Now consider a random metric on the unit disc, in which the length of a path is defined to be the integral along that path of e^h, where h is continuous function. (By the generalized Riemann mapping theorem, all compact, two dimensional Riemannian manifolds that are topologically equivalent to the disc can be represented in this way, since they can all be conformally mapped onto a disc, and we may take h to be the log of the magnitude of the inverse derivative of this map.) When h is constant, the geodesics are ordinary Euclidean lines.
Random two dimensional Riemannian geometries arise in models of quantum gravity. Such a random geometry corresponds to a random function h. Conformal Markov properties suggest that it may be interesting to analyze the case that h is a multiple of the Gaussian free field (see Liouville gravity ). In all of the figures, h is some constant alpha times the projection of the GFF onto the space of continuous functions which are linear on each triangle in an 80000-triangle grid, and the geodesics pictured begin at the center point. (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; we also set a fixed maximum Euclidean length and stop drawing flow lines if they reach this length before hitting the boundary --- thus, in figures depicting geodesic flows that don't hit the boundary, all non-boundary-hitting flow lines have the same Euclidean length).
A local "bump" in h (i.e., a local maximum of h that is strictly bigger than points around it) may correspond to a "mushroom" shape in the random surface (if one views it as embedded in three dimensional space, say), and it is possible for a taut geodesic flow to wind several times around such a mushroom before continuing onward. Based on this intuition, one should not find it too surpising that the geodesic flows in the figures intersect themselves and sometimes loop around an area many times before leaving it.
Parallel transport on this geometry affects the tangent space by a combination of dilations and rotations. Because dilations and rotations commute with each other, an affine connection which acts only by dilation and rotation may be understood as a pair of vector fields---one for the rotations, one for the dilations. In the Levi-Cevita connection (i.e., the unique torsion-free, metric preserving connection), the dilation vector field is the gradient of h (which ensures metric preservation) and the rotation vector field is a ninety-degree rotation of the gradient of h (which ensures zero torsion). In such a connection, the curvature is the Laplacian of h, and the holonomy group is the group of rotations (assuming the Laplacian of h is not identically zero).
There is a natural "dual" torsion-free connection in which one takes the rotation vector field to be the gradient of h and the dilation vector field to be a minus 90 degree rotation of the gradient of h. The auto-parallels of this dual connection (whose holonomy group is the group of dilations) are the rays of the "AC-geometries" described elsewhere on this site. However, while it is understood that continuum Gaussian free field analogs of the AC-geometry rays are forms of SLE, it is not known how to describe continuum Gaussian free field analogs of the geodesics pictured below.
