The geometric phase diagram, diffusion, and spectra of random planar maps

The explanation

Draw a network on a sphere with one link. There are two ways: a single dot with a loop, or two dots joined by a line. Now open or close the link. That gives four pictures. Give each picture a weight of √q for every boundary loop that separates open clusters from closed ones. The loop map with the link closed has one such loop, and with it open has two. The two-dot map has two loops when closed and one when open. With q = 9, the weights are 3, 9, 9 and 3. So the four pictures have chances 1/8, 3/8, 3/8 and 1/8. With q = 1, all four are equally likely.

This is a Fortuin–Kasteleyn planar map, or FK map, at parameter qSource 1. Both the network and the open links are random, so q changes the shape of the network itself. The question is what a huge FK map looks like from far away. Shrink distances so the whole map has a fixed size, and let the number of links grow without bound.

Sheffield's work encoded these maps as walks and found a transition at q = 4Source 2. His parameter is √q/(√q + 2). It is below one half for q below 4, and above one half for q above 4Source 3. In the example, q = 9 gives 3/5. Below the transition the maps were predicted to look like a random surface, called a Liouville quantum gravity sphere. Above it they were predicted to look like a random tree. But a limit of the encoding walk does not by itself give the shape of the finite mapSource 4.

The papers claim the full picture. For every q between 0 and 4, FK maps converge to Liouville quantum gravity spheres, decorated by random loopsSource 5. Distances, area and the loops all converge together. Another paper claims the distance part of a conjecture by Gwynne and Miller for these mapsSource 6. At exactly q = 4, the maps converge to a critical version of the sphereSource 7. For every q above 4, distances shrink by a factor of the square root of the size. The map then looks like the Brownian continuum random tree, a standard random tree shapeSource 8,Source 9.

Three papers study a walker that moves along a random link at each step. The shape of a surface does not by itself fix how a walker moves on itSource 10. For q = 2, the FK–Ising case, the walk converges to Liouville Brownian motion, the natural random motion on the limiting sphereSource 11. The same holds for maps weighted by their number of spanning treesSource 12. A spectral paper claims that the vibration frequencies of these maps converge tooSource 13. That result uses stated inputs about Brownian motion and the sphere, taken from other workSource 14.

The single obstacle is that the claims concern one finite map of exact size, measured with all its links. An encoding walk without that size condition does not see thisSource 4. The tree paper finds an exact tree of blocks inside the map. It then controls the distances that each block adds along every path to the root.

How sure is this? A computer proof checker called Lean has a formal proof of the tree limit for every q above 4. The surface results, the q = 4 result and the random-walk and spectral results lie outside that check. The spectral result also depends on outside inputs that it states. No specialist review of any of the seven papers has been published.

What it doesn't show. This does not give a Lean check of the surface, q = 4, walk or spectral results. The spectral result depends on stated outside inputs.