Student Probability Seminar

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Self-Avoiding Walks on the Honeycomb Lattice

Speaker: Guillaume Dubach

Location: Warren Weaver Hall 1314

Date: Wednesday, October 18, 2017, 10 a.m.

Synopsis:

While in general there is no exact formula that counts self-avoiding walks on a given infinite graph, the number of self-avoiding walks of length $k$ on lattices is known to be logarithmically equivalent to $c^k$, where $c$ is called the connective constant of the lattice, and usually can only be approximated. In the specific case of the honeycomb lattice (i.e. the hexagonal structure one finds in beehives), physical heuristics led to the conjecture that the connective constant is $ \sqrt{2 + \sqrt{2} } $. Duminil-Copin and Smirnov proved this conjecture a few years ago in a sensational thirteen-page-long paper. We will sketch their proof, discuss why it cannot be easily extended to other settings, and what else can be asked or expected from a probabilistic point of view.