Cédric Boutillier

LPSM — Sorbonne Université

Mathematical gallery

Here are results of my attempts to simulate random combinatorical objects related to my research interests.

Tiling with dominos and rhombi

aztec diamond — A uniform tiling of an Aztec diamond of size 160 with dominos. Click on the image for a vector version.

aztec rectangle — A tiling of an Aztec rectangle with special boundary conditions with periodic weights along each row. This is an example of a Schur process. Click on the image for a vector version.

boxed plane partition — A uniform boxed plane partition with a large number of boxes, generated using Boltzmann samplers and Pak’s bijection.¹

path in a double dimer configuration — Two independent uniform domino configurations of a 141x141 rectangle deprived from a corner: the blue one does not cover the lower left corner, and the red one does not cover the upper right corner. A path is obtained by following alternatively red and blue dominos from the lower left to the upper right corner. This path is conjectured to converge to a SLE4 in the scaling limit. Click on the image for a larger example in a 201x201 rectangle or here for just just the curve in a 1001x1001 rectangle.

tiling of a hexagon with rhombi — A uniform tiling of a 30x30x30 hexagon by lozenges, generated using Propp and Wilson coupling from the past algorithm.

Temperley's bijection — Temperley’s bijection between a spanning tree of a 20x15 rectangle (in black) and a domino tiling of 41x31 rectangle deprived from its lower left corner. The grey arrows represent the dual of the black tree which is also a tree if consider the outside of the rectangle as a vertex of that dual tree. Both trees are oriented towards their roots: the removed corner and the outer dual vertex for the primal and dual tree respectively.

Other models from combinatorics and statistical mechanics

grove — A uniform grove of size 100 generated by the grove shuffling algorithm ². This random object exhibits an arctic circle phenomenon ³.

isoradial graph — A piece of an infinite isoradial graph (in black) and the underlying quadgraph, providing a tiling of the plane with rhombi.

percolation exploration process — The exploration of a critical site percolation configuration in a “rectangle” in the triangular lattice. The discrete curve converges to a SLE6.

Spectral curves

The evolution of a family of amoebas of genus 1 curves

The amoeba of a genus 1 Harnack curve blowing up, related to the temperature change in the Ising model.

References

Random Sampling of Plane Partitions, O. Bodini, É. Fusy and C. Pivoteau, Combinatorics, Probability and Computing (2010), 19: 201–226) ↩
The Cube Reccurrence, Gabriel D. Carroll, David E Speyer ↩
An arctic circle theorem for groves, T. K. Petersen, D. Speyer ↩