Take
me back!
Spatial random permutations:
Some theory and a few pictures
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.
![](data:image/png;base64,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)
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.
![psychedelic image](pictures/srp.png)
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.
![psychedelic image](pictures/srp.png)