The statistical mechanics of the interlacement point process
In this project, we will lay the foundations for a new theory of statistical mechanics for point processes of random paths in Euclidean space of unbounded and even infinite lengths. The main novelty will be a proper treatment of the level-three large deviations for the empirical stationary field in large boxes, including a description of the rate function as a new kind of limiting entropy density. This will make possible a mathematically sound geometric description of the interacting Bose gas in the condensation regime in the thermodynamic limit -- this is our main example and the driving force of this research. We plan to describe the condensate rigorously in terms of an interacting interlacement process. The understanding we expect to obtain might also be useful in the context of the open problem of finding a proof for the occurrence of Bose--Einstein condensation.