**Lecture plans:**

This set of two lectures covered the probabilistic aspect of analysis on fractals, complementing the tutorials by Bob Strichartz.
It covers the essential ideas needed to read, for instance, Martin Barlow's St. Flour lecture notes *"Diffusions on fractals"*,
and Takashi Kumagai's St. Flour lecture notes *"Random walks on disordered media and their scaling limits"*.

Using the language of Markov chains, I explained the significance of harmonic functions, Green's functions, and heat kernels in the study of
diffusions on fractals. I will also touched on various types of stopping times in Markov processes (hitting times, commute times, etc.) and their
connections to electric network theory.

**Lecture 1 (06/15/2017):** Covered the basics of Markov chains, focusing on symmetric random walks on unweighted graphs. Using one-step calculations we
connected hitting probabilities to harmonic functions, and expected hitting times to Laplacian-1 functions (integrals of Green's functions).
Motivating example: gambler's ruin (in 1D).

**Lecture 2 (06/16/2017):** Generalization to symmetric random walks on weighted graphs. Green's functions and heat kernels. Explicit calculations on SG (showing the time scale factor 5).
Connections to effective resistance: escape probability, commute-time identity.

Besides the tutorial, I gave a 30-minute talk (on June 14) entitled
*"Strong shape theorems in cellular automata models on the Sierpinski gasket,"*
covering recent results obtained with my Colgate undergraduate mentee Jonah Kudler-Flam. PDF Talk slides

Sponsored by the Bielefeld-Seoul IRTG 2235

(IRTG event listings here)

**Dates:**
Mi. 12.07, Fr. 14.07, Mo. 17.07, Mi. 19.07, Fr. 21.07.

**Times:** Mo. & Mi. 12:15~14:00; Fr. 10:15~12:00.

**Room:** D5-153.

**Lecture outline** (will be updated after each lecture. Actual coverage in blue.)

**Lecture 1 (12.07.2017)**: Reversible Markov chains on a countable state space = symmetric random walks on a weighted graph. Laplacian and Dirichlet energy. Harmonic functions. Hitting probabilities and times. Green's functions.**Lecture 2 (14.07.2017)**: Green's function and Green's density. Worked examples: gambler's ruin in 1D, Sierpinski gasket. Electric network theory: potential, current, and resistance. Thomson's (Dirichlet's) principle (proved using analysis ideas). Connection between electric networks and reversible Markov chains: escape probability, on-diagonal Green's function.**Lecture 3 (17.07.2017)**: Commute time identity. Example calculation: Vicsek tree (showing the time scale factor is \(15^N\) on a level-\(N\) pre-fractal graph). Network reduction (via Schur complements). Thomson's principle proved using network reduction.**Lecture 4 (19.07.2017)**: Exclusion process basics. Octopus inequality (Caputo, Liggett, Richthammer). Moving particle lemma (C.). A rough proof sketch of the hydrodynamic limit of the exclusion process on the Sierpinski gasket (C., Hinz, Teplyaev).

PDF Slides used in this lecture**Lecture 5 (21.07.2017)**: Limit shape of internal DLA on the Sierpinski gasket (C., Huss, Sava-Huss, Teplyaev). Some potential theoretic ideas behind the proof: harmonic measure, elliptic Harnack inequality. Limit shape universality of abelian networks on SG (C., Kudler-Flam).

**Synopsis:**

I will start by discussing the basics of particle systems on graphs and their connections to
the random walk process and the relevant potential theory. Emphasis will be on results and techniques which are applicable to non-transitive graphs.

After the introductory lecture(s), I will describe two classes of particle systems for which the scaling limits are mostly known on Euclidean lattices,
but whose generalizations to fractal lattices have only been established recently.

- The boundary-driven exclusion process.
- Sandpile and aggregation models.

For concreteness, I will use the Sierpinski gasket as the model graph, and describe the limit theorems for the aforementioned processes. Though I will say a few words about (variants of) these models on other fractal (or general weighted) graphs, and mention connections to aspects of non-equilibrium statistical mechanics.

This course assumes working knowledge of real analysis and probability theory at the advanced undergraduate / intro graduate level. Exposure to Markov chains is useful but not necessary.

**References:**

Introductory materials are drawn from a variety of sources.

- PDF Doyle and Snell, "Random Walks and Electric Networks." (A classic, suitable for undergrads.)
- PDF Levin, Peres, and Wilmer, "Markov chains and mixing times." See Chapters 9 and 10.
- Lyons and Peres, "Probability on Trees and Networks." See Chapter 2.

- Caputo, Liggett, and Richthammer, "Proof of Aldous' spectral gap conjecture." Journal of the AMS (2010). Here is Aldous' original statement of the spectral gap conjecture.
- My recent papers, see arXiv:1606.01577 (moving particle lemma) and arXiv:1705.10290 (local ergodicity). For an overview of results see arXiv:1702.03376 (joint work with Hinz and Teplyaev).
- Patricia Gonçalves' lectures at the Institut Henri Poincaré "Hydrodynamics for symmetric exclusion in contact with reservoirs" (May 2017). YouTube Lectures: Part 1, Part 2, Part 3. PDF Slides.
- For an earlier work (on the hydrodynamics of the zero-range process on SG), see Jara, Comm. Math. Phys. (2009); arXiv:0805.0380.

- Levine and Peres, "Laplacian growth, sandpiles and scaling limits." Bulletin of the AMS (2017).
- Lawler, Bramson, and Griffeath, "Internal Diffusion limited Aggregaton." Ann. Probab. (1992)
- Duminil-Copin, Lucas, Yadin, and Yehudayoff, "Containing internal diffusion limited aggregation." Electron. Commun. Probab. (2013)
- Levine and Peres, "Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile." Pot. Anal. (2009); arXiv:0704.0688.
- See Lionel Levine's page for more background info and animations.
- For shape theorems on SG, see arXiv:1702.04017 (IDLA, joint work with Huss, Sava-Huss, and Teplyaev), arXiv:1702.08370 (divisible sandpiles, Huss and Sava-Huss). The limit shape universality result (joint work with Kudler-Flam) will be advertised.

**Acknowledgements.** I am grateful to Prof. Dr. Moritz Kassmann and Dr. Michael Hinz for arranging this opportunity.

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