MTH 507. Advanced Topics in Probability: First- and Last- Passage Percolation


Registration: The class is up on cdcs and you may register for it.

Time: T-Thu 9.40am - 10.55am

Location: online on zoom: nine96 3745 0795 (See blackboard if registered or email me for the password)

Office hours: M-W 12-1pm on zoom: seven34 845 9846

First-Passage Percolation (FPP) is a simple shortest path problem on a graph: pick a graph; put some random weights on its edges to mimic the time to cross edges; pick any two vertices and find the shortest time (weight) path between them. The passage time between vertices is a random metric on the graph, and the shortest-time paths are geodesics for this metric. One usually studies the statistics of the passage time and the behavior of geodesics on $\mathbb{Z}^d$.

This simple model has incredible connections to 1) the zeros of the zeta function 2) random matrix theory 3) representation theory and combinatorics 4) Hamilton-Jacobi PDEs. The model was proposed in the 60s, but the main questions in the field are almost entirely open.

In this course, I will loosely follow the review “50 years of first-passage percolation”, and then move on to the research level questions in towards the end of the course. Along the way, I intend to cover:

  1. Some basic ergodic theory,

  2. Shortest path and flow algorithms (Dijkstra, Max-Flow/Min-Cut),

  3. Geodesic dynamics through ergodic theory, and

  4. Hamilton-Jacobi PDEs and nonlinear eigenvalue problems.

The course will be mostly probabilistic and analytic. The prerequisites are the basic real analysis and probability courses: 471 and 406. However, I think you can easily manage if you’ve only taken one of these.

Textbooks

  1. 50 years of first-passage percolation. (Auffinger, Damron, and Hanson 2015)

  2. Aspects of first-passage percolation. (Kesten 1986)

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References

Auffinger, Antonio, Michael Damron, and Jack Hanson. 2015. "50 Years of First Passage Percolation." arXiv:1511.03262 [Math-Ph], November. arxiv/1511.03262

Kesten, Harry. 1986. "Aspects of First Passage Percolation." In École d'été de Probabilités de Saint-Flour, XIV---1984, 1180:125--264. Lecture Notes in Math. Springer, Berlin. mathscinet/876084