### Abstract:

The path W[0,t] of a Brownian motion on a d -dimensional torus Td run for time t is a random compact subset of Td . We study the geometric properties of the complement Td∖W[0,t] as t→∞ for d≥3 . In particular, we show that the largest regions in Td∖W[0,t] have a linear scale φd(t)=[(dlogt)/(d−2)κdt]1/(d−2) , where κd is the capacity of the unit ball. More specifically, we identify the sets E for which Td∖W[0,t] contains a translate of φd(t)E , and we count the number of disjoint such translates. Furthermore, we derive large deviation principles for the largest inradius of Td∖W[0,t] as t→∞ and the ε -cover time of Td as ε↓0 . Our results, which generalise laws of large numbers proved by Dembo et al. (Electron J Probab 8(15):1–14, 2003), are based on a large deviation estimate for the shape of the component with largest capacity in Td∖Wρ(t)[0,t] , where Wρ(t)[0,t] is the Wiener sausage of radius ρ(t) , with ρ(t) chosen much smaller than φd(t) but not too small. The idea behind this choice is that Td∖W[0,t] consists of “lakes”, whose linear size is of order φd(t) , connected by narrow “channels”. We also derive large deviation principles for the principal Dirichlet eigenvalue and for the maximal volume of the components of Td∖Wρ(t)[0,t] as t→∞ . Our results give a complete picture of the extremal geometry of Td∖W[0,t] and of the optimal strategy for W[0,t] to realise extreme events.