage percolation (FPP) was introduced by Hammersley and Welsh in 1965 (see [26]) as a model of fluid flow through a randomly porous material. Envision a fluid injected into the material at a fixed site: as time elapses, the portion of the materia
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1
Introduction ............................................... 126
1.1 1.2
The Basic Model and Some Fundamental Questions ................. 126 Notation .................................................... 128
2
The Time Constant ......................................... 129
2.1 2.2 2.3
The Fundamental Processes of Hammersley and Welsh ............... 129 About p, .......•..........................•...........•...•. 131 Minimizing Paths ............................................. 133
3
Asymptotic Shape and Shape Fluctuations ................... 134
3.1 3.2 3.3 3.4 3.5 3.6
Shape Theorems for Standard FPP .............................. About the Asymptotic Shape for Lattice FPP ...................... FPP Based on Poisson Point Processes ........................... Upper Bounds on Shape Fluctuations .. , ......................... Some Related Longitudinal Fluctuation Exponents .................. Monotonicity ................................................
4
Transversal Fluctuations and the Divergence of Shape Fluctuations ....................................... 154
4.1 4.2 4.3 4.4 4.5
Transversal Fluctuation Exponents .............................. Upper Bounds on~ ........................................... Lower Bounds on x ........................................... Lower Bounds on ~ ........................................... Fluctuations for Other Related Models ...........................
5
Infinite Geodesics and Spanning Trees ....................... 161
5.1 5.2 5.3
Semi-Infinite Geodesics and Spanning Trees ....................... 161 Coalescence and Another Spanning Tree in 2 Dimensions ............. 165 Doubly-Infinite Geodesics ...................................... 167
6
Summary of Some Open Problems ........................... 168
134 138 140 143 150 151
154 155 157 158 160
References ..................................................... 170 * Research supported by NSF Grant DMS-02-03943. H. Kesten (ed.), Probability on Discrete Structures © Springer-Verlag Berlin Heidelberg 2004
126
C. Douglas Howard
1 Introduction 1.1 The Basic Model and Some Fundamental Questions First-passage percolation (FPP) was introduced by Hammersley and Welsh in 1965 (see [26]) as a model of fluid flow through a randomly porous material. Envision a fluid injected into the material at a fixed site: as time elapses, the portion of the material that is wet expands in a manner that is a complicated function of the material's random structure. In the standard FPP model, the spatial randomness of the material is represented by a family of non-negative i.i.d. random variables indexed by the nearest neighbor edges of the zd lattice. (We take d :::0: 2 throughout this chapter.) If edge e has endpoints u, v E zd (so In- vi = 1, where 1·1 denotes the usual Euclidean norm) then the associated quantity r(e) represents the time it takes fluid to flow from site u to site v, or the reverse, along the edge e. If the sequence of edges r = (e1, ... , en) forms a path from u E zd to v E zd, then T(r) = Li r(ei) repres
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