Papers in preparation (2019+) 

P1.  Semilinear evolution equations on resistance spaces, with Michael Hinz and Alexander Teplyaev. 
P2.  Large deviations in the boundarydriven exclusion process on the Sierpinski gasket, with Michael Hinz. 
P3.  Nonequilibrium and stationary fluctuations in the boundarydriven 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 cyclerooted 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 boundarydriven 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 OrnsteinUhlenbeck 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 coarsegraining arguments needed to pass from the microscopic observables (in the exclusion process) to the corresponding macroscopic averages.
The twoblocks 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 KudlerFlam. 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 rotorrouter 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 selfsimilar sandpile tiles.
In the process we also prove the identity elements of the sandpile groups of subgraphs of SG (with two different boundary conditions).


13.  Internal DLA on Sierpinski gasket graphs, with Wilfried Huss, Ecaterina SavaHuss, 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 diffusionlimited 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. 

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 nonsymmetric particle systems to nonlinear 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. 503513. 
This is a summary of a series of four papers (HydroSG Parts I~IV) concerning the hydrodynamic limit of the boundarydriven exclusion process on a resistance space (the Sierpinski gasket being the model space).


10.  Regularized Laplacian determinants of selfsimilar fractals, with Alexander Teplyaev and Konstantinos Tsougkas. Lett. Math. Phys. 108 (2018) 15631579. (See also correction: pp. 15811582.) 


2017 

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


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) 


7.  Wave equations on onedimensional 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) 9941027. 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 selfsimilar Laplacian on the halfline, 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 highdimensional Sierpinski carpet graphs, with Baris Evren Ugurcan. Stoch. Proc. Appl. 125 (2015) 46324673. 
2.  Periodic billiard orbits of selfsimilar Sierpinski carpets, with Robert Niemeyer. J. Math. Anal. Appl. 416 (2014) 969994. 
1.  Quantum Theory of CavityAssisted Sideband Cooling of Mechanical Motion, with Florian Marquardt, Aashish Clerk, and Steven M. Girvin. Phys. Rev. Lett. 99 093902 (2007). 