RANDOM PERMUTATIONS OF A REGULAR LATTICE

Abstract. Spatial random permutations were originally studied due to their con-nections to Bose-Einstein condensation, but they possess many interesting prop-erties of their own. For random permutations of a regular lattice with periodic boundary conditions, we prove existence of the infinite volume limit under fairly weak assumptions. When the dimension of the lattice is two, we give numerical evidence of a Kosterlitz-Thouless transition, and of long cycles having an almost sure fractal dimension in the scaling limit. Finally we comment on possible con-nections to Schramm-Löwner curves