Space-time random walk loop measures

Recently Markovian Loop measures have become an active field in probability (LeJan,Sznitman) with its origin going back to Symanzik [Sym69] and Brydges et al. [BFS82]. In this work we are going to investigate a novel setting of these loop measures, namely we consider first loop soups with varying intensity $ \mu\le 0 $ (chemical potential in physics terms). Secondly, we study Markovian loop measures on graphs with an additional time dimension leading to so-called space-time random walks and their loop measures and Poisson point loop processes. Interesting phenomena appear when the additional coordinate of the space-time process is on a discrete torus with non-symmetric jumping rates. The projection of these space-time random walk loop measures onto the space dimensions are loop measures on the spatial graph, and in the scaling limit of the discrete torus these loop measures converge to the so-called Bosonic loop measures. These novel loop measures have similarities with the standard Markovian loop measures. The Bosonic loop measures not only have the probabilistic motivation as outlined, a second major interest in these objects stems from the fact the the total weights of the Bosonic loop measure for a finite graph is exactly the logarithm of the grand-canonical partition function of non-interacting Bose gases in thermodynamic equilibrium at inverse temperature $ \beta>0 $ and chemical potential $ \mu\le 0 $. We complement our study with generalised versions of Dynkin's isomorphim theorem as well as Symanzik's moment formulae for complex Gaussian measures. Due to the lacking symmetry of our space-time random walks, the distributions of the occupation time fields is given in terms of a complex Gaussian measures over complex-valued random fields [B92,BIS09].