149 citations · 841 across the 44 of their papers we have counts for
30 papers · 1 filter
Universality Conjectures for Activated Random Walk
Lionel Levine, Vittoria Silvestri
Activated Random Walk is a particle system displaying Self-Organized Criticality, in that the dynamics spontaneously drive the system to a critical state. How universal is this cri…
Exact sampling and fast mixing of Activated Random Walk
Lionel Levine, Feng Liang
Activated Random Walk (ARW) is an interacting particle system on the -dimensional lattice . On a finite subset it defines a Markov chain o…
Branching in a Markovian Environment
Lila Greco, Lionel Levine
A branching process in a Markovian environment consists of an irreducible Markov chain on a set of "environments" together with an offspring distribution for each environment. At e…
How far do Activated Random Walkers spread from a single source?
Lionel Levine, Vittoria Silvestri
Unlike many particle systems, Activated Random Walk has nontrivial behavior even in one spatial dimension. We prove inner and outer bounds on the spread of n activated random walke…
Random walks with local memory
Swee Hong Chan, Lila Greco, Lionel Levine +1
We prove a quenched invariance principle for a class of random walks in random environment on , where the walker alters its own environment. The environment consists…
How long does it take for Internal DLA to forget its initial profile?
Lionel Levine, Vittoria Silvestri
Internal DLA is a discrete model of a moving interface. On the cylinder graph , a particle starts uniformly on and perfo…
Formation of large-scale random structure by competitive erosion
Shirshendu Ganguly, Lionel Levine, Sourav Sarkar
We study the following one-dimensional model of annihilating particles. Beginning with all sites of uncolored, a blue particle performs simple random walk from unt…
Double jump phase transition in a soliton cellular automaton
Lionel Levine, Hanbaek Lyu, John Pike
In this paper, we consider the soliton cellular automaton introduced in [Takahashi 1990] with a random initial configuration. We give multiple constructions of a Young diagram desc…
Sandpiles on the square lattice
Bob Hough, Dan Jerison, Lionel Levine
We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice . We also determine the asymptotic spectral ga…
Laplacian growth, sandpiles and scaling limits
Lionel Levine, Yuval Peres
Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We…
Interpolating between random walk and rotor walk
Wilfried Huss, Lionel Levine, Ecaterina Sava-Huss
We introduce a family of stochastic processes on the integers, depending on a parameter and interpolating between the deterministic rotor walk (p=0) and the simple ra…
Mixing time and eigenvalues of the abelian sandpile Markov chain
Daniel C. Jerison, Lionel Levine, John Pike
The abelian sandpile model defines a Markov chain whose states are integer-valued functions on the vertices of a simple connected graph . By viewing this chain as a (nonreversib…
Formation of an interface by competitive erosion
Shirshendu Ganguly, Lionel Levine, Yuval Peres +1
In 2006, the fourth author proposed a graph-theoretic model of interface dynamics called competitive erosion. Each vertex of the graph is occupied by a particle that can be either…
The divisible sandpile at critical density
Lionel Levine, Mathav Murugan, Yuval Peres +1
The divisible sandpile starts with i.i.d. random variables ("masses") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an…
The range of a rotor walk
Laura Florescu, Lionel Levine, Yuval Peres
In a \emph{rotor walk} the exits from each vertex follow a prescribed periodic sequence. On an infinite Eulerian graph embedded periodically in , we show that any simple roto…
Oil and water: a two-type internal aggregation model
Elisabetta Candellero, Shirshendu Ganguly, Christopher Hoffman +1
We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting with n "oil" and n "water" particles at the origin, the particles diffuse in…
Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle
Lionel Levine
We prove a precise relationship between the threshold state of the fixed-energy sandpile and the stationary state of Dhar's abelian sandpile: In the limit as the initial condition…
Internal DLA for cylinders
David Jerison, Lionel Levine, Scott Sheffield
We show that space-time averages of the fluctuations of internal DLA on a cylinder Z_N x Z converge in law to the Gaussian free field.
Escape rates for rotor walk in Z^d
Laura Florescu, Shirshendu Ganguly, Lionel Levine +1
Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z^d for the initial configuration of all rotors aligned. If n particles…
The looping constant of Z^d
Lionel Levine, Yuval Peres
The looping constant is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in . Poghosyan, Priezzhev and Ruelle, and inde…
Internal DLA and the Gaussian free field
David Jerison, Lionel Levine, Scott Sheffield
In previous works, we showed that the internal DLA cluster on \Z^d with t particles is a.s. spherical up to a maximal error of O(\log t) if d=2 and O(\sqrt{\log t}) if d > 2. This…
Internal DLA in Higher Dimensions
David Jerison, Lionel Levine, Scott Sheffield
Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d > 2. We show that A(t) is approximately spherical, up…
Logarithmic fluctuations for internal DLA
David Jerison, Lionel Levine, Scott Sheffield
Let each of n particles starting at the origin in Z^2 perform simple random walk until reaching a site with no other particles. Lawler, Bramson, and Griffeath proved that the resul…
Fast simulation of large-scale growth models
Tobias Friedrich, Lionel Levine
We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a "least action principle" which ch…