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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. Here is a link to a vector graphics rendering of the same image.

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. Here is a link to a vector graphics rendering of the same image.

hexagon

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

boxed plane partition

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

Other models from combinatorics and statistical mechanics

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

grove

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