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Random Geometry

Periodic Reporting for period 3 - RANDGEOM (Random Geometry)

Reporting period: 2019-01-01 to 2020-06-30

The objective of this project is an investigation of the geometric structure of random spaces that arise in critical models of statistical physics. We are motivated by inspiring yet non-rigorous predictions from the physics community and the models studied are some of the most popular models in contemporary probability theory such as percolation, random planar maps and random walks. One topic of study is random planar maps and quantum gravity, a thriving field on the intersection of probability, statistical physics, combinatorics and complex analysis. Our goal is to develop a rigorous theory of these maps viewed as surfaces (rather than metric spaces) via their circle packing. The circle packing structure was recently used by the PI and Gurel-Gurevich to show that these maps are a.s. recurrent, re- solving a major conjecture in this area. The main concrete goal of this project is a rigorous proof of the mysterious KPZ correspondence, a conjectural formula from the physics literature allowing to compute dimensions of certain random sets in the usual square lattice from the corresponding dimension in the random geometry. We hope to gain progress on the most central problems in two-dimensional statistical physics, such as estimating the typical displacement of the self-avoiding walk, proving conformal invariance for critical percolation on the square lattice and many others.
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.
"1. A complete geometric and spectral analysis of ""random causal map"" - a natural and popular model of random planar maps.

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:"
1. A proof that (weighted) random walk on circle packings with mesh size going to 0 converges to Brownian motion.
2. A proof that connectivity of the free uniform spanning forest is a 0-1 event.
3. A proof that in a certain model of random maps (namely, mated-CRT map) there are no macroscopic circles.
4. A proof that the spectral dimension of uniform random quadrangulation is 2.
An orthodiagonal map