## Interacting particle systems, sandpiles, (stochastic) PDEs, and fractals

#### 2018-19:Research lectures at Bielefeld (Autumn 2018) and Lisboa (Winter 2019)2017:Research tutorials & lectures at Cornell (June) and Bielefeld (July)

My research focuses on 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.

Models which I have studied include: Exclusion processes (9, 11, 15, 16, P2, P3, P4), internal diffusion-limited aggregation (13), rotor-router aggregation (14), abelian sandpiles (14), and Gaussian free fields (3).

In order to gain a deeper understanding of stochastic processes on resistance spaces, I continue to study spectral analysis of Laplacian and Schrödinger operators on self-similar fractal spaces, on which there are a number of peculiar features (4, 5, 17). 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 (17).

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.

One of my main focus projects is to provide microscopic derivations of (non)linear (S)PDEs from interacting particle systems on spaces which are bounded in the resistance metric. These higher dimensional spaces include trees, fractals, and random environments, and do not necessarily possess translational invariance like Euclidean spaces.

While the models I study may seem disparate, my approach is ultimately grounded in the study of analysis on graphs, 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, 17 involved notable contributions from undergraduate students in the form of numerical findings, conjectures, and lemmas, which were then formally turned into theorems by the senior author(s).

### Funding

As of July 1, 2019, my research is supported in part by the National Science Foundation grant DMS-1855604, "Stochastic dynamics on large-scale networks" (2019-22).

I also acknowledge partial support from the Simons Foundation (via a Collaboration Grant for Mathematicians, 2017-19) and the Research Council of Colgate University (Major grant 2017-18, Picker Research Fellowship 2019-20).

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

### Papers in preparation (2019+)

P1. Semilinear evolution equations on resistance spaces, with Michael Hinz and Alexander Teplyaev.
P2. Large deviations in the boundary-driven exclusion process on the Sierpinski gasket, with Michael Hinz.
P3. Nonequilibrium and stationary fluctuations in the boundary-driven exclusion process on the Sierpinski gasket, with Chiara Franceschini, Patrícia Gonçalves, and Otávio Menezes.
P4. Scaling limits of density fluctuations in asymmetric exclusion processes on resistance spaces.

### Submitted

17. Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Hofstadter's butterfly, determinants, and loop soup entropy, with Ruoyu (Tony) Guo. 47 pages, submitted (2019+).

In this paper we solve the magnetic spectrum on the Sierpinski gasket under uniform magnetic field, revealing the true analog of Hofstadter's butterfly. Among other results, we also compute determinantal quantities associated with the magnetic Laplacian on the gasket, which are connected to a random spatial process called cycle-rooted spanning forests.

16. Phase transition in the exclusion process on the Sierpinski gasket with slowed boundary reservoirs, with Patrícia Gonçalves. 44 pages, submitted (2019+).

In this paper we demonstrate a phase transition in the boundary-driven exclusion process on the Sierpinski gasket. Depending on the strength of the boundary reservoirs' dynamics, the particle density evolves according to the heat equation with Dirichlet, Robin, or Neumann boundary condition. In the equilibrium setting, we also prove a central limit theorem for the density fluctuations, which satisfy the Ornstein-Uhlenbeck equation with the corresponding boundary condition.

15. Local ergodicity in the exclusion process on an infinite weighted graph. 36 pages, submitted (2018+).

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.

### 2019+

14. Laplacian growth and sandpiles on the Sierpinski gasket: limit shape universality & exact solutions, with Jonah Kudler-Flam. 54 pages, to appear in Ann. Inst. Henri Poincaré Comb. Phys. Interact. (2019+). Click here for Jonah's AutomataSG program and animations on GitHub.

In this paper we solve the rotor-router aggregation problem and the abelian sandpile growth problem on the graphical Sierpinski gasket (SG) when particles are launched from the corner vertex. The sandpile growth problem is solved exactly via a renormalization scheme involving self-similar sandpile tiles. In the process we also prove the identity elements of the sandpile groups of subgraphs of SG (with two different boundary conditions).

Conclusion: The four Laplacian growth models---IDLA, rotor-router aggregation, divisible sandpiles, and abelian sandpiles---started from the corner vertex of SG satisfy "limit shape universality" in the sense of Levine and Peres (2017).

13. Internal DLA on Sierpinski gasket graphs, with Wilfried Huss, Ecaterina Sava-Huss, and Alexander Teplyaev. 24 pages, to appear in "Analysis and Geometry on Graphs & Manifolds," LMS Lecture Notes, Cambridge University Press (2019+).

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.

12. Fractal AC circuits and propagating waves on fractals, with Eric Akkermans, Gerald Dunne, Luke G. Rogers, and Alexander Teplyaev. 9 pages, to appear in the Cornell conference proceedings "Analysis, Probability, and Mathematical Physics on Fractals," World Scientific Publishing (2019+).

### 2018

11. From non-symmetric particle systems to non-linear PDEs on fractals, with Michael Hinz and Alexander Teplyaev. In A. Eberle et al. (Eds.), Stochastic Partial Differential Equations and Related Fields: In Honor of Michael Röckner SPDERF, Bielefeld, Germany, October 10 -14, 2016, Springer Proceedings in Mathematics & Statistics (2018), pp. 503-513.

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. 108 (2018) 1563-1579. (See also correction: pp. 1581-1582.)

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. 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.

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.

We study wave propagation on the pq-model, a self-similar inhomogeneous wave medium on the unit interval, with an initial approximate delta pulse at the origin. Using the spectral resolution of the Laplacian (via spectral decimation), we prove that the solution of the wave equation can be approximated uniformly in space by a Fourier series, up to some cutoff time. This gives a quantitative glimpse into the "infinite speed of wave propagation" phenomenon on fractals.

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.

In this paper we study a coupled ODE system in two complex dimensions that admits finite-time blow-up solutions. We show analytically and numerically that stabilization can be achieved by adding a suitable Brownian noise, and that the resulting system of SDEs is ergodic. The proof uses Girsanov theorem to effect a time change from our 2D system to a quasi-1D-system similar to the one studied by Herzog and Mattingly.

### 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). (JPhysA Cover image)
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).

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