# Wesley Pegden

## Professor

**Address:**

7105 Wean Hall

Department of Mathematical Sciences

Carnegie Mellon University

5000 Forbes Avenue

Pittsburgh, PA 15213

**P:** 412-268-9782

# Education

Ph.D., Rutgers University

# Awards

- Sloan Research Fellowship

# Research

My research is in the broad area of Discrete Mathematics, including probabilistic combinatorics, combinatorial game theory, graph theory, and discrete geometry.

My recent focus has been on the Abelian sandpile, a simple deterministic diffusion process on the lattice which produces striking fractal limits. My paper with Charles Smart on the convergence of the Abelian sandpile gives a new paradigm—based on the set of quadratic growths attainable by integer superharmonic functions on the integer lattice—for understanding the sandpile's limiting process.

Recent work with Lionel Levine and Charles Smart has used this paradigm to analyze the fractal geometry of the sandpile process, through a computationally established Apollonian property of such quadratic growths.

In very recent work, we prove this Apollonian property underlying the sandpile process, through a recursive construction of integer superharmonic functions which assigns discrete tilings of the plane to circles in the Apollonian band circle packing. There is hope that our construction may ease progress on problems unrelated to the sandpile, including number-theoretic conjectures related to Apollonian circle packings.

# Select Publications

Pegden, W., & Smart, C. K. (2013). Convergence of the Abelian sandpile. *Duke Mathematical Journal*, 162(4), 627-642.

Levine, L., Pegden, W., & Smart, C. K. (2016). Apollonian structure in the Abelian sandpile. *Geometric and functional analysis*, 26(1), 306-336.

Levine, L., Pegden, W., & Smart, C. K. (2017). The Apollonian structure of integer superharmonic matrices. *Annals of Mathematics,* 1-67.

Chikina, M., Frieze, A., & Pegden, W. (2017). Assessing significance in a Markov chain without mixing. *Proceedings of the National Academy of Sciences*, 114(11), 2860-2864.

Frieze, A., & Pegden, W. (2017). Looking for vertex number one. *The Annals of Applied Probability*, 27(1), 582-630.

Frieze, A., & Pegden, W. (2016, June). Separating subadditive Euclidean functionals. *In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing*, (pp. 22-35). ACM.

Pegden, W. (2011). Highly nonrepetitive sequences: winning strategies from the Local Lemma. *Random Structures & Algorithms*, 38(1‐2), 140-161.

Pegden, W. (2014). An extension of the Moser--Tardos algorithmic local lemma.* SIAM Journal on Discrete Mathematics*, 28(2), 911-917.

Pegden, W. (2013). Critical graphs without triangles: an optimum density construction. *Combinatorica*, 33(4), 495-512.