Selected Works
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This page contains short descriptions of some of my mathematical works.
For a full list of my publications and preprints, see my CV or my works on arXiv.
The Exponential Random Graph Model
The ERGM is a Gibbs measure on graphs, where the probability of a particular graph is tilted according to subgraph counts,
which has the effect of encouraging or discouraging the presence of certain subgraphs, like triangles.
This model exhibits a phase transition and with high-temperature and low-temperature phase analogous to many other statistical mechanics models.
In my paper Concentration via Metastable Mixing, with Applications to the Supercritical Exponential Random Graph Model,
I prove that the metastable wells in the low-temperature phase behave qualitatively like copies of the full high-temperature phase, through the lens of concentration of measure.
This validates a particular case of the folklore belief that low-temperature models can effectively be decomposed into regions exhibiting high-temperature behavior.
Independent Sets in a Percolated Hypercube
Independent sets in a graph are sets of vertices which contain no neighbors.
They are important from a computer science perspective as counting the number of independent sets and finding the largest independent sets in general graphs are both NP-hard problems.
Additionally, they are used in the hard-core model of a gas in statistical mechanics.
In the '80s, Sapozhenko's graph container method was developed in order to approximately count the number of independent sets in the Boolean hypercube, and more recently interest has developed
in a disordered media version of this problem, where the edges of the hypercube are removed independently with probability \(p\) each.
In our paper Gaussian to log-normal transition for independent sets in a percolated hypercube
(joint work with Mriganka Basu Roy Chowdhury and Shirshendu Ganguly), we develop a new probabilistic framework to study the number of independent sets in this random graph.
We prove that this random variable obeys a central limit theorem when \(p>\tfrac{2}{3}\), and has a log-normal distributional limit when \(p=\tfrac{2}{3}\).
Moreover, we use our techniques to extract information about typical geometric structure of independent sets in a percolated hypercube for all \(p\geq\tfrac{2}{3}\).