The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to countable Markov chains. Subject to general conditions, we prove that many natural quantities associated with the rotor walk (including normalized hitting frequencies, hitting times and occupation frequencies) concentrate around their expected values for the random walk. Furthermore, the concentration is stronger than that associated with repeated runs of the random walk; the discrepancy is at most C/n after n runs (for an explicit constant C), rather than c / √ n. 1

Abstract. A wired tree is a graph obtained from a tree by collapsing the leaves to a single vertex. We describe a pair of short exact sequences relating the sandpile group of a wired tree to the sandpile groups of its principal subtrees. In the case of a regular tree these sequences split, enabling us to compute the full decomposition of the sandpile group as a product of cyclic groups. This resolves in the affirmative a conjecture of E. Toumpakari concerning the ranks of the Sylow p-subgroups. 1.

Abstract. 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 neighbouring 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 towards the root. For i.i.d. 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 walker’s 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 o(1) functions. No larger discrepancy is possible, while for regular trees the discrepancy is at most logarithmic. 1.

Abstract. We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpile group of LG by its k-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. 1.

An abelian sandpile is a collection of indistinguishable chips distributed among the vertices of a graph. More precisely, it is a function from the vertices to the nonnegative integers, indicating how many chips are at each vertex. A vertex is called unstable if it has at least as many chips as its degree, and an unstable vertex can topple by sending one chip to each neighboring vertex. Note that toppling one vertex may cause neighboring vertices to become unstable. If the graph is connected and infinite, and the number of chips is finite, then all vertices become stable after finitely many topplings. An easy lemma says that the final stable configuration is independent of the order of topplings (this is the reason

Abstract. In Dhar’s model of abelian distributed processors, finite automata occupy the vertices of a graph and communicate via the edges. A local commutativity condition ensures that the output of such a network does not depend on the order in which the automata process their inputs. In this paper (the first of a series) we consider the halting problem for such networks and the crticial group, an invariant that governs the behavior of the network on large inputs. Our main results are 1. A finite abelian network halts on all inputs if and only if its Laplacian is positive definite; 2. The critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network; 3. The critical group is a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. 1.

The broad goal of my research is to understand how large-scale forms and complex patterns emerge from simple local rules. My approach is to analyze specific mathematical models which isolate just one or a few features of pattern formation. A good model is one that captures some aspect of “scaling up ” from local to global, yet is tractable enough to prove theorems about! Some of the models I’m working on include • Abelian sandpiles [BTW87], a model of self-organization and pattern formation. • Parallel chip-firing [BG92], a model of mode-locking and synchronization. • Internal DLA [LBG92], a model of fluid flow and random interfaces. • Rotor-router aggregation [LP09], a model based on derandomizing random walks. Abelian networks, invented by Deepak Dhar [Dha06], tie these models and many others together in a common mathematical framework. The mathematics involved is a mixture of probability and combinatorics, draws on techniques that originated in the study of partial differential equations, and has close connections with statistical physics and computer science. Figure 1. Pattern formation in a stable sandpile (left) and an exploding sandpile (right) in Z 2. Each site is colored according to the number of sand grains present.

We prove the existence of recurrent initial configurations for the rotor walk on many graphs, including Z d, and planar graphs with locally finite embeddings. We also prove that recurrence and transience of rotor walks are invariant under changes in the starting vertex and finite changes in the initial configuration.

The sandpile group of a graph G is an abelian group whose order is the number of spanning trees of G. We find the decomposition of the sandpile group into cyclic subgroups when G is a regular tree with the leaves are collapsed to a single vertex. This result can be used to understand the behavior of the rotor-router model, a deterministic analogue of random walk studied first by physicists and more recently rediscovered by combinatorialists. Several years ago, Jim Propp simulated a simple process called rotor-router aggregation and found that it produces a near perfect disk in the integer lattice Z 2. We prove that this shape is close to circular, although it remains a challenge to explain the near perfect circularity produced by simulations. In the regular tree, we use the sandpile group to prove that rotor-router aggregation started from an acyclic initial condition yields a perfect ball.

Abstract. In rotor walk on a finite directed graph, the exits from each vertex follow a prescribed periodic sequence. Here we consider the case of rotor walk where a particle starts from a designated source vertex and continues until it hits a designated target set, at which point the walk is restarted from the source. We show that the sequence of successively hit targets, which is easily seen to be eventually periodic, is in fact periodic. We show moreover that reversing the periodic patterns of all rotor sequences causes the periodic pattern of the hitting sequence to be reversed as well. The proofs involve a new notion of equivalence of rotor configurations, and an extension of rotor walk incorporating time-reversed particles. 1.