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Conformal Geometry Program Lecture Series: Jesper Jacobsen, ENS Paris

Jesper Jacobsen, ENS Paris

Title: Critical Manifolds for Percolation and Potts Models from Graph Polynomials
Speaker:  Jesper Jacobsen, ENS Paris
Date: Wednesday, January 30, 2013
Time: 10:00am – 11:00am
Place: Seminar Room 313, Simons Center




Abstract: The first parameter to be fixed when studying a phase transition is the critical temperature. Somewhat surprisingly, this parameter is only known analytically for the simplest two-dimensional models (Ising model), or for more complicated models (Potts and O(n) vector models) on the simplest possible lattices. The known critical temperatures are invariably given by simple algebraic curves. Some of these results have very recently been proved mathematically by the technique of discrete holomorphicity.

For Potts and (bond or site) percolation models on any desired two-dimensional lattice we define a graph polynomial whose roots turn out to give very accurate approximations to the critical temperatures, or even yield the exact result in the exactly solvable cases. This polynomial depends on a basis (unit cell) and its embedding into the infinite lattice. As the size of the basis is increased the approximation becomes increasingly accurate. This, on the one hand, gives strong evidence that the critical temperature for the lattices with no known analytical solution may not be algebraic numbers, and that conformal invariance will not have any counterpart in finite size (discrete holomorphicity). On the other hand, the method determines the critical temperature to unprecedented accuracy, typically 12-13 significant digits for bond percolation thresholds. It also shows that the phase diagram of the Potts model in the antiferromagnetic regime has an intricate and highly lattice-dependent structure.