Intersections of random walks

Reference:

Phetpradap, P., 2011. Intersections of random walks. Thesis (Doctor of Philosophy (PhD)). University of Bath.

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Abstract

We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below th etypical value. This is a result analogous to that obtained by van den Berg, Bolthausen and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visitedby two independent simple random walks starting at the origin. We call this theintersection of ranges. We derive a large deviation principle for the probability thatthe intersection of ranges by time n exceeds a multiple of n. This is also an analogousresult of the intersection volume of two independent Wiener sausages.

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