Abstract: 

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 the hypercubic lattice will be introduced and specific cases corresponding to models such as the $O(N)$-loops and  -spin model, the dimer and double dimer model, and spatial random permutations will be explained.

A general monotonicity result for correlations in this model will be presented 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, but long standing conjecture in the case $N\geq 3$.