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Given a graph G with i.i.d. 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. Here the weight 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 mean d(u,v)^\alpha where d(.,.) is
the graph distance and \alpha>0 is a fixed parameter. Standard
first-passage percolation corresponds to \alpha=\infty. The growth set
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 \alpha. For \alpha2d+1 the rate is linear
like the standard model. This is a joint work with Shirshendu Chatterjee.
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