Georgia Institute of Technology College of Sciences

In the study of random growth models belonging to the Kardar--Parisi--Zhang (KPZ) universality class, a notably successful approach has been to analyze stationary initial conditions defined by the Busemann functions. Recently, this perspective has been extended to handle multiple asymptotic directions simultaneously, but the joint distribution of the Busemann process is more difficult to access, and many aspects of this process remain elusive. In particular, the remarkable independence property present in the exactly solvable setting fails when considering Busemann functions across different directions. In the corner growth model, also known as exponential last-passage percolation (LPP), we prove that, regardless of their different directions, Busemann functions along a down-right path are always negatively associated across each individual direction. In other words, increasing the value of Busemann functions in one direction tends to probabilistically decrease the values of neighboring ones. As an application, we obtain an exponential concentration inequality on the diffusive scale for Busemann functions along a down-right path, in the absence of independence. Joint work with Erik Bates.

Spin glasses are disordered statistical physics system with both ferromagnetic and anti-ferromagnetic spin interactions. The Gibbs measure belongs to the exponential family with parameters, such as inverse temperature $\beta>0$ and external field $h\in R$.  A fundamental statistical problem is to estimate the system parameters from a single sample of the ground truth. In 2007, Chatterjee first proved that under reasonable conditions, for spin glass models with $h=0$, the maximum pseudo-likelihood estimator for $\beta$ is $\sqrt{N}$-consistent. This is in contrast to the existing estimation results for classical non-disordered models. However, Chatterjee's approach has been restricted to the single parameter estimation setting.  The joint parameter estimation of $(\beta,h)$ for spin glasses has remained open since then. In this talk, I will introduce a new idea to show that under some easily verifiable conditions,  the bi-variate maximum pseudo-likelihood estimator is jointly $\sqrt{N}$-consistent for a large collection of spin glasses, including the Sherrington-Kirkpatrick model and its diluted variants. Based on joint work with Wei-Kuo Chen, Arnab Sen. 

We study a broad class of space–time continuous directed polymers in a Gaussian environment that is white in time and spatially correlated (Itô sense). Under Dalang’s condition, we prove key properties of the partition function—positivity, stationarity, scaling, homogeneity, and a Chapman–Kolmogorov relation—and establish pathwise regularity of the polymer (Hölder continuity and quadratic variation). We give a sharp singularity criterion: the polymer measure is singular w.r.t. Wiener measure iff the spectral measure has infinite total mass. Finally, for d≥3, we prove diffusive large-time behavior in the high-temperature regime, providing a unified framework for polymers driven by singular Gaussian noise.

Joint work with Cheng Ouyang (UIC), Samy Tindel (Purdue), and Panqiu Xia (Cardiff).

Partial identification provides an alternative to point identification: instead of pinning down a unique parameter estimate, the goal is to characterize a set guaranteed to contain the true parameter value. Many partial identification approaches take the form of linear optimization problems, which seek the "best- and worst-case scenarios" of a proposed model subject to the constraint that the model replicates correct observable information. However, such linear programs become intractable in settings with multivalued or continuous variables. This paper introduces a novel method to overcome this computational and statistical curse of cardinality: an entropy penalty transforms these potentially infinite-dimensional linear programs into general versions of multi-marginal Schrödinger bridges, enabling efficient approximation of their solutions. In the process, we establish novel statistical and mathematical properties of such multi-marginal Schrödinger bridges---including an analysis of the asymptotic distribution of entropic approximations to infinite-dimensional linear programs. We illustrate this approach by analyzing  instrumental variable models with continuous variables, a setting that has been out of reach for existing methods.