Code

My GitHub page provides tools for working with common discrete statistical physics models: the dimer model, the loop-erased random walk, the uniform spanning tree, and the discrete Gaussian free field. The packages are written in Julia, for two reasons:

  • It's simple. Julia's syntax is readable and standard, so experience with Julia is not a prerequisite for using and modifying the code. Furthermore, Julia can be run online at JuliaBox, so you don't need to install anything.
  • It's fast. Speed is one of the Julia developers' top priorities, so generating very large simulations is feasible.
Instructions for getting started are below.

Examples

The SLE fan

using Graphics2D
using GaussianFreeFields
n = 250
h = fix_boundary_values(DGFF(n))
κ = 3
χ = 2/sqrt(κ) - sqrt(κ)/2
z0 = (n+1)/2 + im*(n+1)/2
fan = GraphicElement[]
for θ=0.0:0.05:2π
push!(fan,Line(flowline(h, z0, χ, θ); color = θ/(2π)*"green" + (1-θ/(2π))*"blue"))
end
showgraphics([fan;[Line([1 n; n n; n 1; 1 1; 1 n])]])

SLE fan

GFF level lines

using Graphics2D
using Contour
using GaussianFreeFields
n = 250
h = DGFF(n)
h0 = fix_boundary_values(h)
showgraphics([Line(c;linesize=1.0,color=0.3*"green")
    for c in contour(1.0:n,1.0:n,h0,0.0).lines]; dim = 512)

GFF level lines

A Gaussian free field surface plot

using GaussianFreeFields
n = 20
h = DGFF(n)
h0 = fix_boundary_values(h)
using PyPlot # run Pkg.add("PyPlot") if you don't already have it
X = [x for x=1:n,y=1:n]
Y = [y for x=1:n,y=1:n]
plot_surface(X, Y, h0, rstride = 1, cstride = 1, cmap="autumn")

GFF

Uniform dimer sampling

using Graphics2D
using Dimers
showgraphics(drawgraph(dimersample(20)))

Dimer Sample

Uniform spanning tree sampling

using Dimers
G = grid_graph(20)
roots = [[true];[false for i=1:length(G.vertices)-1]]
showgraphics(draw_graph(Wilson(G,roots)))

Uniform Spanning Tree sample

Conformal maps

using Graphics2D
using ConformalMaps
slitdomain = [0.0, 0.495, 0.5 + 0.25*im, 0.505, 1.0, 1.0 + 1.0*im, im]
f = ConformalMap(slitdomain,0.5+0.5im;resolution=40)
g = inv(f)
showgraphics([visualize(g;rings=12,rays=20)[2];domain(g)])

Conformal Map

Getting Started

  1. Use your Google account to sign in at https://www.juliabox.org, or install Julia on your own machine.
  2. Open a new notebook and install the packages by running

    Pkg.clone("git://github.com/sswatson/Graphics2D.jl.git")
    Pkg.clone("git://github.com/sswatson/Dimers.jl.git")
    Pkg.clone("git://github.com/sswatson/GaussianFreeFields.jl.git")
    Pkg.clone("git://github.com/sswatson/ConformalMaps.jl.git")

  3. That's it! You can learn about the language on the Julia learning page, and you can read the package documentation and see more examples on my GitHub page. To inspect the source code, open the src directory on GitHub and look at PackageName.jl.