My research specialty surrounds the **analysis of probability models**---in particular, **interacting particle systems** and **Laplacian growth models**---on state spaces which are bounded in distance determined by **electrical resistance**.
By studying explicit models and taking the right **space-time scaling limits**, I can rigorously derive **(stochastic) partial differential equations** or prove **limit shape theorems** on these spaces.
These mathematical results capture various laws of nature, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.

**My current focus project** is to establish microscopic derivations of **(non)linear (S)PDEs** from **interacting particle systems**---the main model being the **(weakly a)symmetric exclusion process** with variable speed boundary---on spaces which are bounded in the resistance metric.
The two tunable "knobs" are the bulk antisymmetric jump rates and the rates of the boundary birth-and-death dynamics.
By tuning these knobs we obtain a large variety of (S)PDEs with appropriate boundary conditions, describing the macroscopic behavior of particles in and out of equilibrium.
The underlying state spaces include trees, fractals, and random environments, and do not necessarily possess translational invariance like Euclidean spaces.
See papers 9, 11, 16, 17, P1, P2, P3.

Other probability models I have studied include: internal diffusion-limited aggregation (13), rotor-router aggregation (14), abelian sandpiles (14), and Gaussian free fields (3).
In all of these models I have established optimal or near-optimal results on the corresponding state spaces.
For instance, paper 14 is the first to rigorously establish that abelian sandpile growth on a geometrically self-similar state space exhibits power law modulated by log-periodic oscillations.

In order to gain a deeper understanding of stochastic processes on resistance spaces, I have also studied **spectral analysis of Laplacian and Schrödinger operators** on self-similar fractal spaces, on which there are a number of peculiar features (4, 5, 15).
The results have implications for wave propagation (7), and for the asymptotic complexity of combinatorial structures such as spanning trees (10) and cycle-rooted spanning forests (15).

My paper 9 is an example where these two aspects of my work (probability theory + spectral analysis) meet: spectral gap inequality + electric network reduction = **moving particle lemma** in the exclusion process.

While the models I study may seem disparate, my approach is ultimately grounded in the study of **analysis on graphs** (Laplacian, heat kernels, and spectral analysis), **Markov processes** (such as random walks, Markov chains, Poisson processes, and Brownian motions), and **potential theory** (using techniques such as Dirichlet forms and related functional inequalities).

Papers 1, 6, 8, 14, 15 involved notable contributions from undergraduate students in the form of numerical findings, conjectures, and lemmas, which were then turned into theorems by the senior author(s).