Moderate deviations for the volume of the Wiener sausage

For a > 0, let W-a(t) be the a-neighbourhood of standard Brownian motion in R-d starting at 0 and observed until time t. It is well-known that E\W-a(t )\ similar to kappa (a)t (t --> infinity) for d greater than or equal to 3, with kappa (a) the Newtonian capacity of the ball with radius a. We prove that lim(t --> infinity) 1/t((d-2))/d log P(\W-a(t)\ less than or equal to bt) = -I-kappaa (b) is an element of (-infinity ,0) for all 0 < b < kappa (a) and derive a variational representation for the rate function I-kappaa. We show that the optimal strategy to realise the above moderate deviation is f or W-a(t) to 'look like a Swiss cheese': W-a(t) has random holes whose size s are of order I and whose density varies on scale t(1/d). The optimal stra tegy is such that t(-1/d)W(a)(t) is delocalised in the limit as t --> infin ity. This is markedly different from the optima]. strategy for large deviat ions {\W-a(t)\ less than or equal to f(t)} with f(t) = o(t), where W-a(t) i s known to fill completely a ball of volume f(t) and nothing outside, so th at W-a(t) has no holes and f(t)W--1/d(a)(t) is localised in the limit as t --> infinity. We give a detailed analysis of the rate function I-kappaa, in particular, i ts behaviour near the boundary points of (0, kappa (a)) as well as certain monotonicity properties. It turns out that I-kappaa has an infinite slope a t m, and, remarkably, for d greater than or equal to 5 is nonanalytic at so me critical point in (0, kappa (a),), above which it follows a pure power l aw. This crossover is associated with a collapse transition in the optimal strategy. We also derive the analogous moderate deviation result for d = 2. In this c ase E\W-a(t)\ similar to 2 pit/log t (t --> infinity), and we prove that lim(t --> infinity) 1/log t log P(\W-a(t)\ less than or equal to bt/log t) = -I-2 pi(b) is an element of(-infinity, 0) for all 0 < b < 2 pi. The rate function I-2 pi has a finite slope at 2 pi.