Macroscopic self-avoiding walk interacting with lattice permutations and uniformly-positive monomer-monomer correlations in the dimer model in Z^d, d > 2
7th February 2020, 3:30 pm – 4:30 pm
Fry Building, 2.41
We consider lattice permutations, namely permutations of the vertices of a box in Z^d such that every vertex is mapped to a nearest neighbour. Such objects can be viewed as ensembles of mutually-disjoint oriented self-avoiding loops and are sampled according to the uniform measure or to a measure which rewards the total number of loops. We consider a version of this model in which one of these loops is 'open', namely it is a self-avoiding walk. The motivation for studying lattice permutations when d > 2 is twofold: firstly, they are related to (loop representation of) the interacting quantum Bose gas, for which the existence of a ''regime with a long walk'' is an important open conjecture. Secondly, the two-point function of lattice permutations equals the monomer-monomer correlation of the dimer model, a classical statistical mechanics model which is mostly unexplored when d > 2 (contrary to the two-dimensional dimer model, which is integrable and well-studied). We prove that the two-point function of lattice permutations is uniformly positive and that the "typical" distance of the end-points of the walk is of the same order of magnitude as the diameter of the box when d > 2.