# Student Probability Seminar

#### The Law of Large Numbers for First-Passage Percolation

Speaker: Arjun Krishnan

Location: Warren Weaver Hall 1314

Date: Thursday, March 28, 2013, 3:30 p.m.

Synopsis:

First Passage Percolation is a simple generalization of percolation introduced in the 1960s. The model can be described as follows: let $$\tau$$ be a random variable such that $$0 < a \leq \tau \leq b$$, and let's attach i.i.d copies of $$\tau$$ to the edges of the square-lattice $$\mathbb{Z}^d$$. That is, we have a set of random, positive, edge-weights on the lattice. A path from the origin to $$x \in \mathbb{Z}^d$$ is a nearest neighbour walk along the lattice, and the time taken to traverse the path is just the sum of edge-weights along the path. Let the first-passage time $$T(x)$$ be the least time required to get from $$0$$ to $$x$$ (it's an inf over all paths). If you know the subadditive ergodic theorem, it's relatively easy to establish that $$\lim_{n \to \infty} T(nx)/n = \mu(x)$$ exists. However, it's fairly difficult to say anything quantitative about the limit; solutions are known only in a handful of very special edge-weight distributions. We will use a few simple techniques from optimal control theory and PDEs to establish a formula for $$\mu(x)$$. No prior knowledge (other than basic probability) will be assumed, and the talk will be pedagogical.