Title: The Shortest Crossing of a Box in Percolation

Abstract: On the two-dimensional square lattice, call each nearest-neighbor bond "open" with probability 1/2 and "closed" with probability 1/2, each independently. Conditioned on the existence of an open left-right crossing path of a box of side-length n, it was shown by Aizenman-Burchard that, with high probability, the shortest crossing has at least n^{1+\epsilon} edges, for some \epsilon>0. It was also shown by Morrow-Zhang that the *lowest* crossing has order n^{4/3} edges. In 1992, Kesten and Zhang asked if the shortest crossing has the same length as the lowest crossing. Specifically, conditioned on the existence of an open crossing, does the ratio of the length of the shortest crossing to the length of the lowest crossing go to zero in probability as n tends to infinity? I will talk about these questions, and joint work with J. Hanson and P. Sosoe in which we show that the answer to the Kesten-Zhang question is yes.

About the speaker: Michael Damron’s work is in statistical mechanics models, including percolation models (Bernoulli and invasion percolation, first-passage percolation) and spin systems (coarsening models and spin glasses). He received his Ph.D. at NYU in 2009, moved to Princeton until 2013 as a postdoc, then to Indiana University until 2015 and now Georgia Tech, both as an Assistant Professor. Michael is the recipient of an NSF CAREER award.

The ISU Mathematics Department Colloquium is co-organized by David P. Herzog (dherzog@iastate.edu) and Pablo Stinga (stinga@iastate.edu).

See the website for more information.