TY - JOUR
T1 - Multiple Phase Transitions in Long-Range First-Passage Percolation on Square Lattices
AU - Chatterjee, Shirshendu
AU - Dey, Partha S.
N1 - Publisher Copyright:
© 2015 Wiley Periodicals, Inc.
PY - 2016/2/1
Y1 - 2016/2/1
N2 - We consider a model of long-range first-passage percolation on the d-dimensional square lattice Zd in which any two distinct vertices x,y∈Zd are connected by an edge having exponentially distributed passage time with mean x-y α+o(1), where α>0 is a fixed parameter and . is the l1-norm on Zd. We analyze the asymptotic growth rate of the set ßt, which consists of all x∈Zd such that the first-passage time between the origin 0 and x is at most t as t→∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α<d, (ii) stretched exponential growth for α∈(d,2d), (iii) superlinear growth for α∈(2d,2d+1), and finally (iv) linear growth for α>2d+1 like the nearest-neighbor first-passage percolation model corresponding to α=∞.
AB - We consider a model of long-range first-passage percolation on the d-dimensional square lattice Zd in which any two distinct vertices x,y∈Zd are connected by an edge having exponentially distributed passage time with mean x-y α+o(1), where α>0 is a fixed parameter and . is the l1-norm on Zd. We analyze the asymptotic growth rate of the set ßt, which consists of all x∈Zd such that the first-passage time between the origin 0 and x is at most t as t→∞. We show that depending on the values of α there are four growth regimes: (i) instantaneous growth for α<d, (ii) stretched exponential growth for α∈(d,2d), (iii) superlinear growth for α∈(2d,2d+1), and finally (iv) linear growth for α>2d+1 like the nearest-neighbor first-passage percolation model corresponding to α=∞.
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U2 - 10.1002/cpa.21571
DO - 10.1002/cpa.21571
M3 - Article
AN - SCOPUS:84955196422
SN - 0010-3640
VL - 69
SP - 203
EP - 256
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
IS - 2
ER -