Professor of Mathematics and Statistics
Stanford University, USA

Email: souravc@stanford.edu

Mailing address:
Department of Statistics
390 Jane Stanford Way
Stanford, CA 94305, USA

Selected publications

I am interested in probability theory, statistics, and mathematical physics. The following are some of my favorite works, and some recent preprints. (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.

• A Lorentzian construction of timelike Liouville field theory on the cylinder. Preprint. 2026. [pdf]

  Timelike Liouville field theory is a candidate model for positive curvature two-dimensional quantum gravity, but a mathematically controlled Lorentzian formulation has remained elusive. In this paper we construct the theory on the cylinder $\mathbb{R}\times \mathbb{S}^1$ in the integer screening sector for a natural algebra of renormalized exponential observables. Starting from a renormalized finite-volume torus regularization, we construct infinite-volume Euclidean correlation functions, prove analytic continuation in the time variables, and identify the resulting Lorentzian boundary values by explicit contour formulas. This yields exact Lorentzian correlators for a natural class of exponential observables. We then prove locality: spacelike separated vertex operators commute in the Lorentzian theory. For smeared observables generated by the integer-charge fields $e^{2nb\phi}$, these Lorentzian expectation values define a vacuum functional on an ordered $*$-algebra and support an AQFT-type quantization without positivity. More precisely, we obtain isotone local algebras, a complete locally convex space $\mathcal H$ with dense algebraic subspace $\mathcal H_0$ carrying a nondegenerate Hermitian form (shown to be indefinite for $b<8^{-1/2}$), a continuous cyclic representation, operator-topologically closed represented local algebras, an action of cylinder translations by continuous linear homeomorphisms, and locality for the represented local net. The construction does not produce a Hilbert space or a Haag-Kastler net of local von Neumann algebras in the usual sense, but it shows that a substantial part of the Euclidean-to-Lorentzian and algebraic reconstruction mechanism survives in this nonpositive setting for timelike Liouville theory on the cylinder.

• Exact calculations beyond charge neutrality in timelike Liouville field theory. Preprint. 2026. [pdf] [Slides]

  Timelike Liouville field theory (also known as imaginary Liouville theory or imaginary Gaussian multiplicative chaos) is expected to describe two-dimensional quantum gravity in a positive-curvature regime, but its path integral is not a probability measure and rigorous exact computations are currently available only in the charge-neutral (integer screening) case. In this paper we show that at the special coupling $b=1/\sqrt{2}$, the Coulomb-gas expansion of the timelike path integral becomes explicitly computable beyond charge neutrality. The reason is that the $n$-fold integrals generated by the interaction acquire a Vandermonde/determinantal structure at $b=1/\sqrt{2}$, which allows exact evaluation in terms of classical special functions. We derive Mellin-Barnes type representations (involving the Barnes $G$-function and, in a three-point case, Gauss hypergeometric functions) for the zero- and one-point functions, for an antipodal two-point function, and for a three-point function with a resonant insertion $\alpha_2=b$. We then address the subtle zero-mode integration: after a Gaussian regularization we obtain an explicit renormalized partition function $C(1/\sqrt{2},\mu)=e(4\pi\sqrt{2}\mu)^{-1}$, identify distributional limits in the physically relevant regime $\alpha_j=\frac{1}{2}Q+\mathrm{i}P_j$, and compare with the Hankel-contour prescription recently proposed in the physics literature. These results provide the first rigorously controlled family of exact calculations in timelike Liouville theory outside charge neutrality.

• Rigorous results for timelike Liouville field theory. Forum of Math., Sigma, 2026. [pdf] [Slides]

  Liouville field theory has long been a cornerstone of two-dimensional quantum field theory and quantum gravity, which has attracted much recent attention in the mathematics literature. Timelike Liouville field theory is a version of Liouville field theory where the kinetic term in the action appears with a negative sign, which makes it closer to a theory of quantum gravity than ordinary (spacelike) Liouville field theory. Making sense of this "wrong sign" requires a theory of Gaussian random variables with negative variance. Such a theory is developed in this paper, and is used to prove the timelike DOZZ formula for the 3-point correlation function when the parameters satisfy the so-called "charge neutrality condition". Expressions are derived also for the k-point correlation functions for all k ≥ 3, and it is shown that these functions approach the correct semiclassical limits as the coupling constant is sent to zero.

• Spin glass phase at zero temperature in the Edwards-Anderson model. Preprint. 2023. [pdf] [Slides]

  Mean field spin glass models have undergone substantial mathematical development, but finite dimensional short range spin glasses remain much less understood. This paper proves several rigorous zero temperature signatures of glassy behavior for the Edwards--Anderson model with Gaussian couplings, in finite boxes in arbitrary dimension. First, the ground state is sensitive to small perturbations of the disorder: after a perturbation of size $p$, the new ground state is nearly orthogonal to the original one in site overlap once $p$ is sufficiently larger than the inverse system size. Second, the droplets generated by such perturbations have large interfaces; in the macroscopic-droplet regime, their boundaries satisfy lower bounds consistent with a fractal dimension strictly greater than $d-1$. Third, there exist macroscopic spin excitations whose energy cost is negligible compared with the size of their interface, in sharp contrast with ferromagnetic behavior. Fourth, the expected size of the critical droplet associated with a typical bond grows at least as a power of the volume. Finally, a natural boundary condition sensitivity for nearest-neighbor spin products cannot decay faster than order the inverse distance to the boundary, contrasting with recent exponential decay results for the two-dimensional random field Ising model. Taken together, these results provide rigorous evidence — and, in the senses made precise in the paper, proof — of zero temperature 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.