# Variational formula for the time-constant of first-passage percolation

18 Oct 2015This was my PhD thesis.

My thesis was on first-passage percolation, a classical probabilistic model for a fluid flowing through a porous medium. Here is a simple description of the model: To each edge of the cubic lattice on $\mathbb{Z}^d$, we attach a random, positive number called the edge-weight, which represents the time it takes for the fluid to flow through the edge. Typically we focus on the planar $d=2$ case. Suppose we release some fluid at point A on the grid, the fluid flows and spreads through all the edges. We’re interested in the time it takes for the fluid to reaches a far away point B, and we’ll denote this $T(A,B)$. It’s easy to see that this function on $\Z^d \times \Z^d$ defines a random metric on the lattice (if the edge-weights are never $0$).

Although the model is easily described in one paragraph, it’s fairly hard to determine the statistical properties of the passage time. Computer simulations indicate that the model has very interesting and universal statistical behaviour. However, our mathematical understanding is so poor that we cannot even analytically determine the average of T(A,B). My thesis research involves of T(A,B), and using it to study various properties of the model.

There are two publications related to it. The first was a preprint on arxiv, posted in Nov 2013, that eventually turned into my thesis. The proof is rather long-winded, and steals homogenization theorems from the theorem to apply it to the discrete problem.

Arjun Krishnan. *Variational formula for the time-constant of first-passage percolation*. ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–New York University. http

The preprint from the arxiv had a huge update and cleanup (that I will post soon), and was accepted by Comm. Pure and Appl. Math. for publication. It corrects a few mistakes and typos in my thesis. It does not rely on any continuum homogenization theorems, and proves it directly.

Arjun Krishnan. *Variational formula for the time-constant of first-passage percolation*. 2015. http