149 citations · 841 across the 44 of their papers we have counts for
14 papers · 1 filter
The devil's staircase for chip-firing on random graphs and on graphons
Viktor Kiss, Lionel Levine, Lilla Tóthmérész
We study the behavior of the activity of the parallel chip-firing upon increasing the number of chips on an Erdős--Rényi random graph. We show that in various situations the result…
Multi-Eulerian tours of directed graphs
Matthew Farrell, Lionel Levine
Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least o…
CoEulerian graphs
Matthew Farrell, Lionel Levine
We suggest a measure of "Eulerianness" of a finite directed graph and define a class of "coEulerian" graphs. These are the graphs whose Laplacian lattice is as large as possible. A…
How to decompose a permutation into a pair of labeled Dyck paths by playing a game
Louis J. Billera, Lionel Levine, Karola Meszaros
We give a bijection between permutations of length 2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate…
Local-to-global principles for rotor walk
Giuliano Pezzolo Giacaglia, Lionel Levine, James Propp +1
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 d…
Rotor-router aggregation on the layered square lattice
Wouter Kager, Lionel Levine
In rotor-router aggregation on the square lattice Z^2, particles starting at the origin perform deterministic analogues of random walks until reaching an unoccupied site. The limit…
Sandpile groups and spanning trees of directed line graphs
Lionel Levine
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 d…
Growth Rates and Explosions in Sandpiles
Anne Fey, Lionel Levine, Yuval Peres
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…
Parallel Chip-Firing on the Complete Graph: Devil's Staircase and Poincare Rotation Number
Lionel Levine
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…
Chip-Firing and Rotor-Routing on Directed Graphs
Alexander E. Holroyd, Lionel Levine, Karola Meszaros +3
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 s…
The Rotor-Router Model on Regular Trees
Itamar Landau, Lionel Levine
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…
The Sandpile Group of a Tree
Lionel Levine
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…
Fractal Sequences and Restricted Nim
Lionel Levine
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 pl…
The Rotor-Router Model
Lionel Levine
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 aggre…