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Probability and Combinatorics

Tuesday, March 20, 2012 - 4:30pm

Partha Sarathi Dey

NYU

Location

University of Pennsylvania

drl 3C6

Given a graph G with non-negative edge weights, first- passage time between two vertices u and v is defined as the minimum passage time over all paths connecting u and v. The passage time of a path is sum of all edge weights in the path. In long-range first-passage percolation, any two vertices u,v in G are connected by a weighted edge with the weight given by an independent exponential r.v. with rate $d(u,v)^{-s}$ where $d(\cdot,\cdot)$ is the graph distance and $s>0$ is a fixed parameter. Standard first-passage percolation corresponds to $s=\infty$. The growth set $B_t$ upto time $t$ is defined as the set of points reachable from a fixed point within time $t$. We consider the case where G is the d-dimensional euclidean square lattice and show that there are four different growth regions depending on the value of $s$. For $s2d+1$ the rate is linear like the standard model. Joint work with Shirshendu Chatterjee.