Conformal Probability in Julia
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. Julia is designed from the ground up to be efficiently just-in-time compiled. This means that, after a small compilation delay on each function’s first call, you can expect to get C-like speed with simple, idiomatic Julia code.
Instructions for getting started are below.
The SLE fan
using AsyPlots using FractionalGaussianFields n = 250 h = zeroboundary(torus_gff(n)) κ = 3 χ = 2/sqrt(κ) - sqrt(κ)/2 z0 = (n+1)/2 + im*(n+1)/2 L = 40 C = Colors.colormap("Blues",L) paths = [Path(flowline(h, z0, χ, θ),color=c) for (θ,c) in zip(linspace(0,2π,L),C)] Plot(paths,bgcolor="Black",border=0)
GFF level lines
using AsyPlots using Contour using FractionalGaussianFields n = 250 h = zeroboundary(torus_gff(n)) levellines = contour(1.0:n,1.0:n,h,0.0) C = Colors.distinguishable_colors(length(levellines), lchoices=linspace(90,100,10)) loops = [Polygon(l,color=c) for (c,l) in zip(C,levellines)] P = Plot(loops,bgcolor="Black")
A Gaussian free field surface plot
using AsyPlots using FractionalGaussianFields h = zeroboundary(torus_gff(50)) plot(h)
using AsyPlots using ConformalMaps slitdomain = 2*[0.0, 0.495, 0.5 + 0.25*im, 0.505, 1.0, 1.0 + 1.0*im, im] f = ConformalMap(slitdomain,1+im) g = inv(f) visualize(g)
using PlanarMaps, AsyPlots draw(UWT(100),fillfaces=true,linecolor="black")
- Use your Google account to sign in at https://www.juliabox.com, or install Julia on your own machine.
Open a new notebook and install the packages by running
Pkg.add("AsyPlots") Pkg.add("ConformalMaps") Pkg.add("FractionalGaussianFields")
- 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
srcdirectory on GitHub and look at