# Random Permutations of a Regular Lattice

@article{Betz2013RandomPO, title={Random Permutations of a Regular Lattice}, author={Volker Betz}, journal={Journal of Statistical Physics}, year={2013}, volume={155}, pages={1222-1248} }

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 regular lattice with periodic boundary conditions, we prove existence of the infinite volume limit under fairly weak assumptions. When the dimension of the lattice is two, we give numerical evidence of a Kosterlitz–Thouless transition, and of long cycles having an almost sure fractal dimension in the…

## 24 Citations

Random permutations without macroscopic cycles

- MathematicsThe Annals of Applied Probability
- 2020

We consider uniform random permutations of length n conditioned to have no cycle longer than nβ with 0 < β < 1, in the limit of large n. Since in unconstrained uniform random permutations most of the…

Loop Correlations in Random Wire Models

- Mathematics, Computer ScienceCommunications in Mathematical Physics
- 2019

It is proved 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.

Loop Correlations in RandomWire Models

- Mathematics, Computer Science
- 2019

It is proved 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.

Scaling limit of ballistic self-avoiding walk interacting with spatial random permutations

- MathematicsElectronic Journal of Probability
- 2019

We consider nearest neighbour spatial random permutations on Z d . In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an…

The number of cycles in random permutations without long cycles is asymptotically Gaussian

- Mathematics
- 2016

For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior…

Interacting self-avoiding polygons

- Computer Science, Mathematics
- 2018

It is proved the existence of a sub-region of the phase diagram where the self-avoiding polygons are space filling and the non-trivial characterization of the regime where the polygon length admits uniformly bounded exponential moments is provided.

Uniformly positive correlations in the dimer model and phase transition in lattice permutations on Z, d > 2, via reflection positivity

- Mathematics
- 2019

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…

The Cycle Structure of Random Permutations without Macroscopic Cycles

- Mathematics
- 2018

We consider the Ewens measure on the symmetric group conditioned on the event that no cycles of macroscopic lengths occur and investigate the resulting cycle structure of random permutations without…

Gibbs measures over permutations of point processes with low density.

- Mathematics
- 2019

We study a model of spatial random permutations over a discrete set of points. Formally, a permutation $\sigma$ is sampled proportionally to the weight $\exp\{-\alpha \sum_x V(\sigma(x)-x)\},$ where…

Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity

- Mathematics
- 2019

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…

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