LARGE-VOLUME ASYMPTOTICS OF BROWNIAN INTEGRALS AND ORBITAL MAGNETISM

We study the asymptotic expansion of a class of Brownian integrals wit h paths constrained to a finite domain as this domain is dilated to in finity. The three first terms of this expansion are explicitly given i n terms of functional integrals. As a first application we consider th e finite size effects in the orbital magnetism of a free electron gas subjected to a constant magnetic field in two and three dimensions. Su m rules relating the volume and surface terms to the current density a long the boundary are established. We also obtain that the constant te rm in the pressure (the third term) of a two dimensional domain with s mooth boundaries is purely topological, as in the non magnetic case. T he effects of corners in a polygonal shape are identified, and their c ontribution to the zero field susceptibility is calculated in the case of a square shaped domain. The second application concerns the asympt otic expansion of the statistical sum for a quantum magnetic billiard in the semiclassical and high temperature limits. In the semiclassical expansion, the occurence of the magnetic field is seen in the third t erm, whereas in the high temperature expansion, it appears only in the fifth term.