<p class=strong> Registration: The class is up on cdcs and you may register for it. </p>
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:
Some basic ergodic theory,
Shortest path and flow algorithms (Dijkstra, Max-Flow/Min-Cut),
Geodesic dynamics through ergodic theory, and
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.
50 years of first-passage percolation. (Auffinger, Damron, and Hanson 2015)
Aspects of first-passage percolation. (Kesten 1986)
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