Simulation of random processes and mathematical images

(the pictures are in postscript or pdf)

You can use these drawings for your own publications/talks free of charges (attribution demanded, creative commons licence CC-BY). I am always pleased to see that these pictures are used, so if you don't mind telling me, with a short email for example...

If you desire a personal picture, you may also ask for it: if my program is not lost, it should be possible...

Uniform quadrangulations (the map structure is lost, only the underlying graph structure is kept.

Pictures in pdf : size-150, size 2500, size 5000, size 7500, size 10 000, size 20 000, size 25 000a, b, c, d, e, f, g , size 30 000, size 50 000

(size= number of vertices)

Uniform quadrangulations where separating cycles have been deleted

Pictures in pdf. size-150 000, size 150 000, size 300 000, size 400 000, size 500 000,

(size= number of vertices before deleting the separating cycles.)

Uniform planar trees (drawing style 1) size = number of nodes, pictures in pdf, size 50, size 100, size 500, size 1000, size 5 000, size 15 000

Uniform planar trees (drawing style 2)

size = number of nodes, pictures in pdf, size 1000, size 5 000, size 10 000, size 25 000, size 50 000, size 100 000

(combinatorial) minimum spanning tree of a complete graph

size = number of nodes, pictures in pdf, size 1 000, size 5 000, size 10 000, size 25 000, size 50 000, size 100 000

Diffusion limited aggregation (DLA)

size = number of particles, pictures in pdf or ps, size 1 000, size 5 000, size 10 000, size 50 000, size 100 000, size 500 000, size 1 000 000

Internal DLA

The first step out of the cluster is drawn, giving a tree. Here, size = number of particles, pictures in ps, size 1 000, size 10 000, size 100 000, size 1 000 000

First passage percolation On the edges, i.i.d. uniform [0,1] random variables. Minimal paths to each vertex is drawn: again, this forms a tree.

size = temporal radius radius 10, size 50 , size 100, size 200

First passage percolation On the edges, i.i.d. uniform exponential random variables with parameter 1. Minimal paths to each vertex is drawn: again, this forms a tree.

size = temporal radius radius 10, size 50 , size 100, size 200

Sandpile models on the grid Z

nb of grains 10, nb of grains 10

Sandpile models on the grid Z

nb of grains 10, nb of grains 10

Sandpile models on the grid Z

nb of grains 10

Some processes(pictures in postscript)

Brownian motion | 1 | 2 | 3 | |

Brownian motion in R^2 | 1 | 2 | 3 | 4 |

Brownian bridge | 1 | 2 | 3 | 4 |

Brownian excursion | 1 | 2 | 3 | 4 |

Brownian meander | 1 | 1 | 3 | 4 |

Uniform loop erased random walk (killed the first time at which 100 steps are present) | 1 | 2 | 3 | 4 |

Uniform loop erased random walk (killed at 1 000 steps) | 1 | 2 | 3 | 4 |

Uniform loop erased random walk (killed at 10 000 steps) | 1 | 2 | 3 | 4 |

Uniform loop erased random walk (killed at 100 000 steps) | 1 | 2 | 3 | 4 |

Uniform spanning tree: grid 50 x 50 | 1 | 2 | 3 | |

Uniform spanning tree: grid 100 x 100 | 1 | 2 | 3 | |

Uniform spanning tree: grid 200 x 200 | 1 | 2 | 3 |

Bessel processes with parameter 2 3 20 100 1000

Local times

Local time at time 1 of Brownian motion

Local time at time 1 of Brownian bridge

Local time at time 1 of the normalised Brownian excursion

Branching random walks

Discrete branching random walk (size 58 000) with positive mean displacment (and finite variance)

Discrete branching random walk (size 58 000) with gaussian spatial increments N(0,1).

Discrete branching random walk (size 58 000) with Cauchy spatial increments

Discrete branching random walk (size 180 000) with spatial increments distribution 5/(4(1+|x|)^7/2)

Brownian snakes

Brownian snake with lifetime the normalized Brownian excursion and R valued spatial process

Brownian snake with lifetime the normalized Brownian excursion and R

Random walk in random environment (critical: site uniform on [0.3,0.7], random walk +-1)

(Each site s received a random variable x(s) , uniform in [a,b], independent from the other sites: the walk at site s goes up with proba x(s) and down with proba 1-x(s) )

Random walk in random environment (critical: site uniform on [0.2,0.8], random walk +-1)

Random walk in random environment (site uniform on [0.2,0.9])

Quicksort limiting distribution

Density of limiting quicksort distribution (fix point of the law equality Y = U Y'+(1-U)Y''+2(U log(U)+(1-U) log(1-U)) +1, where Y' and Y'' are i.i.d. copies of Y, and are independent of U, uniform on [0,1]