Alexander E. Holroyd

Chip-Firing and Rotor-Routing on Directed Graphs

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Integrals, partitions, and cellular automata

We prove that ∫ 1 0 -log f(x)/x dx = π 2 /3ab, where f(x) is the decreasing function that satisfies f a - f b = x a - x b , for 0 < a < b. When a is an integer and b = a + 1 we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having a consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.

Extra heads and invariant allocations

Let n be an ergodic simple point process on E d and let n* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of Π and n*; that is, one can select a (random) point Y of n such that translating n by -Y yields a configuration whose law is that of Π*. We construct shift couplings in which Y and Π* are functions of Π, and prove that there is no shift coupling in which n is a function of Π*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R d ) to the points of n. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let Γ be an ergodic random element of {0, 1} Zd and let Γ* be r conditioned on r(0) = 1. A shift coupling X of r and Γ* is called an extra head scheme. We show that there exists an extra head scheme which is a function of r if and only if the marginal E[r(0)] is the reciprocal of an integer. When the law of Γ is product measure and d > 3, we prove that there exists an extra head scheme X satisfying E exp c∥X∥ d < ∞; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].

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.

Edge routing with ordered bundles

We propose a new approach to edge bundling. At the first stage we route the edge paths so as to minimize a weighted sum of the total length of the paths together with their ink. As this problem is NP-hard, we provide an efficient heuristic that finds an approximate solution. The second stage then separates edges belonging to the same bundle. To achieve this, we provide a new and efficient algorithm that solves a variant of the metro-line crossing minimization problem. The method creates aesthetically pleasing edge routes that give an overview of the global graph structure, while still drawing each edge separately, without intersecting graph nodes, and with few crossings.

Slow convergence in bootstrap percolation

In the bootstrap percolation model, sites in an L by L square are initially infected independently with probability p. At subsequent steps, a healthy site becomes infected if it has at least 2 infected neighbours. As (L; p) ! (1; 0), the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter p log L, occurring asymptotically at = 2 =18 [15]. We prove that the discrepancy between the critical parameter and its limit is at least ((log L) 1=2 ). In contrast, the critical window has width only ((log L) 1 ). For the so-called modied model, we prove rigorous explicit bounds which imply for example that the relative discrepancy is at least 1% even when L = 10 3000 . Our results shed some light on the observed dierences between simulations and rigorous asymptotics.

Entanglement in Percolation

We study the existence of finite and infinite entangled graphs in the bond percolation process in three dimensions with density p. After a discussion of the relevant definitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small p, and it is shown that this graph has radius whose tail decays at least as fast as exp(−αn/ log n); indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant α. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation.

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

Tail Bounds for the Stable Marriage of Poisson and Lebesgue

o(1)

Geometry of Lipschitz percolation

functions. No larger discrepancy is possible, while for regular trees, the discrepancy is at most logarithmic.