Research

Spectrum of Random d-regular Graphs Up to the Edge

with Horng-Tzer Yau.

Communications on Pure and Applied Mathematics, 2024.

We develop local resampling methods to study the spectrum of random regular graphs down to microscopic scales. These yield complete eigenvector delocalization up to the spectral edge and polynomially accurate eigenvalue rigidity.

Ramanujan Property and Edge Universality of Random Regular Graphs

with Theo McKenzie and Horng-Tzer Yau.

Preprint, 2024.

We develop microscopic edge loop equations and use them to establish edge universality for random regular graphs of every fixed degree d ≥ 3. As a consequence, approximately 69% of these graphs are asymptotically Ramanujan, achieving optimal spectral expansion.

Bulk Universality of Random Regular Graphs

with Yukun He and Xiaoyu Wang.

Preprint, 2026.

We prove bulk universality for random regular graphs of every fixed degree d ≥ 3. Our two-step framework combines quantitative eigenvector moment estimates, which establish Gaussian-wave limits, with microscopic bulk loop equations.

Local Law and Mesoscopic Fluctuations of Dyson Brownian Motion for General β and Potentials

with Benjamin Landon.

Probability Theory and Related Fields, 2019.

We develop the method of characteristics for Dyson Brownian motion with general potentials and β ≥ 1, proving local laws and nearly optimal rigidity in the bulk. The method also yields a central limit theorem for mesoscopic linear statistics.

Dynamical Loop Equation

with Vadim Gorin.

The Annals of Probability, 2024.

We introduce dynamical loop, or Dyson–Schwinger, equations for broad families of two-dimensional interacting particle systems. These equations describe global fluctuations and yield Gaussian free field limits for a family of weighted lozenge tilings.

Loop Equations Characterize Random Matrix Statistics

with Paul Bourgade.

Preprint, 2026.

We characterize the universal Sine_β and Airy_β point processes through their loop equation hierarchies for every rational β > 0. These uniqueness results provide a direct route to universality through approximate loop equations.

Strong Characterization for the Airy Line Ensemble

with Amol Aggarwal.

Inventiones mathematicae, 2026.

We characterize the Airy line ensemble, up to an independent affine shift, through the Brownian Gibbs property and parabolic behavior of its top curve. This resolves longstanding uniqueness predictions and provides a new route to Airy limits in the KPZ universality class.

A Convergence Framework for Airy_β Line Ensemble via Pole Evolution

with Lingfu Zhang.

Advances in Mathematics, 498, 111028, 2026.

We provide a framework for proving convergence to the Airy_β line ensemble via a characterization through the pole evolution of meromorphic functions satisfying certain stochastic differential equations.

Edge Statistics for Lozenge Tilings of Polygons, I: Concentration of Height Function on Strip Domains

Probability Theory and Related Fields, 2023.

We establish nearly optimal concentration of the height function and arctic boundary for lozenge tilings of suitable strip domains. These estimates form a key input for understanding universal edge statistics in polygonal domains.

Edge Statistics for Lozenge Tilings of Polygons, II: Airy Line Ensemble

with Amol Aggarwal.

Forum of Mathematics, Pi, 2025.

We prove convergence to the Airy line ensemble near regular arctic-boundary points of simply connected polygons under suitable limit-shape assumptions. The proof combines height-function concentration with a local comparison to the well-understood tilings of a hexagon.

Height Fluctuation for Lozenge Tilings of Polygons

Preprint, 2026.

We introduce a tiling action whose critical points encode the limit-shape geometry and use it to construct explicit contour-integral approximations to the inverse Kasteleyn matrix. This yields Gaussian free field fluctuations for a broad class of polygons whose sides cycle through the three lattice directions.

Efficient Derivative-Free Bayesian Inference for Large-Scale Inverse Problems

with Daniel Zhengyu Huang, Sebastian Reich, and Andrew M. Stuart.

Inverse Problems, 2022.

We develop derivative-free methods for approximate Bayesian inference in large-scale inverse problems with expensive forward models. The framework uses Kalman methodology and a mean-field dynamical system to build practical ensemble algorithms.

Efficient, Multimodal, and Derivative-Free Bayesian Inference with Fisher–Rao Gradient Flows

with Yifan Chen, Daniel Zhengyu Huang, Sebastian Reich, and Andrew M. Stuart.

Inverse Problems, 2024.

We combine Fisher–Rao gradient flows with Gaussian mixture approximations and derivative-free Kalman updates. The resulting method efficiently represents multiple posterior modes in challenging Bayesian inverse problems.

Convergence Analysis of Probability Flow ODE for Score-Based Generative Models

with Daniel Zhengyu Huang and Zhengjiang Lin.

IEEE Transactions on Information Theory, 2025.

We establish convergence guarantees for deterministic probability-flow ODE samplers in score-based generative models under L² control of the score error. The analysis quantifies sampling errors from both score approximation and numerical time discretization.

Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models

with Daniel Zhengyu Huang and Zhengjiang Lin.

Preprint, 2025.

We prove convergence guarantees for high-order ODE solvers in diffusion models, including Runge–Kutta and exponential Runge–Kutta methods. The results connect rigorous error analysis with fast sampling algorithms, accounting for both score and discretization errors.