**Take me back!**

Spatial random permutations

Spatial random permutations

**:**

**Some theory and a few pictures**

The basic model is very simple:
conside a planar square lattice, with

periodic boundary conditions. On the set of all permutations of the points

of the lattice, define a Gibbs measure with energy given by the sum of the

squared distances between every point and its image: i.e.

The following pictures are from a square lattice with side length 1000, and show the

points belonging to the 10 longest cycles of a typical random permutation obtained by

Markov chain Monte Carlo simulation. The three longest cycles are red, blue, green, in that

order. Clearly visible is the fractal nature of the random sets obtained, as well as the dependence

on the parameter beta.

Click on the small images for a larger image of the cycles!

Finally, below you can see a nice psychedelic work of art - this is what happens for really low beta,

i.e. around 0.1, on a really large grid of side length 2500.

periodic boundary conditions. On the set of all permutations of the points

of the lattice, define a Gibbs measure with energy given by the sum of the

squared distances between every point and its image: i.e.

The following pictures are from a square lattice with side length 1000, and show the

points belonging to the 10 longest cycles of a typical random permutation obtained by

Markov chain Monte Carlo simulation. The three longest cycles are red, blue, green, in that

order. Clearly visible is the fractal nature of the random sets obtained, as well as the dependence

on the parameter beta.

beta = 0.4 |
beta = 0.5 | beta = 0.6 |
beta = 0.7 |

Click on the small images for a larger image of the cycles!

Finally, below you can see a nice psychedelic work of art - this is what happens for really low beta,

i.e. around 0.1, on a really large grid of side length 2500.