linear foam

Apply random Y-Delta and Delta-Y moves to a planar network with 3 pinned nodes.

area-1 rectangulation

area-1 rectangulation

area-1 mappings

random triangulation

square spiral

integrable resistor network

double cover

You think your way in

Aztec diamond tiling with semifrozen phases

Integrable dynamics on circle patters

Critical percolation on the honeycomb conditioned to have fixed boundary connections.

Unbalanced grove a uniform spanning tree on the honeycomb with fixed boundary connections

Care to dance?

Dance with more partners

integrable dynamics? 10^5 points

integrable dynamics? 10^6 points

integrable dynamics? 10^6 points

Iterates of a point on the real line under the piecewise isometry

f(z) = exp(it)(z+2sin(t)) if Im(z)>0

f(z) = exp(it)(z-2sin(t)) else.

Here t is an irrational multiple of pi.

LERW in Z^2, Half-plane LERW n Z^2 Edge probabilities for the loop-erased random walk on Z^2.

CRSF A simulation of a cycle-rooted spanning forest process. This is a determinantal measure on the edges of a graph generalizing the uniform spanning tree process.

Eigenvalues of the discrete-time TASEP

The totally asymmetric exclusion process on a circle of length 7.

Uniform random branched polymer construction (Joint work with Peter Winkler)

Uniform random branched polymer with 95 balls(Joint work with Peter Winkler)

Branched polymer A uniformly chosen banched polymer with 1100 disks.

Branched polymer A uniformly chosen banched polymer with 500 disks.

Dancing triangles See the paper Dimers, tilings and trees joint with Scott Sheffield

packing segments Random packing of segments in a torus. Here I took about 200 segments, of angles evenly spaced around a circle. This packing is apparently in an "ordered" phase where nearby segments have generally nearby directions. Conjecturally there is a phase transition when the length or number of segments increases, from a disordered phase to an ordered phase.

Doubling tiles (Pdf version here ) Compact sets with interior which can be tiled with two similar copies of themselves (with non-overlapping interiors). These are (conjecturally) all 17 of them.

Freezing amoeba An amoeba "freezing": tending to its tropical version.

Mobius Linkage a linkage whose configuration space is (an open set in) the Mobius group.

frozen boundaries for a 9-gon

frozen boundaries for a 12-gon

melting An amoeba melting away from its spine

"Octic" circle: the frozen/temperate boundary for "diabolo" tilings of an Aztec-diamond-like region. This shape was originally discovered by Henry Cohn and Robin Pemantle. New techniques (joint with Andrei Okounkov) yield a simple proof.

Volume constrained boxed-plane-partition The frozen/temperate boundaries for a boxed-plane partition as a function of the volume. The curves are logarithms of ellipses, that is, images of a one-parameter family of ellipses under the map (x,y)->(log(x),log(y)).

Simulation of a random tiling of a regular hexagon ("boxed plane partition").

The arctic/temperate boundary for lozenge tilings of an indented hegaxon. It is a rational curve of degree 5.

simulation of actual random tiling in octagonal region, where the arctic boundary is a cardiod.

Another example with a degree-4 curve.

Mobius-invariant Sierpinski carpet A random fractal with distribution invariant under the action of the group of Mobius transformations of the 2-sphere. lower-density example

aztec armadillo The triangle tiling associated to an Aztec diamond of order 25.

amoeba The phase diagram for a dimer problem with 8 distinct phases (4 solid, 1 liquid, 3 gaseous).

The discrete Sqrt[z] function.

Random triangulation of the nXn square with "unit" integer triangles

Other examples 2 3 Degree-6 triangulations of the plane in which all triangles have circumcircles of radius 1. Such an embedding is generated by three maps Z->R. For these three images I took three triples of sinusoidal mappings of different incommensurable periods.

Self-similar tiling with expansion factor a root of x^4+x+1=0.

dark blue -> red -> yellow -> light blue -> {purple, dark blue}

green-> purple -> {yellow, green}

Resonance in the non-intersecting lattice path model. This is the normalized logarithm of the grand canonical partition function of the critical non-intersecting monotone lattice path model with periodic boundaries. Here m,n are the dimension of the torus, and A=mn.

Pattern probabilities in a random domino tiling

Poincaré recurring Under 121 iterations of the linear map {{1,1},{1,0}}, this image of Henri Poincaré magically returns to its original state. How many pixels are there in the image?

Energy level spectrum for electrons in Z^2 as a function of magnetic field strength

(vertical coordinate ranging from 0 to 2Pi, horizontal from -4 to 4). This is the eigenvalue spectrum for n large of the tridiagonal nXn matrix

whose entries one above and below the diagonal are 1, and diagonal entries are 2, 2Cos[t],2Cos[2t],... where t is the vertical coordinate.

Entropy for diabolo tilings with weights (a,b,c,d)=(a,1/a,c,1/c), as function of a and c.

(Range a>1,c>1. The function is symmetric under a -> 1/a and c -> 1/c.)

Probability of an `a' edge in a diabolo tiling with weights a,b=1/a,c,d=1/c, as function of a and c.

Contour in a random domino tiling

Take a domino tiling of an annulus, with a certain (small) probability there is a height contour which winds around the annulus. It is conjectured that these contours are conformally invariant self-avoiding curves.

Long contours in a random domino tiling

Conformal subdivision of a triangle into 7 copies

This combinatorial subdivision rule is "conformal" in the sense of Cannon, Floyd, Parry. In particular there is a conformal structure on the disk such that the subdision rule is realized by conformal maps. In this case (as is usually the case) the conformal structure is essentially unique. There it is.

Square wave

Map the infinite checkerboard to the plane so that black squares map to squares and white squares map to quadrilaterals. Such a map is a disguised version of a discrete analytic function. This is a discrete approximation to the exponential map f(z)=e^z.

Self-similar tiling

Square tiling of an ellipse

The Smith diagram is a geometric realization of a discrete harmonic function on a planar graph. Each edge of the graph becomes a square whose upper and lower sides have coordinates given by the value of the harmonic function at its endpoints. The right and left sides have coordinates given by the conjugate harmonic function. In this picture, the underlying graph is Cannon's "combinatorial hyperbolic plane". I chose some simple Dirichlet boundary conditions.

Equilateral exponential function

Another generalization of the Smith diagram , a discrete version of the function f(z)=exp(c*z).

Equilateral Weierstrass function

As above but this time the harmonic function is on a finite graph on a torus.

Arctan(z)

Smith diagram for the discrete arctangent function.