Omer Angel

Uniform Infinite Planar Triangulations

The existence of the weak limit as n→∞ of the uniform measure on rooted triangulations of the sphere with n vertices is proved. Some properties of the limit are studied. In particular, the limit is a probability measure on random triangulations of the plane.

The stationary measure of a 2-type totally asymmetric exclusion process

We give a combinatorial description of the stationary measure for a totally asymmetric exclusion process (TASEP) with second class particles, on either Z or on the cycle ZN. The measure is the image by a simple operation of the uniform measure on some larger finite state space. This reveals a combinatorial structure at work behind several results on the TASEP with second class particles.

Random sorting networks

Abstract A sorting network is a shortest path from 12 ⋯ n to n ⋯ 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the space–time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Holder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.

Percolations on random maps I: Half-plane models

We study Bernoulli percolations on random lattices of the half-plane obtained as local limit of uniform planar triangulations or quadrangulations. Using the characteristic spatial Markov property or peeling process [5] of these random lattices we prove a surprisingly simple universal formula for the critical threshold for bond and face percolations on these graphs. Our techniques also permit us to compute off-critical and critical exponents related to percolation clusters such as the volume and the perimeter.

Amenability of linear-activity automaton groups

We prove that every linear-activity automaton group is amenable. The proof is based on showing that a sufficiently symmetric random walk on a specially constructed degree 1 automaton group — the mother group — has asymptotic entropy 0. Our result answers an open question by Nekrashevich in the Kourovka notebook, and gives a partial answer to a question of Sidki.

Random orderings and unique ergodicity of automorphism groups

We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss’s example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.

Localization for linearly edge reinforced random walks

We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.

The TASEP speed process.

In the multi-type totally asymmetric simple exclusion process (TASEP), each site ofZ is occupied by a labeled particle, and two neighboring particles are int erchanged at rate one if their labels are in increasing order. Consider the process with the initial con figuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [−1, 1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is a stationary measure for the multi-type TASEP dynamics. Consequently, we can describe every ergodic stationary measure as a projection of the speed process measure. This generalizes previous descriptions restricted to finitely many classes. By combining this result with known stationary measures for TASEPs with finitely many types we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribut ion is that two speeds are equal with positive probability and for any given particle there are infinitely m any others with the same speed. We also study the (partially) asymmetric simple exclusion process (ASEP). We prove that the states of the ASEP with the above initial configuration, seen as permutations ofZ, are symmetric in distribution. This allows us to extend some of our results, i ncluding the stationarity and description of all ergodic stationary measures, also to the ASEP.

Classification of half-planar maps

We characterize all translation invariant half planar maps satisfying a certain natural domain Markov property. For p-angulations with p \ge 3 where all faces are simple, we show that these form a one-parameter family of measures H^{(p)}_{alpha}. For triangulations we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well known half plane uniform infinite planar maps. The sub-critical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.

Rotor Walks on General Trees

The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighboring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially toward the root. For independently and identically distributed random initial rotor directions on a regular tree, the limiting proportion of escapes is either zero or the random walk escape probability, and undergoes a discontinuous phase transition between the two as the distribution is varied. In the critical case there are no escapes, but the walkers maximum distance from the root grows doubly exponentially with the number of visits to the root. We also prove that there exist trees of bounded degree for which the proportion of escapes eventually exceeds the escape probability by arbitrarily large