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. 
View on Springer
link.springer.com
12 Citations
Highly Influential Citations
3
Background Citations
3
Methods Citations
5
Results Citations
2

12 Citations

Macroscopic loops in the Bose gas, Spin O(N) and related models
TLDR
This work approximate Z by finite boxes and proves that, given any two vertices whose distance is proportional to the diameter of the box, the probability of observing a loop visiting both is uniformly positive.
Uniformly positive correlations in the dimer model and phase transition in lattice permutations on Z, d > 2, via reflection positivity
Our first main result is that correlations between monomers in the dimer model in Z do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in…
Uniformly Positive Correlations in the Dimer Model and Macroscopic Interacting Self‐Avoiding Walk in ℤd, d ≥ 3
  • Lorenzo Taggi
  • Mathematics
    Communications on Pure and Applied Mathematics
  • 2021
Our first main result is that correlations between monomers in the dimer model in ℤd do not decay to 0 when d>2 . This is the first rigorous result about correlations in the dimer model in dimensions…
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…
An elementary proof of phase transition in the planar XY model
Using elementary methods we obtain a power-law lower bound on the twopoint function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and…
Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems
We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the…
Macroscopic loops in the $3d$ double-dimer model
The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of selfavoiding…
Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity
Our first main result is that correlations between monomers in the dimer model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first rigorous result about correlations in the dimer…
Exponential decay of transverse correlations for O(N) spin systems and related models
We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when…
Exponential decay of transverse correlations for spin systems with continuous symmetry and non-zero external field
We prove exponential decay of transverse correlations in the Spin O(N) model for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N > 1. Our result is new when…
...
...

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…
...
...