Hyperbolic and Parabolic Unimodular Random Maps Omer Angel Tom Hutchcroft Asaf Nachmias Gourab Ray

Hyperbolic and Parabolic Unimodular Random Maps Omer Angel Tom Hutchcroft Asaf Nachmias Gourab Ray

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply co...

Asaf Nachmias ’ Lectures in PIMS Probability Summer School 2014 Random Walks on Random Fractals

Geometry and percolation on half planar triangulations

We analyze the geometry of domain Markov half planar triangulations. In [5] it is shown that there exists a one-parameter family of measures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition in view of their geometry at α = 2/3. For α < 2/3, the maps form a...

The critical random graph, with martingales

We give a short proof that the largest component of the random graph G(n, 1/n) is of size approximately n. The proof gives explicit bounds for the probability that the ratio is very large or very small.

Tessellation and Lyubich-Minsky laminations associated with quadratic maps II: Topological structures of 3-laminations

We investigate topological and combinatorial structures of Lyubich and Minsky’s affine and hyperbolic 3-laminations associated with the hyperbolic and parabolic quadratic maps. We begin by showing that hyperbolic rational maps in the same hyperbolic component have quasi-isometrically the same 3-laminations. Then we describe the topological and combinatorial changes of laminations associated wit...