First Passage Percolation - Adam Oberman Math

Introduction

These notes have been made because it is an interesting (if fairly mature subject) and because of possible connections with Homogenization of metric Hamilton-Jacobi Equations

References

• Feature article from the AMS
  • http://www.ams.org/featurecolumn/archive/percolation.html
• Wikipedia article on percolation
  • http://en.wikipedia.org/wiki/Percolation
  • http://en.wikipedia.org/wiki/Schramm-Loewner_evolution
• Books
  • Percolation By Geoffrey Grimmett Google Books
  • Grimmet Web Page http://www.statslab.cam.ac.uk/~grg/preprints.html
• Pictures on Martin Barlow web site http://www.math.ubc.ca/~barlow/rwp/index.html

Possible connections to Homogenization

Percolation Theory

Number of work, including fields metal on critical exponents in percolation theory.

What can we learn about a relaxed version of the problem. By this we mean, instead of 0-1 values for connections, introduce a speed/cost ratio

Consider a random medium.

• Percolation Theory: 0 1 with probablity p
• Relaxed Version: c 1 with prob p.

Obviously, this is a different problem. Question: what can we learn from this problem? With this problem, we can look at the HJ equation for the distance in the metric. This is useful, and computationally orders of magnitude simpler. Numerical computations show that the resulting metric is Isotropic, so it is characterized by one parameter, C.

Question 2: Is there a scaling between c and p? In other words, if we consider the function

C = C(c,p)

Is this actually a function of

C = C(c^ap^b)

If so then we could explore the limit in an even easier fashion.