I am trying to present on this page various simulations I did over the past years, and especially to give a few details about the SLE pictures I produced. Some of the images produced are on this page. Help yourself, use them as you see fit, just let me know. If you are interested in the code, you can download the full source for the last versions with complete history and my current experiments if you have git. Just say

git clone git://github.com/vbeffara/Simulations.git

Try them out, modify them, everything is under the GPL license. I develop essentially on Mac OS X, but once in a while I check that everything compiles on Linux. I tend to like experimenting with new features (typically the functional stuff in C++11/C++14, and more recently OpenMP) so you mostly need a recent compiler if you want to try things out.

Included in the source code is a separate library called libvb which is used for both easily displaying pictures on screen during the simulation, and producing a picture file in the end. It is not very well documented at this point, but it should still be reasonably easy to use, reasonably cross-platform, and it can save you a lot of time if you want to produce pictures yourself.

SLE - Schramm-Loewner Evolution

The SLE pictures are obtained quite brutally by solving Loewner’s equation (using an Euler scheme) starting from various points in the upper-half plane; the two shades of grey correspond to the sign of the real part of $g(z)$ at time $1$ and the black curve is the boundary between them, that is, the SLE trace. That is very easy to program, and it gives very pleasing pictures at a controlled resolution; plus it lends itself to various generic optimizations, especially knowing that the boundary thus obtained is connected: Not all points have to be computed.

The “tail” in the SLE pictures is due to a trick in the simulation; what is represented is really obtained by using Loewner’s equation driven by a Brownian motion stopped at time $1$, and the SLE cluster at time $1$ is whatever appears on the picture, minus the smooth tail. (The trick is that looking at the picture at time infinity, the complement of the cluster has a right side and a left side, which is neat (see below); if you know of any way to locate the point where the tail meets the cluster, I am very interested in knowing how to do that …)

There are of course other, more “complex analytical” ways to draw pictures of SLE, the first one to come to mind being to use the reverse flow and look at the images of the origin for various values of the time parameter. The basic version of that is even easier to program, and the main advantage is that one gets a list of points that are distributed along the trace of the process, which can be plotted using any software. Tom Kennedy has a nice way of optimizing the procedure. It is my (debatable) opinion that the pictures obtained that way look somewhat nicer for small values of $\kappa$ but are not as satisfying when $\kappa$ is greater than $4$, in particular at the multiple points …

One other way to proceed is to use the so-called zipper algorithm introduced by Don Marshall, though I couldn’t find any SLE pictures produced this way online …

Walks

There are two very different kinds of walks present on the page. The easiest one to describe, though apparently the most difficult to study, is the (uniform) self-avoiding walk, which is just the uniform measure on the set of discrete, self-avoiding paths of a given length. The only efficient way to simulate that is known as the pivot algorithm, which is a Monte-Carlo method. Besides the spectral gap in this case is not known, so it is not clear how precise the process is …

The other kind present here is the loop-erased walk, which is just a simple random walk from which all the loops are removed as soon as they appear. This does not lead to a uniform distribution on the set of paths; in fact, it is expected to have scaling exponent 5/4 whereas the SAW should have exponent 4/3.

A third walk-related object is DLA, obtained as follows: Start on the lattice $\mathbb{Z}^2$, with a core at the origin. Launch a simple random walk “from infinity” (or equivalently, with a known distribution on some large square), until it touches the core, and stop it there, so that it sticks to the core. Then launch a second walk, which will stick either to the core or to the final position of the first walk, and so on. This generates a tree-like structure, about which not much is known at all, but it provides with nice pictures anyway.

Lattice models

The easiest one to set up is percolation: on the lattice $\mathbb{Z}^2$, declare each edge open with some probability $p$, and closed with probability $1-p$, independently of all the others; then, look for connected clusters of open edges. For all but a critical value of the parameter, this model has a finite correlation length and thus has no (non-trivial) scaling limit; but at criticality (in this case, when $p=1/2$), one expects a non-trivial limit to exist. Represented are two samples of large clusters at criticality.

The gradient percolation model is the same thing except that the parameter $p$ depends on the location of the edge. In other words, it will be subcritical at some places and supercritical at others, and interesting things happen around the boundary between these two regions.

The Ising model describes the interaction of ferromagnetic spins on a lattice. The picture on the page is a sample of this model taken at criticality, on a $1000 \times 1000$ grid, with boundary conditions that are white on the left and black on the right, generated by perfect simulation, using the “coupling from the past” technique. The interface is supposed to look like an SLE for $\kappa=3$ … You can try to generate larger samples, but CFTP takes forever: this one took about 4 hours.

More details

These are a few separate pages for a few of the programs, with additional pictures and details.

Glauber dynamics of the 3D Ising model

This is a video of the zero-temperature Glauber dynamics of the Ising model in the 3-dimensional cubic lattice, with “hexagonal” boundary conditions (which is equivalent to the usual flip dynamics on rhombus tilings of the hexagon). It converges to an asymptotic shape exhibiting an arctic circle.