Mathematical gallery
Here are results of my attempts to simulate random combinatorical objects related to my research interests.
Tiling with dominos and rhombi
A uniform tiling of a 30x30x30 hexagon by lozenges, generated using Propp and Wilson coupling from the past algorithm.
A uniform boxed plane partition with a large number of boxes, generated using Boltzmann samplers and Pak’s bijection [^1]
A uniform tiling of an Aztec diamond of size 160 with dominos. Click on the image for a vector version.
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.
Other models from combinatorics and statistical mechanics
A piece of an infinite isoradial graph (in black) and the underlying quadgraph, providing a tiling of the plane with rhombi.
A uniform grove of size 100 generated by the grove shuffling algorithm [^2]. This random object exhibits an arctic circle phenomenon [^3].
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
[1] Random Sampling of Plane Partitions, O. Bodini, É. Fusy and C. Pivoteau, Combinatorics, Probability and Computing (2010), 19: 201–226)
[2] The Cube Reccurrence, Gabriel D. Carroll, David E Speyer
[3] An arctic circle theorem for groves, T. K. Petersen, D. Speyer