On the spatial Markov property of soups of unoriented and oriented loops

TL;DR

This paper explores the spatial Markov property in soups of unoriented and oriented Markov loops, linking these properties to well-known probabilistic models like uniform spanning trees and the Gaussian Free Field.

Contribution

It introduces a simple interpretation of the spatial Markov property for both unoriented and oriented loop soups, connecting them to classical models and identities in probability theory.

Findings

The Markov property of unoriented loop soups relates to the Gaussian Free Field.

The Markov property of oriented loop soups connects to uniform spanning trees.

Provides a unified framework for understanding loop-soups and their properties.

Abstract

We describe simple properties of some soups of unoriented Markov loops and of some soups of oriented Markov loops that can be interpreted as a spatial Markov property of these loop-soups. This property of the latter soup is related to well-known features of the uniform spanning trees (such as Wilson's algorithm) while the Markov property of the former soup is related to the Gaussian Free Field and to identities used in the foundational papers of Symanzik, Nelson, and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan.