Space-time random walk loop measures

TL;DR

This paper introduces Bosonic loop measures derived from space-time random walks with non-symmetric jump rates, providing a probabilistic framework and new insights into quantum correlation functions and loop measure convergence.

Contribution

It defines Bosonic loop measures via space-time random walks, explores their properties, and connects them to quantum correlations and classical loop measures.

Findings

Bosonic loop measures arise as limits of space-time random walk loop measures.

The measures assign weights to loops of lengths multiple of a parameter β.

The study extends Dynkin's isomorphism and Symanzik's formulae to complex Gaussian fields.

Abstract

In this work, we investigate a novel setting of Markovian loop measures and introduce a new class of loop measures called Bosonic loop measures. Namely, we consider loop soups with varying intensity $ \mu\le 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 $ \beta> 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 ([B92,BIS09]. Our space-time setting allows obtaining quantum correlation functions as torus limits of space-time correlation functions.