Moderate deviations for the volume of the Wiener sausage

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For a>0,let Wᵃ(t) be the a-neighbourhood of standard Brownian motion in Rᵈ starting at 0 and observed until time t.It is well-known that E|Wᵃ(t)| kappaₐ t (t->infty) for d >= 3,with kappaₐ the Newtonian capacity of the ball with radius a. We prove that limt->infty 1/t⁽ᵈ⁻²⁾/ᵈlog P(|Wᵃ(t)|<=bt) = -Ikappaₐ(b) in (-infty,0) for all 0<b<kappaₐ and derive a variational representation for the rate function Ikappaₐ.We show that the optimal strategy to realise the above moderate deviation is for Wᵃ(t) to look like a Swiss cheese: Wᵃ(t) has random holes whose sizes are of order 1 and whose density varies on scale t¹/ᵈ.The optimal strategy is such that t-1/d Wᵃ(t) is delocalised in the limit as t->infty.This is markedly different from the optimal strategy for large deviations |Wᵃ(t)|<=f(t) with f(t)=o(t),where Wᵃ(t) is known to fill completely a ball of volume f(t) and nothing outside,so that Wᵃ(t) has no holes and f(t)⁻¹/ᵈWᵃ(t) is localised in the limit as t->infty.We give a detailed analysis of the rate function Ikappaₐ,in particular,its behaviour near the boundary points of (0,kappaₐ).It turns out that Ikappaₐ has an infinite slope at kappaₐ and,remarkably,for d>=5 is nonanalytic at some critical point in (0,kappaₐ),above which it follows a pure power law.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 case E|Wᵃ(t)| 2pi t/log t (t->infty),and we prove that limt->infty 1/log t log P(|Wᵃ(t)|<=bt/log t) =-I2pi(b)in (-infty,0) for all 0<b<2pi.