Asaf Nachmias

Critical random graphs: Diameter and mixing time

Let C 1 denote the largest connected component of the critical Erdos-Renyi random graph G(n, 1/n). We show that, typically, the diameter of C 1 is of order n 1/3 and the mixing time of the lazy simple random walk on C 1 is of order n. The latter answers a question of Benjamini, Kozma and Wormald. These results extend to clusters of size n 2/3 of p-bond percolation on any d-regular n-vertex graph where such clusters exist, provided that p(d-1)≤ 1+O(n- 1/3 ).

Arm exponents in high dimensional percolation

We study the probability that the origin is connected to the sphere of radius r (an arm event) in critical percolation in high dimensions, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We prove that this probability decays like 1/r^2. Furthermore, we show that the probability of having k disjoint arms to distance r emanating from the vicinity of the origin is 1/r^2k.

Testing the expansion of a graph

We study the problem of testing the expansion of graphs with bounded degree d in sublinear time. A graph is said to be an @a-expander if every vertex set U@?V of size at most 12|V| has a neighborhood of size at least @a|U|. We show that the algorithm proposed by Goldreich and Ron [9] (ECCC-2000) for testing the expansion of a graph distinguishes with high probability between @a-expanders of degree bound d and graphs which are -far from having expansion at least @W(@a^2). This improves a recent result of Czumaj and Sohler [3] (FOCS-07) who showed that this algorithm can distinguish between @a-expanders of degree bound d and graphs which are -far from having expansion at least @W(@a^2/logn). It also improves a recent result of Kale and Seshadhri [12] (ECCC-2007) who showed that this algorithm can distinguish between @a-expanders and graphs which are -far from having expansion at least @W(@a^2) with twice the maximum degree. Our methods combine the techniques of [3], [9] and [12].

Colouring powers of cycles from random lists

Let C n k be the kth power of a cycle on n vertices (i.e. the vertices of C n k are those of the n-cycle, and two vertices are connected by an edge if their distance along the cycle is at most k). For each vertex draw uniformly at random a list of size c from a base set S of size s=s(n). In this paper we solve the problem of determining the asymptotic probability of the existence of a proper colouring from the random lists for all fixed values of c, k, and growing n.

The evolution of the cover time

The cover time of a graph is a celebrated example of a parameter that is easy to approximate using a randomized algorithm, but for which no constant factor deterministic polynomial time approximation is known. A breakthrough due to Kahn, Kim, Lovasz and Vu [25] yielded a (log logn)2 polynomial time approximation. We refine the upper bound of [25], and show that the resulting bound is sharp and explicitly computable in random graphs. Cooper and Frieze showed that the cover time of the largest component of the Erdős–Renyi random graph G(n, c/n) in the supercritical regime with c > 1 fixed, is asymptotic to ϕ(c)nlog2n, where ϕ(c) → 1 as c ↓ 1. However, our new bound implies that the cover time for the critical Erdős–Renyi random graph G(n, 1/n) has order n, and shows how the cover time evolves from the critical window to the supercritical phase. Our general estimate also yields the order of the cover time for a variety of other concrete graphs, including critical percolation clusters on the Hamming hypercube {0, 1}n, on high-girth expanders, and on tori ℤdn for fixed large d. This approach also gives a simpler proof of a result of Aldous [2] that the cover time of a uniform labelled tree on k vertices is of order k3/2. For the graphs we consider, our results show that the blanket time, introduced by Winkler and Zuckerman [45], is within a constant factor of the cover time. Finally, we prove that for any connected graph, adding an edge can increase the cover time by at most a factor of 4.

Unimodular hyperbolic triangulations: circle packing and random walk

We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons [1]. As a part of this, we obtain an alternative proof of the Benjamini–Schramm Recurrence Theorem [19]. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the simple random walk has positive speed in the hyperbolic metric.

A power law of order 1/4 for critical mean field Swendsen-Wang dynamics

The Swendsen-Wang dynamics is a Markov chain widely used by physicists to sample from the Boltzmann-Gibbs distribution of the Ising model. Cooper, Dyer, Frieze and Rue proved that on the complete graph K_n the mixing time of the chain is at most O(n^{1/2}) for all non-critical temperatures. In this paper we show that the mixing time is Theta(1) in high temperatures, Theta(log n) in low temperatures and Theta(n^{1/4}) at criticality. We also provide an upper bound of O(log n) for Swendsen-Wang dynamics for the q-state ferromagnetic Potts model on any tree with n vertices.

Hyperbolic and Parabolic Unimodular Random Maps

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 connected unimodular random rooted map is sofic, that is, a Benjamini–Schramm limit of finite maps.

Random walks on stochastic hyperbolic half planar triangulations

We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in Angel and Ray [3]. We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like

Nonconcentration of return times

\exp(-cn^{1/3})