Loop Correlations in Random Wire Models

@article{Benassi2019LoopCI,
  title={Loop Correlations in Random Wire Models},
  author={Costanza Benassi and Daniel Ueltschi},
  journal={Communications in Mathematical Physics},
  year={2019},
  volume={374},
  pages={525-547}
}
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. 
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References

SHOWING 1-10 OF 53 REFERENCES
A numerical study of the 3D random interchange and random loop models
We have studied numerically the random interchange model and related loop models on the three-dimensional cubic lattice. We have determined the transition time for the occurrence of long loops. The
Random loop representations for quantum spin systems
We describe random loop models and their relations to a family of quantum spin systems on finite graphs. The family includes spin 1/2 Heisenberg models with possibly anisotropic spin interactions and
Random Permutations of a Regular Lattice
Spatial random permutations were originally studied due to their connections to Bose–Einstein condensation, but they possess many interesting properties of their own. For random permutations of a
Large cycles in random permutations related to the Heisenberg model
We study the weighted version of the interchange process where a permutation receives weight theta(#cycles). For theta = 2 this is Toth's representation of the quantum Heisenberg ferromagnet on the
The random walk representation of classical spin systems and correlation inequalities
Ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion. By using a mixed spinrandom walk representation we present a unified approach to
Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths
We study random spatial permutations on ℤ3 where each jump x↦π(x) is penalized by a factor $\mathrm{e}^{-T\| x-\pi (x)\|^{2}}$. The system is known to exhibit a phase transition for low enough T
Length distributions in loop soups.
TLDR
The resulting joint length distribution for macroscopic loops is Poisson-Dirichlet with a parameter θ fixed by the loop fugacity and by symmetries of the ensemble.
The interchange process with reversals on the complete graph
We consider an extension of the interchange process on the complete graph, in which a fraction of the transpositions are replaced by `reversals'. The model is motivated by statistical physics, where
Lectures on the Spin and Loop O(n) Models
The classical spin O(n) model is a model on a d-dimensional lattice in which a vector on the \((n-1)\)-dimensional sphere is assigned to every lattice site and the vectors at adjacent sites interact
Decay of Correlations in 2D Quantum Systems with Continuous Symmetry
We study a large class of models of two-dimensional quantum lattice systems with continuous symmetries, and we prove a general McBryan–Spencer–Koma–Tasaki theorem concerning algebraic decay of
...
...