Sandpile groups and spanning trees of directed line graphs (2009)
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,...
Internal diffusion-limited aggregation is a growth model based on random walk in Z^d. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks...
Growth Rates and Explosions in Sandpiles (2009)
Fey, Anne, Levine, Lionel, Peres, Yuval
We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in Z^d. Any site with at least 2d particles then topples by sending one...
Parallel Chip-Firing on the Complete Graph: Devil's Staircase and Poincare Rotation Number (2008)
We study how parallel chip-firing on the complete graph K_n changes behavior as we vary the total number of chips. Surprisingly, the activity of the system, defined as the average number of firings...
Much of my research is centered on aggregation models, an area at the interface of combinatorics, probability theory and partial differential equations. An aggregation model in the integer lattice Zd...
Chip-Firing and Rotor-Routing on Directed Graphs (2008)
Holroyd, Alexander E., Levine, Lionel, Meszaros, Karola, Peres, Yuval, Propp, James, Wilson, David B.
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...
Limit Theorems for Internal Aggregation Models (2007)
We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router...
Limit Theorems for Internal Aggregation Models (2007)
We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router...
Scaling Limits for Internal Aggregation Models with Multiple Sources (2007)
We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which...
The Rotor-Router Model on Regular Trees (2007)
Landau, Itamar, Levine, Lionel
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this...
Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile (2007)
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is...
The Sandpile Group of a Tree (2007)
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...
The sandpile group of a tree (2007)
Abstract. We describe a short exact sequence relating the sandpile group of a tree to those of its principal subtrees. In the case of a regular tree this sequence splits, enabling us to compute the...
Spherical Asymptotics for the Rotor-Router Model in Z^d (2005)
The rotor-router model is a deterministic analogue of random walk invented by Jim Propp. It can be used to define a deterministic aggregation model analogous to internal diffusion limited...
Mixing for Markov Chains and Spin Systems (2005)
Yuval Peres, Ra Kliem, Lionel Levine, Yun Long, Asaf Nachmias, Alex Skorokhod, ...
for help preparing these notes. These notes have not been subjected to the usual scrutiny reserved for formal publications. –Yuval Lecture 1: Introduction and total variation distance
Polynomial recurrences and cyclic resultants (2004)
Hillar, Christopher J., Levine, Lionel
Let $K$ be an algebraically closed field of characteristic zero and let $f \in K[x]$. The $m$-th {\it cyclic resultant} of $f$ is \[r_m = \text{Res}(f,x^m-1).\] A generic monic polynomial is...
Building on earlier work of Diaconis and Fulton (1991) and Lawler, Bramson, and Griffeath (1992), Propp in 2001 defined a deterministic analogue of internal diffusion-limited aggregation. This growth...
Fractal Sequences and Restricted Nim (2004)
The Grundy number of an impartial game G is the size of the unique Nim heap equal to G. We introduce a new variant of Nim, Restricted Nim, which restricts the number of stones a player may remove...
Chip-Firing and Rotor-Routing on Directed Graphs
Er E. Holroyd, Lionel Levine, Karola Mészáros, Yuval Peres, David B. Wilson
Abstract. 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...