Ben Lees

Monotonicity of correlations for general loop soups

Loop soup models involve one dimensional objects (loops/walks/dimers) living in a higher dimensional space. These models are connected to various spin and particle systems from statistical mechanics.

A general loop soup model on $\mathbb Z^d$ lattice will be introduced and specific cases corresponding to models such as the $O(N)$-loop and -spin model, the dimer and double dimer model, and spatial random permutations will be explained.

We will see a general monotonicity result for correlations in this model and, in the special case of the $O(N)$-spin model it will be shown that this provides a proof of monotonicity for two point correlations as a function of the distance between particles, providing a positive answer to a well-accepted conjecture in the case $N\geq 3$.