Professor of Mathematics and Statistics
Stanford University, USA
Department of Statistics
390 Jane Stanford Way
Stanford, CA 94305, USA
I am interested in probability theory, statistics, and mathematical physics. The following are some of my favorite works. (This list is subject to change from time to time.) For the complete list of publications and preprints, click here or visit my page in Google Scholar.
Spin glass phase at zero temperature in the Edwards-Anderson model. Preprint. 2023. [pdf] [Slides]
While the analysis of mean-field spin glass models has seen tremendous progress in the last twenty years, lattice spin glasses have remained largely intractable. This article presents the solutions to a number of questions about the Edwards-Anderson model of short-range spin glasses (in all dimensions) that were raised in the physics literature many years ago. First, it is shown that the ground state is sensitive to small perturbations of the disorder, in the sense that a small amount of noise gives rise to a new ground state that is nearly orthogonal to the old one with respect to the site overlap inner product. Second, it is shown that one can overturn a macroscopic fraction of the spins in the ground state with an energy cost that is negligible compared to the size of the boundary of the overturned region - a feature that is believed to be typical of spin glasses but clearly absent in ferromagnets. The third result is that the boundary of the overturned region in dimension d has fractal dimension strictly greater than d - 1, confirming a prediction from physics. The fourth result is that the correlations between bonds in the ground state can decay at most like the inverse of the distance. This contrasts with the random field Ising model, where it has been shown recently that the correlation decays exponentially in distance in dimension two. The fifth result is that the expected size of the critical droplet of a bond grows at least like a power of the volume. Taken together, these results comprise the first mathematical proof of glassy behavior in a short-range spin glass model.
A new coefficient of correlation. J. Amer. Statist. Assoc. 2021. [pdf] [Slides]
Is it possible to define a coefficient of correlation which is (a) as simple as the classical coefficients like Pearson's correlation or Spearman's correlation, and yet (b) consistently estimates some simple and interpretable measure of the degree of dependence between the variables, which is 0 if and only if the variables are independent and 1 if and only if one is a measurable function of the other, and (c) has a simple asymptotic theory under the hypothesis of independence, like the classical coefficients? This article answers this question in the affirmative, by producing such a coefficient.
Rigorous solution of strongly coupled SO(N) lattice gauge theory in the large N limit. Comm. Math. Phys. 2019. [pdf] [Slides]
Gauge-string duality is a general notion in physics which claims that gauge theories — which are theories of the quantum world — are sometimes dual to certain string theories, which are theories of gravity. The main result of this paper is a rigorous computation of Wilson loop expectations in strongly coupled SO(N) lattice gauge theory in the large N limit, in any dimension. The formula appears as an absolutely convergent sum over trajectories in a kind of string theory on the lattice, demonstrating an explicit gauge-string duality.
Nonlinear large deviations. Adv. Math. 2016.
(Coauthored with Amir Dembo.) [pdf] [Slides]
This paper develops techniques for understanding large deviations of sparse random graphs. The techniques for dense graphs, based on Szemerédi's regularity lemma, are not applicable in the sparse regime. The technology developed here applies more broadly to large deviations for nonlinear functions of independent random variables, going beyond classical methods which cater mostly to linear functions.
Superconcentration and Related Topics. Springer, Cham. 2014. [Download link] [Slides]
This monograph studies three features of Gaussian random fields, called superconcentration, chaos, and multiple valleys, and explores the relations between them. It is shown that superconcentration is equivalent to chaos, and chaos implies multiple valleys. Superconcentration has been a known feature in probability theory for a while (under different names). This book connects it to chaos and multiple valleys. Two main results in the book are proofs of the disorder chaos and multiple valley conjectures for mean-field spin glasses.
The universal relation between scaling exponents in first-passage percolation. Ann. Math. 2013. [pdf] [Slides]
It is a longstanding conjecture that in the model of first-passage percolation on a lattice, two important numbers, known as the fluctuation exponent and the wandering exponent, are related through a universal relation that does not depend on the dimension. This is sometimes called the KPZ relation. This paper gives a rigorous proof of the KPZ relation assuming that the exponents exist in a certain sense.
The large deviation principle for the Erdős-Rényi random graph. European J. Comb. 2011.
(Coauthored with S. R. S. Varadhan.) [pdf] [Slides]
This paper develops a theory of large deviations for random graphs, using Szemerédi's regularity lemma and the theory of graph limits.
A new method of normal approximation. Ann. Probab. 2008. [pdf] [Slides]
This paper introduces a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, a general CLT is proved for functions that are obtained as combinations of many local contributions, where the definition of "local" itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by P. Bickel.