We study Linial-Meshulam random 2 2 -complexes Y ( n , p ) Y(n,p) , which are 2 2 -dimensional analogues of Erdős-Rényi random graphs. We find the threshold for simple connectivity to be p = n − 1 / 2 p = n^{-1/2} . This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log ⁡ n / n p = 2 \log n / n . We use a variant of Gromov’s local-to-global theorem for linear isoperimetric inequalities to show that when p = O ( n − 1 / 2 − ϵ p = O( n^{-1/2 -\epsilon } ), the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2 2 -dimensional simplicial complexes and establish isoperimetric inequalities for such complexes. These intermediate results do not involve randomness and may be of independent interest.

Abstract. Consider the following method of card shuffling. Start with a deck of N cards numbered 1 through N . Fix a parameter p between 0 and 1. In this model a "shuffle" consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p. If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p = 1/2, the mixing time of this card shuffling is O(N 2 ), as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

Let $\Xi$ be a discrete set in ${\mathbb{R}}^d$. Call the elements of $\Xi$ centers. The well-known Voronoi tessellation partitions ${\mathbb{R}}^d$ into polyhedral regions (of varying sizes) by allocating each site of ${\mathbb{R}}^d$ to the closest center. Here we study ``fair'' allocations of ${\mathbb{R}}^d$ to $\Xi$ in which the regions allocated to different centers have equal volumes. We prove that if $\Xi$ is obtained from a translation-invariant point process, then there is a unique fair allocation which is stable in the sense of the Gale--Shapley marriage problem. (I.e., sites and centers both prefer to be allocated as close as possible, and an allocation is said to be unstable if some site and center both prefer each other over their current allocations.) We show that the region allocated to each center $\xi$ is a union of finitely many bounded connected sets. However, in the case of a Poisson process, an infinite volume of sites are allocated to centers further away than $\xi$. We prove power law lower bounds on the allocation distance of a typical site. It is an open problem to prove any upper bound in $d>1$.Comment: Published at http://dx.doi.org/10.1214/009117906000000098 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Coalescence of semi-infinite geodesics remains a central question in planar first passage percolation. In this paper we study finer properties of the coalescence structure of finite and semi-infinite geodesics for exactly solvable models of last passage percolation. Consider directed last passage percolation on Z 2 with i.i.d. exponential weights on the vertices. Fix two points v 1 = (0, 0) and v 2

We consider the nearest-neighbor simple random walk on Z d , d ≥ 2, driven by a field of bounded random conductances ωxy ∈ [0, 1]. The conductance law is i.i.d. subject to the condition that the probability of ωxy > 0 exceeds the threshold for bond percolation on Z d . For environments in which the origin is connected to infinity by bonds with positive conductances, we study the decay of the 2n-step return probability P 2n ω (0, 0). We prove that P 2n ω (0, 0) is bounded by a random constant times n −d/2 in d = 2, 3, while it is o(n −2 ) in d ≥ 5 and O(n −2 log n) in d = 4. By producing examples with anomalous heat-kernel decay approaching 1/n 2 we prove that the o(n −2 ) bound in d ≥ 5 is the best possible. We also construct natural n-dependent environments that exhibit the extra log n factor in d = 4.