Periodic Reporting for period 4 - RANDGEOM (Random Geometry)
Berichtszeitraum: 2020-07-01 bis 2021-12-31
Another set of problems is investigating aspects of universality in critical percolation in various high-dimensional graphs. These graphs include lattices in dimension above 6, Cayley graphs of finitely generated non-amenable groups and also finite graphs such as the complete graph, the Hamming hypercube and expanders. It is believed that critical percolation on these graphs is universal in the sense that the resulting percolated clusters exhibit the same mean-field geometry.
2. A proof that the uniform spanning forest on any proper planar map with bounded degrees is almost surely connected. This answers a question of Benjamini, Lyons, Peres and Schramm 2001.
3. A proof that the exit measure of the random walk on a unimodular hyperbolic triangulation exists (that is, the random walker converges) and has full support and no atoms.
4. A proof that almost surely impossible to distinguish the connected components of the uniform spanning forest from each other by invariantly defined graph properties, resolving a conjecture of Benjamini, Lyons, Peres and Schramm 2001.
5. A proof that the diameter of the uniform spanning tree on high-dimensional graph grows like square root of the volume.
6. A proof that the local limit of the uniform spanning tree on regular graph with degree tending to infinity is the Poisson(1) branching process conditioned to survive.
7. A proof that the random walk on orthodiagonal maps converges to Brownian motion.
8. A proof that the circle packing on the so-called mated-CRT random triangulation has no macroscopic circles.
9. A full characterization of parabolicity/hyperbolicity of random planar maps.
10. The book "Planar maps, random walks and circle packing" which constitutes a comprehensive introduction to the research topics of this ERC project.