Finding thresholds and structure in discrete random models
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chin-byronc-phd-math-2026-thesis.pdf
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Author(s)
Chin, Byron
Advisor(s)
Mossel, Elchanan
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
This thesis studies a variety of discrete random models from two perspectives – the location of thresholds and the emergence of structure. These two perspectives turn out to be often intertwined. The structure of the model informs the location of the threshold, and the phase diagram determines regimes in which distinct behaviors emerge.
Part I of the thesis focuses on locating thresholds. Chapter 2 initiates the study of weak recovery for stochastic block models in a new regime where the number of communities grows with the size of the graph. When the number of communities is sufficiently large, we push past the usual barrier for efficient weak recovery and ask if the new barrier we uncover is the true threshold. Chapter 3 identifies the exact threshold for the appearance of nonlinear Hamilton cycles in random hypergraphs. This confirms a conjecture of Narayanan and Schacht, and along the way we compute the limiting distribution of the number of such cycles at the threshold. Chapter 4 introduces a model of corrupted Glauber dynamics in which some vertices are updated by an adversary. We study the quantitative relationship between the size of the corruption and the effect on global features of the system.
Part II of the thesis focuses on results related to structural properties. Chapter 5 presents the first proof of typical structure in a sparse Erdős–Rényi random graph conditioned on a lower tail subgraph count event. We show that the conditioned graph resembles the mean-field prediction in certain regimes. Chapter 6 proves the first vanishing upper bound on the probability that a random instance of the stable roommates problem has a solution by understanding the structure of the solution set. This resolves a long-standing conjecture of Gusfield and Irving from 1989. Chapter 7 studies clade (fringe subtree) counts for beta-splitting trees in the critical regime via the precise asymptotic analysis of a related recursive sequence. These results answer a number of questions of Aldous, Janson, and Pittel.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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