Space–time random walk loop measures

In this work, we introduce and investigate two novel classes of loop measures, space–time Markovian loop measures and Bosonic loop measures, respectively. We consider loop soups with intensity μ≤0 (chemical potential in physics terms), and 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 jump rates. The projection of these space–time random walk loop measures onto the space dimensions is 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. This provides a natural probabilistic definition of Bosonic loop measures. These novel loop measures have similarities with the standard Markovian loop measures only that they give weights to loops of certain lengths, namely any length which is multiple of a given length β>0 which serves as an additional parameter. We complement our study with generalised versions of Dynkin’s isomorphism theorem (including a version for the whole complex field) 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 are given in terms of complex Gaussian measures over complex-valued random fields [8,10]. Our space–time setting allows obtaining quantum correlation functions as torus limits of space–time correlation functions.