A large-deviation result for the range of random walk and for the Wiener sausage

Abstract

Let {Sn} be a random walk on ℤd and let Rn be the number of different points among 0, S1, . . . , Sn-1. We prove here that if d ≥ 2, then ψ(x) := limn→∞(-1/n) log P {Rn ≥ nx} exists for x ≥ 0 and establish some convexity and monotonicity properties of ψ(x). The one-dimensional case will be treated in a separate paper. We also prove a similar result for the Wiener sausage (with drift). Let B(t) be a d-dimensional Brownian motion with constant drift, and for a bounded set A ⊂ ℝd let A1 = A1(A) be the d-dimensional Lebesgue measure of the 'sausage' ∪0≤s≤1 (B(s) + A). Then ø(x) := lim1→∞(-1/t) log P{ A1 ≥ tx} exists for x ≥ 0 and has similar properties as ψ.