Research

“What is it that drives some people to do mathematics in the pursuit of knowledge so abstract it can be communicated to but a small group of people? The only answer I can give employs two words: mathematics is beautiful, and it is true.“

— James Simons, Renaissance Technologies

My current research interests are (1) exchangeable structures, Bayesian nonparametric statistics, and machine learning (2) Coulomb gases and related dependent point processes (3) heavy-tailed random matrices. Below is a list of all of my research papers sorted by topic. They are not necessarily in chronological order.

Bayesian Statistics and Neural Networks

• “Over-parameterised Shallow Neural Networks with Asymmetrical Node Scaling: Global Convergence Guarantees and Feature Learning”
  Submitted (arXiv:2302.01002)
  with François Caron, Fadhel Ayed, Hoil lee, Juho Lee, and Hongseok Yang.
• “Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility”
  Submitted (arXiv:2205.08187)
  with Hoil lee, Fadhel Ayed, Juho Lee, Hongseok Yang, and François Caron.
• “-Stable convergence of heavy-tailed infinitely-wide neural networks”
  Advances in Applied Probability (to appear, issue 55.4, December 2023) (arXiv:2106.11064)
  with Hoil Lee, Jiho Lee, and Hongseok Yang.
• “A Generalization of Hierarchical Exchangeability on Trees to Directed Acyclic Graphs”
  Annales Henri Lebesgue (2021) 4:325-368
  with Jiho Lee, Sam Staton, and Hongseok Yang.
  Honorable mention poster, 12th International Conference on Bayesian Nonparametrics.

Coulomb Gases

• “At the edge of a one-dimensional jellium”
  Bernoulli (2022) 28(3): 1784-1809 (arXiv:2012.04633)
  with Djalil Chafaï and David García-Zelada.
• “Large deviations in the quantum quasi-1D jellium”
  Probability and Mathematical Physics (2022) 3(2):381-429 (arXiv:2009.14144)
  with Christian Hirsch and Sabine Jansen.
• “Macroscopic and edge behavior of a planar jellium”
  Journal of Mathematical Physics, (2020) 61(3):033304 (arXiv:1909.00613)
  with Djalil Chafaï and David García-Zelada.
• “Wigner Crystallization in the Quantum 1D Jellium at All Densities.”
  Communications in Mathematical Physics, (2014) 331:1133-1154
  with Sabine Jansen.
• “Symmetry Breaking in Quasi-1D Coulomb Systems”
  Annales Henri Poincaré, (2010) 11(8):1453-1485
  with Michael Aizenman and Sabine Jansen.

Random Matrices

• “Extreme eigenvalue statistics of m-dependent heavy-tailed matrices”
  Annales de l’Institut Henri Poincaré (2021) 57(4):2100-2127 (arXiv:1910.08511)
  with Bojan Basrak, Yeonok Cho, and Johannes Heiny.
• “Delocalization and Limiting Spectral Distribution of Erdős-Rényi Graphs with Constant Expected Degree”
  Electronic Communications in Probability, (2018) 23(92):1-13
  with Jaehun Lee.
• “Lévy-Khintchine random matrices and the Poisson weighted infinite skeleton tree”
  Transactions of the AMS (2018), 370:641-668.

Chaotic Billiards

• “Necessary and sufficient condition for -convergence to a Lévy process for billiards with cusps at flat points”
  Nonlinearity, (2019) 33:807-839 (arXiv:1902.08958)
  with Ian Melbourne, Françoise Pène, Paulo Varandas, and Hongkun Zhang.
• “Convergence to stable Lévy motion for chaotic billiards with several cusps at flat points”
  Nonlinearity, (2019) 33:807-839
  with Françoise Pène and Hongkun Zhang.
• “Stable Laws for Chaotic Billiards with Cusps at Flat Points”
  Annales Henri Poincaré, (2018) 19:3815-3853
  with Hongkun Zhang.

Stable Processes/Fractional Brownian Motion

• “Functional central limit theorem for negatively dependent heavy-tailed stationary infinitely divisible processes generated by conservative flows”
  Annals of Probability (2017), 45:2087-2130
  with Takashi Owada and Gennady Samorodnitsky.
  Korean Mathematical Society Best Paper Award, 2018
• “Hölder Continuity and Occupation-Time Formulas for fBm Self-Intersection Local Time and Its Derivative”
  Journal of Theoretical Probability (2015) 28:299-312
  with Greg Markowsky.
• “On the Tanaka formula for the derivative of self-intersection local time of fBm”
  Stochastic Processes and Their Applications (2014), 124:3846-3868
  with Greg Markowsky.
• “Random-time Isotropic Fractional Stable Fields”
  Journal of Theoretical Probability, (2014) 27:618-633.
• “A Lindeberg-Feller theorem for stable laws”
  Statistics and Probability Letters, (2014) 84:198-203 (arXiv:1302.4011)
  with Clément Dombry.
• “Random walks at random times: Convergence to iterated Lévy motion, fractional stable motions, and other self-similar processes”
  Annals of Probability (2013) 41:2682-2708.
  with Greg Markowsky.
• “Indicator fractional stable motions”
  Electronic Communications in Probability, (2011) 16:165-173.

Interacting Particle Systems

• “On the Critical Behavior at the Lower Phase Transition of the Contact Process”
  ALEA, (2007) 3:301-320
  with Michael Aizenman.
• “Two phase transitions for the contact process on small worlds”
  Stochastic Processes and Their Applications, (2007) 117(12):1910-1929
  with Rick Durrett.
• “The Critical Value of the Contact Process with Added and Removed Edges”
  Journal of Theoretical Probability, (2005) 18(4):949-955.
• “The noisy voter-exclusion process”
  Stochastic Processes and Their Applications, (2005) 115(12):1979-2005.
• “Perturbations of the Symmetric Exclusion Process”
  Markov Processes and Related Fields, (2004) 10(4):564:584. (arXiv:0309237)
• “Extremal Reversible Measures for the Exclusion Process”
  Journal of Statistical Physics, (2003) 112(1):165-191.
• Invariant measures of the exclusion process and related processes.
  Ph.D. Thesis (2003) 1-112.

Miscellaneous

• “The Life and Mathematical Legacy of Thomas M. Liggett”
  Notices of the American Mathematical Society, (2021) 68(1):67-79
  with David Aldous, Pietro Caputo, Rick Durrett, Alexander Holroyd, and Amber Puha.
• “A note on invariance of the Cauchy and related distributions”
  Statistics and Probability Letters, (2020) 158:108668 (arXiv:1908.04006)
  with Wooyoung Chin and Greg Markowsky.
• “Remarks on the recurrence and transience of non-backtracking random walks”
  Journal of Combinatorics, (2020) 11(3):549-555 (arXiv:1905.07863)
  with Greg Markowsky.
• “On the speed and spectrum of mean-field random walks among random conductances”
  Stochastic Processes and Their Applications, (2020) 130(6):3477-3498 (arXiv:1702.04087)
  with Andrea Collevecchio.
• “Target Frequency Analysis of Functional MRI Data.”
  International Journal of Clinical Biostatistics and Biometrics (2015), 1:2
  with Michael Froelich and Shannon Starr.
• “The -Extent of “
  Electronic Journal of Undergraduate Mathematics, (1995) 1:1-11
  with Tucker McElroy and Jason Samuels.