## Lionel LevineAssociate ProfessorDepartment of Mathematics Cornell University |

Hello, I'm Lionel! I study abelian networks. These are

If you want to learn a bit about this fascinating nexus of math, computer science, and statistical physics, I recommend starting with WHAT IS a sandpile? for a non-technical overview, and Laplacian growth, sandpiles, and scaling limits for a more recent survey.

Some highlights of my research are the scaling limit of the abelian sandpile in Z^2 where an Apollonian circle packing makes a surprise appearance, the devil's staircase for parallel chip-firing, refuting the density conjecture for sandpiles, logarithmic fluctuations for internal DLA, asynchronous circuits with integer input and output, fast simulation of growth models, a generalization of Knuth's formula for spanning trees, and word equations in uniquely divisible groups.

I thank the National Science Foundation, Sloan Foundation, Simons Foundation, and Institute for Advanced Study for supporting my research.

I'm an editor of Combinatorial Theory, a open-source journal with no author fees and double-blind refereeing. Please submit your work to CT!

- Lionel Levine and Feng Liang

Exact sampling and fast mixing of Activated Random Walk

*Submitted*

- Lila Greco and Lionel Levine

Branching in a Markovian environment

*Submitted*

- Viktor Kiss, Lionel Levine, and Lilla Tóthmérész

The devil's staircase for chip-firing on random graphs and on graphons

*Submitted*

- Swee Hong Chan and Lionel Levine

Abelian networks IV. Dynamics of nonhalting networks

*Memoirs of the American Mathematical Society, to appear*

- Lionel Levine and Vittoria Silvestri

How far do Activated Random Walkers spread from a single source?

*Journal of Statistical Physics, to appear*

- Lionel Levine, Hanbaek Lyu and John Pike

Double jump phase transition in a random soliton cellular automaton

*International Math Research Notices (2022) 665--727*

- Swee Hong Chan, Lila Greco, Lionel Levine, and Peter Li
- Random walks with local memory

*Journal of Statistical Physics 184 (2021), Article 6, 28 pp.*

- Lionel Levine and Vittoria Silvestri

How long does it take for internal DLA to forget its intitial profile?

*Probability Theory and Related Fields (2019) 174:1219--1271*

- Shirshendu Ganguly, Lionel Levine and Sourav Sarkar

Formation of large-scale random structure by competitive erosion

*Annals of Probability (2019) 47:3649--3704*

- Bob Hough, Daniel C. Jerison and Lionel Levine

Sandpiles on the square lattice

*Communications in Mathematical Physics (2019) 367:33--87*

- Daniel C. Jerison, Lionel Levine and John Pike

Mixing time and eigenvalues of the abelian sandpile Markov chain

*Transactions of the American Mathematical Society (2019) 372:8307--8345*

- Alexander E. Holroyd, Lionel Levine and Peter Winkler

Abelian logic gates

*Combinatorics, Probability, and Computing (2019) 28:388--422*

- Wilfried Huss, Lionel Levine and Ecaterina Sava-Huss

Interpolating between random walk and rotor walk

*Random Structures & Algorithms (2018) 52:263--282*

- Lionel Levine and Yuval Peres

Laplacian growth, sandpiles and scaling limits

*Bulletin of the American Mathematical Society (2017) 54:355--382*

- Shirshendu Ganguly, Lionel Levine, Yuval Peres and James Propp

Formation of an interface by competitive erosion

*Probability Theory and Related Fields (2017) 168:455--509*

- Elisabetta Candellero, Shirshendu Ganguly, Christopher Hoffman and Lionel Levine

Oil and water: a two-type internal aggregation model

*Annals of Probability (2017) 45:4019--4070*

- Lionel Levine, Wesley Pegden and Charles K. Smart

Apollonian structure of integer superharmonic matrices

*Annals of Math (2017) 186:1--67*

- Lionel Levine and
Ramis Movassagh

The gap of the area-weighted Motzkin spin chain is exponentially small

*Journal of Physics A: Mathematical and Theoretical 50.25 (2017)*

- Lionel Levine, Wesley Pegden and Charles K. Smart

Apollonian structure in the abelian sandpile

*Geometric and Functional Analysis (2016) 26(1):306--336*

- Benjamin Bond and Lionel Levine

Abelian networks III. The critical group

*Journal of Algebraic Combinatorics (2016) 43:635--663*

- Benjamin Bond and Lionel Levine

Abelian networks II. Halting on all inputs

*Selecta Mathematica (2016) 22:319--340*

- Benjamin Bond and Lionel Levine

Abelian networks I. Foundations and examples

*SIAM Journal on Discrete Mathematics (2016) 30:856--874*

- Matthew Farrell and Lionel Levine

CoEulerian graphs

*Proceedings of the American Mathematical Society (2016) 144:2847--2860*

- Matthew Farrell and Lionel Levine

Multi-Eulerian tours of directed graphs

*Electronic Journal of Combinatorics (2016) 23:P2.21*

- Lionel Levine, Mathav Murugan, Yuval Peres and
Baris
Ugurcan

The divisible sandpile at critical density

*Annales Henri Poincare (2016) 17(7):1677-1711*

- Laura Florescu, Lionel Levine and Yuval Peres

The range of a rotor walk

*American Mathematical Monthly (2016) 123(7):627--642*

- Lionel Levine

Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle

*Communications in Mathematical Physics (2015) 335(2):1003–-1017*

- Louis J. Billera, Lionel Levine and Karola Mészáros

How to decompose a permutation into a pair of labeled Dyck paths by playing a game

*Proceedings of the American Mathematical Society (2015) 143:1865-–1873*

- David Jerison, Lionel Levine and Scott Sheffield

Internal DLA and the Gaussian free field

*Duke Mathematical Journal (2014) 163(2):267–-308*

- Lionel Levine and Yuval Peres

The looping constant of Z^d

*Random Structures & Algorithms (2014) 45:1--13*

- Laura Florescu, Shirshendu Ganguly, Lionel Levine and Yuval Peres

Escape rates for rotor walks in Z^d

*SIAM Journal on Discrete Mathematics (2014) 28(1):323--334*

- Lionel Levine, Scott Sheffield and Katherine E. Stange

A duality principle for selection games

*Proceedings of the American Mathematical Society (2013) 141(12): 4349--4356*

- Christopher J. Hillar, Lionel Levine and Darren Rhea

Equations solvable by radicals in a uniquely divisible group

*Bulletin of the London Mathematical Society (2013) 45(1): 61--79*

- Tobias Friedrich and Lionel Levine

Fast simulation of large-scale growth models

*Random Structures & Algorithms (2013) 42: 185–-213*

- David Jerison, Lionel Levine and Scott Sheffield

Internal DLA in higher dimensions

*Electronic Journal of Probability (2013) 18(98): 1--14*

- David Jerison, Lionel Levine and Scott Sheffield

Logarithmic fluctuations for internal DLA

*Journal of the American Mathematical Society (2012) 25: 271--301*

- Giuliano Giacaglia, Lionel Levine, James Propp and Linda Zayas-Palmer

Local-to-global principles for the hitting sequence of a rotor walk

*Electronic Journal of Combinatorics (2012) 19: P5*

- Lionel Levine

Sandpile groups and spanning trees of directed line graphs

*Journal of Combinatorial Theory A (2011) 118: 350-–364*

- Lionel Levine

Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number

*Ergodic Theory and Dynamical Systems (2011) 31: 891--910*

- Wouter Kager and Lionel Levine

Rotor-router aggregation on the layered square lattice

*Electronic Journal of Combinatorics (2010) 17: R152*

- Anne Fey, Lionel Levine and David B. Wilson

Driving sandpiles to criticality and beyond

*Physical Review Letters (2010) 104: 145703*

- Anne Fey, Lionel Levine and David B. Wilson

The approach to criticality in sandpiles

*Physical Review E (2010) 82: 031121*

- Wouter Kager and Lionel Levine

Diamond Aggregation

*Mathematical Proceedings of the Cambridge Philosophical Society (2010) 149: 351--372*

- Anne Fey, Lionel Levine and Yuval
Peres

Growth rates and explosions in sandpiles

*Journal of Statistical Physics (2010) 138: 143--159*

- Lionel Levine and Yuval Peres

Scaling limits for internal aggregation models with multiple sources

*Journal d'Analyse Mathematique (2010) 111: 151--219*

- Itamar Landau and Lionel Levine

The rotor-router model on regular trees

*Journal of Combinatorial Theory A (2009) 116: 421--433*

- Lionel Levine and Yuval
Peres

Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile

*Potential Analysis 30 (2009), 1--27*

- Lionel Levine

The sandpile group of a tree

*European Journal of Combinatorics 30 (2009) 1026--1035*

- Alexander E. Holroyd,
Lionel Levine,
Karola Mészáros,
Yuval Peres,
James Propp and
David B. Wilson

Chip-firing and rotor-routing on directed graphs

*in In and Out of Equilibrium II, Progress in Probability vol. 60 (Birkhauser, 2008)*

- Lionel Levine and Yuval Peres

Spherical asymptotics for the rotor-router model in Z^d

*Indiana University Mathematics Journal 57 (2008), no. 1, 431--450*

- Christopher J. Hillar and Lionel Levine

Polynomial recurrences and cyclic resultants

*Proceedings of the American Mathematical Society 135 (2007), 1607--1618*

- Lionel Levine

Fractal sequences and restricted Nim

*Ars Combinatoria 80 (2006), 113--127*

- Lionel Levine and Katherine E. Stange

How to make the most of a shared meal: plan the last bite first.

*American Mathematical Monthly 119 (2012) no. 7, 550--565*

- Lionel Levine and James Propp

WHAT IS a sandpile?

*Notices of the American Mathematical Society 57, (2010) no. 8, 976--979*

- Lionel Levine and Yuval Peres

The rotor-router shape is spherical

*Mathematical Intelligencer 27 (2005) no. 3, 9--11*

- Lionel Levine

Fermat's little theorem: a proof by function iteration

*Mathematics Magazine 72, no. 4 (1999), 308--309*

- CoEulerian graphs

- Halting problems for sandpiles and abelian networks: video

- Introduction to abelian networks: slides video

- The future of prediction (Math Awareness Public Lecture)

- Circles in the sand

- Logarithmic fluctuations from circularity

- An algebraic analogue of a formula of Knuth

- Chip-firing and a devil's staircase

- Obstacle problems and lattice growth models

- MATH 7710: Topics in Probability: Limits of discrete random structures, Spring 2022

- MATH 6710: Graduate Probability I, Fall 2021

- MATH 1110: Calculus I, Fall 2021

- MATH 6720: Graduate Probability II, Spring 2021

- MATH 6710: Graduate Probability I, Fall 2020

- MATH 7710: Topics in Probability: Abelian Sandpiles and Abelian Networks, Fall 2020

- MATH 4210: Nonlinear Dynamics and Chaos, Spring 2020

- MATH 6720: Graduate Probability II, Spring 2020

- MATH 6710: Graduate Probability I, Fall 2019

- MATH 6720: Graduate Probability II, Spring 2018

- MATH 6720: Graduate Probability II, Spring 2017

- MATH 1340: Mathematics and Politics, Spring 2016

- MATH 4740: Stochastic Processes, Spring 2015

- MATH 6710: Probability I, Fall 2014

- MATH 4740: Stochastic Processes, Spring 2014

- MATH 4740: Stochastic Processes, Spring 2013

- MATH 7770: Topics in Probability: Laplacian Growth, Fall 2012

- 18.312 Algebraic
Combinatorics, Spring 2011

- (with Yuval Peres) Internal erosion and the
exponent 3/4 describes how an unusual exponent arises from a very simple erosion process in one dimension. The proof we give
is due to Kingman and Volkov (2003), who thought of this not as an erosion process but as a model of a
gunfight (!)

- Orlik-Solomon Algebras of
Hyperplane Arrangements, an expository paper proving the basic theorem
of Orlik-Solomon and Brieskorn on the cohomology ring of the complement of
a complex arrangement, along with some remarks about the associated
combinatorics of the intersection lattice.

- My Ph.D. thesis, advised by Yuval Peres at Berkeley, used ideas from free boundary problems in PDE to prove limit shapes for both random and deterministic growth models. This topic has advanced a lot in the last fifteen years. Our article Laplacian growth, sandpiles and scaling limits surveys some of the advances.

- My senior thesis on the rotor-router model
was advised by Jim Propp.

- Confounding factors for
Hamilton's rule, the final paper for an anthropology class I took in
2002. I found that the rule is surprisingly
sensitive to changes in Hamilton's original hypotheses, which casts some
doubt on the evolutionary stability of kin selection.

- Hall's marriage theorem and Hamiltonian
cycles in graphs, the final paper from a spring 2001 reading course
with Richard Stanley.
A graph on n=24 vertices having no Hamiltonian cycle, in which every set of k<22 vertices is adjacent to at least k+3 vertices:

- Some basic results on Sturmian
words, written before I knew that's what they were called. These
results are all known in some form. Theorem 1 and the surprising
corollary to Theorem 2 go back to Morse and Hedlund (1940).

The beginning of the factor tree of the Sturmian word of slope sqrt(2)/2 and intercept zero: