Intersection properties of simple random walks: A renormalization group approach

We study estimates for the intersection probability, g(m), of two simple random walks on lattices of dimension d=4, 4-epsilon as a problem in Euclidean field theory. We rigorously establish a renormalization group flow equation for g(m) and bounds on the β-function which show that, in d=4, g(m) tends to zero logarithmically as the killing rate (mass) m tends to zero, and that the fixed point, gsup(*), in d=4-epsilon is bounded by const epsilon<=gsup(*)<= const epsilon. Our methods also yield estimates on the intersection probability of three random walks in d=3, 3-epsilon. For epsilon=0, these results were first obtained by Lawler [1]. (orig.)