Visualizing rays in a random geometry

The rays in an AC geometry are the flow lines of the complex vector field e^{i(h + alpha)} where h is a constant multiple of the Gaussian free field and alpha is a number in [0, 2pi). (Some care is needed to make sense of this construction, since h is defined only as a distribution and not a function.) When h is constant, these are the rays of ordinary Euclidean geometry (and alpha corresponds to angle). When h is Sqrt{8 kappa/Pi}/(4 - kappa) times the Gaussian free field, the rays are forms of SLE(kappa).

The following are a few attempts to represent the AC geometry graphically. In all of the figures, h is the projection of the GFF onto the space of continuous functions which are linear on each triangle in an 80000-triangle grid. (So h is a random piecewise linear function whose values on vertices have the law of a discrete Gaussian free field. We used periodic boundary conditions, so the boundary points are not special (but we stop drawing our flow lines when they reach the boundary, instead of letting them wrap around).

Spokes

In the first batch of figures, rays of different angles (alpha values) beginning at the same point are shown, with the angles coded by color.

In the continuum version of this construction, the critical angle is the largest angle for which open rays of that angle beginning at the same point can intersect one another. (Although distinct rays beginning at the same point may intersect, they never cross one another transversely.) When h is a random piecewise linear function, the rays never actually intersect one another (because h is Lipschitz), but they can come very close.

Fewer rays are shown in the figures with larger critical angles. In all cases the angles (which correspond to different alpha choices) are evenly spaced and color coded the same way. When the critical angle is bigger than 2pi, a single flow line (in the continuum version of this construction) may wrap around its starting point and intersect itself. (Paths that intersect themselves are a bit hard to follow on the screen.) When the critical value is n Pi, we have kappa = 4n/(n+1).

In addition to n=infty (kappa=4) the statistical physically most important values of kappa seem to correspond to positive integer values of n. For example, n=1 (uniform spanning tree, loop erased random walk), n=2 (self avoiding walk, percolation hull), n=3 (Ising model), and n=5 (three-state Potts model). Unfortunately, the larger n becomes, the more tangled up the flow lines become and the harder it is to see what is going on in the pictures.

• Critical angle = 0, kappa = 0
• Critical angle = Pi/1000, kappa = 4/1001
• Critical angle = Pi/100, kappa = 4/101
• Critical angle = Pi/10, kappa = 4/11
• Critical angle = Pi/4, kappa = 4/5
• Critical angle = Pi/4, kappa = 4/5 with rays drawn in 200 directions
• Critical angle = Pi/2, kappa = 4/3
• Critical angle = Pi, kappa = 2
• Critical angle = Pi, kappa = 2 with rays drawn in 200 directions
• Critical angle = 2Pi, kappa = 8/3
• Critical angle = 3Pi, kappa = 3
• Critical angle = 5Pi, kappa = 10/3

Grids

Now let's try a different approach to visualizing the random geometry. Shown below are all the east-going lines (alpha = 0) beginning at a row of points on the west side of the box and the north-going lines (alpha = .25) beginning at a row of points on the south side of the box. This forms a "grid" which becomes increasingly distorted as one moves east and north.
• Critical angle = 0, kappa = 0
• Critical angle = Pi/1000, kappa = 4/1001
• Critical angle = Pi/100, kappa = 4/101
• Critical angle = Pi/10, kappa = 4/11
• Critical angle = Pi/2, kappa = 4/3
• Critical angle = Pi, kappa = 2

Trees

Now we try yet another approach. We throw down fifty points uniformly at random and draw the east-going line (alpha = 0) beginning at those points. Then we throw down another fifty points and use a different color to draw the west-going line (alpha = Pi) beginning at those points.
• Critical angle = 0, kappa = 0
• Critical angle = Pi/1000, kappa = 4/1001
• Critical angle = Pi/100, kappa = 4/101
• Critical angle = Pi/2, kappa = 4/3
• Critical angle = Pi, kappa = 2
• Critical angle = 2Pi, kappa = 8/3
• Critical angle = 3Pi, kappa = 3