# Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths

@article{Grosskinsky2012LatticePA,
title={Lattice Permutations and Poisson-Dirichlet Distribution of Cycle Lengths},
author={Stefan Grosskinsky and Alexander A. Lovisolo and Daniel Ueltschi},
journal={Journal of Statistical Physics},
year={2012},
volume={146},
pages={1105-1121}
}
• Published 26 July 2011
• Mathematics
• Journal of Statistical Physics
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 where macroscopic cycles appear. We observe that the lengths of such cycles are distributed according to Poisson-Dirichlet. This can be explained heuristically using a stochastic coagulation-fragmentation process for long cycles, which is supported by numerical data.

### Split-and-Merge in Stationary Random Stirring on Lattice Torus

• Mathematics
Journal of Statistical Physics
• 2020
We show that in any dimension $$d\ge 1$$ d ≥ 1 , the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian

### 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

### 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

### Poisson-Dirichlet asymptotics in condensing particle systems

• Mathematics
• 2021
We study measures on random partitions, arising from condensing stochastic particle systems with stationary product distributions. We provide fairly general conditions on the stationary weights,

### Gibbs measures on permutations over one-dimensional discrete point sets

• Mathematics
• 2013
We consider Gibbs distributions on permutations of a locally finite infinite set $X\subset\mathbb{R}$, where a permutation $\sigma$ of $X$ is assigned (formal) energy $\sum_{x\in X}V(\sigma(x)-x)$.

### The Cycle Structure of Random Permutations without Macroscopic Cycles

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

### 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.

### Loop Correlations in Random Wire Models

• Mathematics, Computer Science
Communications 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.

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

• Mathematics
Electronic 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

## References

SHOWING 1-10 OF 30 REFERENCES

### SPATIAL RANDOM PERMUTATIONS AND POISSON-DIRICHLET LAW OF CYCLE LENGTHS

• Mathematics
• 2011
We study spatial permutations with cycle weights that are bounded or slowly diverging. We show that a phase transition occurs at an explicit critical density. The long cycles are macroscopic and

### The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations

• Mathematics
• 2003
We consider a Markov chain on the space of (countable) partitions of the interval [0,1], obtained first by size-biased sampling twice (allowing repetitions) and then merging the parts (if the sampled

### The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator

• Mathematics
• 1997
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with

### Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

A wide-ranging survey of general kernels of the Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations is attempted.

### Shift in Critical Temperature for Random Spatial Permutations with Cycle Weights

We examine a phase transition in a model of random spatial permutations which originates in a study of the interacting Bose gas. Permutations are weighted according to point positions; the

### Emergence of Giant Cycles and Slowdown Transition in Random Transpositions and k-Cycles

Consider the random walk on the permutation group obtained when the step distribution is uniform on a given conjugacy class. It is shown that there is a critical time at which two phase transitions

### Limiting behavior for the distance of a random walk

• Mathematics, Computer Science
• 2006
This investigation is motivated by a result recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$.

### Ordered cycle lengths in a random permutation

• Mathematics
• 1966
1. Introduction. Problems involving a random permutation are often concerned with the cycle structure of the permutation. Let tY.n be the n! permutation operators on n numbered places, and let a(X) =

### Spatial Random Permutations and Infinite Cycles

• Mathematics
• 2009
We consider systems of spatial random permutations, where permutations are weighed according to the point locations. Infinite cycles are present at high densities. The critical density is given by an

### Poisson–Dirichlet and GEM Invariant Distributions for Split-and-Merge Transformations of an Interval Partition

• J. Pitman
• Mathematics
Combinatorics, Probability and Computing
• 2002
A split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov and another studied by Tsilevich and Mayer-Wolf and Zeitouni and Zerner yields a simple proof of the recent result that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process.