Loop Correlations in Random Wire Models

Abstract

We introduce a family of loop soup models on the hypercubic lattice. The models involve links on the edges, and random pairings of the link endpoints on the sites. We conjecture that loop correlations of distant points are given by Poisson-Dirichlet correlations in dimensions three and higher. We prove that, in a specific random wire model that is related to the classical XY spin system, the probability that distant sites form an even partition is given by the Poisson-Dirichlet counterpart.1991 Mathematics Subject Classification. 60C05, 60K35, 82B05, 82B20, 82B26.

The interchange process with reversals on the complete graph

“…Such models have been considered on trees [9,5] and on the Hamming graph [1]. In very recent work there has been some limited progress in the direction of establishing Poisson-Dirichlet structure in these and related loop models [4,7]. For the Heisenberg model (ν = 1 and ϑ = 2) on the complete graph, the critical point for the appearance of cycles of diverging length was established already in the early 1990's by Tóth and by Penrose [18,22].…”

Exponential decay of transverse correlations for O(N) spin systems and related models

“…For example, the first link on the edge connecting the vertices (1, 1), (2, 1) is coloured red, it is paired at (1, 1) with the third link on the same edge and it is unpaired at (2, 1). Moreover, both links touching the vertex (3,3) are red and they are unpaired at (3,3). Finally, no link is on the edge which connects the vertices (1, 2) and (2,2) we can see from the example in Fig.…”

“…, N } and the measure involves an on-site weight function that penalises large numbers of overlaps. This representation corresponds to a combination of the ones introduced in [3,13], which are in turn related to the one of Brydges, Fröhlich and Spencer [7], and the random current representation of the Ising model [1]. In our representation a ghost vertex, denoted by g, is added to the graph, with edges to each other vertex representing the external field.…”

“…We refer to this representation as the Random Path Model (RPM). This representation corresponds to a combination of the one introduced in [3,13], which was also used in [18] in the study of the dimer model in Z d , d ≥ 3, and of the random current representation of the Ising model [1]. Two key aspects of the representation are pairings and colours, these are two ingredients which are not present (or necessary) in the N = 1 case [1], the well-known Ising model, but which play a crucial role in our analysis, which involves the N > 1 cases.…”

“…Additionally, our proof is also quite flexible and, for example, it holds for any graph of bounded degree; it holds on Z d with finite range (not necessarily translation invariant) coupling constants, and it holds for a class of models with continuous symmetry whose interaction does not necessarily take the form e −H (with H representing the hamiltonian function) -these models are 'less physical' but they lead to interesting random loop models, for example the loop O(N) model [8,9,16,19] (see Sect. 5.2), see also [3,5] for related models.…”