## From particle systems to (S)PDEs on self-similar and random networks

#### Research tutorials & lectures at Cornell (June 2017) and Bielefeld (July 2017)

My research focuses on the analysis of probability models, as well as the relevant (stochastic) PDEs, on self-similar and random geometries. The overarching goal is to prove limit theorems that describe various laws of nature that take place in these nonsmooth spaces, such as heat flow, wave propagation, charged particles in an electromagnetic field, and fluid dynamics.

From the mathematics point of view, I carry out two interconnected lines of research, one analytic and the other probabilistic.
• On the one hand, I analyze variants of the Laplace operator $$\Delta$$ on fractal spaces, focusing on their spectral properties, as well as analyzing the solvability and regularity of the (S)PDEs they generate.
• On the other hand, I study the scaling limits (law of large numbers, central limit theorem) of random processes involving many interacting particles driven by Laplacian-like operators, representing models of magnetism or crystal growth.
From the physics point of view, the example models that interest me fall into one of two broad categories:
• (Non-equilibrium) statistical mechanics models which are many-particle extensions of the (simple) random walk process;
• Quantum mechanics models of a single particle moving under the influence of a (possibly random) scalar or vector potential.
The former category includes the exclusion process (where no two random walkers can occupy the same site at any time); the Gaussian free field (a model random surface); and Laplacian growth models such as sandpiles and aggregation models. The latter category involves the spectral analysis of electric or magnetic Schrödinger operators, as well as the associated PDEs. While the discrete models may seem disparate, all of them are ultimately grounded in the study of calculus on graphs, random walks, Markov chains, and electric network theory.

PDF My research in 1 slide. (September '16, presented at the "meet-and-greet" session during Colgate's NASC divisional colloquium.)

PDF 5-page description of my research, as of February '17. It is NOT up-to-date, as some of the problems in this statement have been solved. But it gives a user-friendly glimpse of the problems I'm working on.

## Research papers & presentations (see also my pages on arXiv, MathSciNet, zbMATH, ORCID)

### 2017+

13. Local ergodicity in the exclusion process on an infinite weighted graph (HydroSG Part II). Submitted (2017+)

In this paper I prove, on every strongly recurrent weighted graph, the coarse-graining arguments needed to pass from the microscopic observables (in the exclusion process) to the corresponding macroscopic averages. The two-blocks estimate is based on the moving particle lemma established in HydroSG Part I.
PDF An hour-long talk on the moving particle lemma and local ergodicity in the exclusion process, delivered at the CUNY Probability Seminar.

HydroSG Parts III & IV, concerning the hydrodynamic limit theorems and existence/uniqueness/regularity of solutions to semilinear PDEs on SG, are being written in collaboration with Michael Hinz and Alexander Teplyaev.
PDF A 20-minute talk on semilinear evolution equations on resistance spaces, delivered at the Northeast Analysis Network conference in Syracuse.

12. Internal DLA on Sierpinski gasket graphs, with Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev. Submitted (2017+)

In this paper we prove that starting from a corner vertex of SG, an internal diffusion-limited aggregation process (where successive i.i.d. random walks deposit upon first exit from the previous cluster) fills balls in the graph metric with probability 1.

See also the shape theorem for divisible sandpiles on SG by Huss and Sava-Huss.
In an upcoming paper with Jonah Kudler-Flam, we show a universal limit shape for four cellular automata models on SG, and identify the precise order of fluctuations in each model.
PDF A 30-minute talk on limit shape universality of Laplacian growth models on SG, delivered at the conference "Analysis & Geometry on Graphs & Manifolds" at Universität Potsdam, Germany. (Open in Adobe Acrobat to run the simulations.)

11. From non-symmetric particle systems to non-linear PDEs on fractals, with Michael Hinz and Alexander Teplyaev. To appear in the Springer Proceedings in Mathematics & Statistics: 2016 conference Stochastic Partial Differential Equations & Related Fields in honor of Michael Röckner's 60th birthday (2017+)

This is a summary of a series of four papers (HydroSG Parts I~IV) concerning the hydrodynamic limit of the boundary-driven exclusion process on a resistance space (the Sierpinski gasket being the model space).

10. Regularized Laplacian determinants of self-similar fractals, with Alexander Teplyaev and Konstantinos Tsougkas. Lett. Math. Phys., Online First (2017+).

In this paper we give rigorous meaning to regularized logarithmic Laplacian determinants on several fractal graphs, using properties of the spectral zeta function. In two examples (the double-sided SG and the double-sided pq model) we find that the logarithmic discrete graph Laplacian determinant has a leading-order term whose coefficient is the asymptotic complexity constant in the enumeration of spanning trees, and whose lagging term corresponds to the regularized logarithmic Laplacian determinant. Our results generalize those of Chinta, Jorgenson, and Karlsson on the discrete tori, and have implications for quantum fields and statistical mechanics on fractals.

### 2017

9. The moving particle lemma for the exclusion process on a weighted graph (HydroSG Part I). Electron. Commun. Probab. 22 (2017), paper no. 47.

I prove in this paper a Sobolev-type energy inequality for the exclusion process, which is analogous to Thomson's (or Dirichlet's) principle for random walks / electric networks. It builds upon the marvelous "octopus inequality" proved by Caputo, Liggett, and Richthammer in 2009.
PDF A 15-minute presentation on this work, targeted towards undergraduates.

8. Power dissipation in fractal AC circuits, with Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew, and Alexander Teplyaev. (2015 UConn math REU fractals group) J. Phys. A: Math. Theor. 50 325205 (2017)

We give a "fractal spin" on the classic infinite ladder circuit discussed in the Feynman Lectures on Physics. For the Feynman-Sierpinski ladder circuit (pictured) we can rigorously prove the convergence of the effective impedances using the dynamics of Möbius transformations.
A refinement of our results (concerning the energy measure) on the F-S ladder circuit was attained recently by Patricia Alonso-Ruiz.

7. Wave equations on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials, with Ulysses Andrews, Grigory Bonik, Richard W. Martin, and Alexander Teplyaev. J. Fourier Anal. Appl. 23 (2017) 994-1027.
(Click here for the wave animations described in the paper.)
6. Stabilization by Noise of a $$\mathbb{C}^2$$-Valued Coupled System, with Lance Ford, Derek Kielty, Rajeshwari Majumdar, Heather McCain, Dylan O'Connell, and Fan Ny Shum. (2015 UConn math REU stochastics group) Stoch. Dyn. 17 (2017) 1750046.

### 2016 and prior

5. Singularly continuous spectrum of a self-similar Laplacian on the half-line, with Alexander Teplyaev. J. Math. Phys. 57 052104 (2016).
4. Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces, with Stanislav Molchanov and Alexander Teplyaev. J. Phys. A: Math. Theor. 48 395203 (2015).
3. Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs, with Baris Evren Ugurcan. Stoch. Proc. Appl. 125 (2015) 4632-4673.
2. Periodic billiard orbits of self-similar Sierpinski carpets, with Robert Niemeyer. J. Math. Anal. Appl. 416 (2014) 969-994.
1. Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion, with Florian Marquardt, Aashish Clerk, and Steven M. Girvin. Phys. Rev. Lett. 99 093902 (2007).

### Papers in preparation (2017+)

P1. Limit shape universality of cellular automata models on the Sierpinski gasket, with Jonah Kudler-Flam.
P2. Semilinear evolution equations on resistance spaces (HydroSG Part III), with Michael Hinz and Alexander Teplyaev.
P3. Hydrodynamic limit of the boundary-driven exclusion process on the Sierpinski gasket (HydroSG Part IV), with Michael Hinz and Alexander Teplyaev.
P4. Anderson localization on infinite fractal lattices, with Stanislav Molchanov and Alexander Teplyaev.

## Funding

As of September 2017, my research is supported in part by the Simons Foundation (via a Collaboration Grant for Mathematicians, 2017-2022) and the Research Council of Colgate University.
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