The path W[0,t] of a Brownian motion on a d-dimensional torus T^d run for time t is a random compact subset of T^d. We study the geometric properties of the complement T^d W[0,t] for t large and d >= 3. In particular, we show that the largest region
s in this complement have a linear scale phi = [(d log t)/(d-2)kt]^{1/(d-2)}, where k is the capacity of the unit ball. More specifically, we identify the sets E for which T^d W[0,t] contains a translate of phi E, and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of T^d W[0,t] for t large and the epsilon-cover time of T^d for epsilon small. Our results, which generalise laws of large numbers proved by Dembo, Peres and Rosen, are based on a large deviation principle for the shape of the component with largest capacity in T^d W_rho[0,t], where W_rho[0,t] is the Wiener sausage of radius rho = rho(t), with rho(t) chosen much smaller than phi but not too small. The idea behind this choice is that T^d W[0,t] consists of lakes, whose linear size is of order phi, connected by narrow channels. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of T^d W_rho[0,t] for t large. Our results give a complete picture of the extremal geometry of T^d W[0,t] and of the optimal strategy for W[0,t] to realise the extremes.
In invasion percolation, the edges of successively maximal weight (the outlets) divide the invasion cluster into a chain of ponds separated by outlets. On the regular tree, the ponds are shown to grow exponentially, with law of large numbers, central
limit theorem and large deviation results. The tail asymptotics for a fixed pond are also studied and are shown to be related to the asymptotics of a critical percolation cluster, with a logarithmic correction.
