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.
Highlights
This conjecture has been rigorously established in a model of spatial permutations related to the quantum Bose gas [10,14,18]
Our random wire model is an extension of the random current representation of the Ising model that was introduced by Griffiths, Hurst, and Sherman [24], and popularised by Aizenman [1]
The Poisson–Dirichlet distribution of random partitions was introduced by Kingman [27]; it is the invariant measure for the split-merge process [7,16,35]
Summary
We list here the main notation used in this article; the precise definitions can be found in subsequent sections. G = (V, E) the graph; V is the set of vertices and E is the set edges. Gb = (V ∪ V , E ∪ E) denotes the graph with a boundary; Vis the set of boundary sites and Eare edges between V and V. GLb is the graph with boundary ∂ L , given by sites of Zd at distance 1 from L. WG = {w = (m, π )} is the set of wire configurations on G, that consists of a link configuration m ∈ MG ⊂ NE0 (with an even number of links touching each site) and a pairing configuration π ∈ PG(m). Nx (m) is the local occupancy (or “local time”); it is equal to the number of times that loops pass by the site x ∈ V
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