# Partha Dey : Central Limit Theorem for First-Passage Percolation along thin cylinders

We consider first-passage percolation on the cylinder graph of length $n$ and width $h_n$ in the d-dimensional square lattice where each edge has an i.i.d.~nonnegative weight. The passage time for a path is defined as the sum of weights of all the edges in that path and the first-passage time between two vertices is defined as the minimum passage time over all paths joining the two vertices. We show that the first-passage time $T_n$ between the origin and the vertex $(n,0,\ldots,0)$ satisfies a Gaussian CLT as long as $h_n=o(n^{1/(d+1)})$. The proof is based on moment estimates, a decomposition of $T_n$ as an approximate sum of independent random variables and a renormalization argument. We conjecture that the CLT holds upto $h_n=o(n^{2/3})$ for $d=2$ and provide support for that. Based on joint work with Sourav Chatterjee.

**Category**: Probability**Duration**: 01:44:57**Date**: November 4, 2010 at 4:10 PM**Views**: 115-
**Tags:**seminar, Probability Seminar

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