It is generally believed that, in the thermodynamic limit, the microcanonical description as a function of energy coincides with the canonical description as a function of temperature. However, various examples of systems for which the microcanonical
and canonical ensembles are not equivalent have been identified. A complete theory of this intriguing phenomenon is still missing. Here we show that ensemble nonequivalence can manifest itself also in random graphs with topological constraints. We find that, while graphs with a given number of links are ensemble-equivalent, graphs with a given degree sequence are not. This result holds irrespective of whether the energy is nonadditive (as in unipartite graphs) or additive (as in bipartite graphs). In contrast with previous expectations, our results show that: (1) physically, nonequivalence can be induced by an extensive number of local constraints, and not necessarily by long-range interactions or nonadditivity; (2) mathematically, nonquivalence is determined by a different large-deviation behaviour of microcanonical and canonical probabilities for a single microstate, and not necessarily for almost all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.
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.
