Lionel Levine


Research Focus

Probability and combinatorics

The broad goal of my research is to understand how and why large-scale forms and complex patterns emerge from simple local rules. My approach is to analyze mathematical models that 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 simple enough to prove theorems about! Recently I have been thinking about abelian networks, a generalization of the Bak-Tang-Wiesenfeld abelian sandpile model.


  • Formation of large-scale random structure by competitive erosion (with S. Ganguly and S. Sarkar), Annals of Probability, to appear.
  • Sandpiles on the square lattice (with B. Hough and D. C. Jerison), Communications in Mathematical Physics (2019) 367:33--87
  • The Apollonian structure of integer superharmonic matrices (with W. Pegden and C. Smart), Annals of Mathematics (2017) 186:1--67
  • Apollonian structure in the abelian sandpile (with Wesley Pegden and Charles K. Smart), Geometric and Functional Analysis 26 (2016), no. 1, 306-336.
  • Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle, Communications in Mathematical Physics 335 no. 2 (2015), 1003–1017.
  • Internal DLA and the Gaussian free field (with D. Jerison and S. Sheffield), Duke Mathematical Journal 163 no. 2 (2014), 267–308.
  • Equations solvable by radicals in a uniquely divisible group (with C.J. Hillar and D. Rhea), Bulletin of the London Mathematical Society 45 (2013), 61–79.
  • Logarithmic fluctuations for internal DLA (with D. Jerison and S. Sheffield), Journal of the American Mathematical Society 25 (2012), 271–301.
  • Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number, Ergodic Theory and Dynamical Systems 31 (2011), 891–910.
  • Driving sandpiles to criticality and beyond (with A. Fey and D. B. Wilson), Physical Review Letters 104 (2010), 145703.

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MATH Courses - Spring 2024

MATH Courses - Fall 2024