High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials

Understand

The critical curves of the q-state Potts model can be determined exactly for regular two-dimensional lattices G that are of the three-terminal type.

• Jacobsen and Scullard have defined a graph polynomial P_B(q,v) that gives access to the critical manifold for general lattices.
• It depends on a finite repeating part of the lattice, called the basis B, and its real roots in the temperature variable v = e^K - 1 provide increasingly accurate approximations to the critical manifolds upon increasing the size of B.
• These authors computed P_B(q,v) for large bases (up to 243 edges), obtaining determinations of the ferromagnetic critical point v_c > 0 for the (4,8^2), kagome, and (3,12^2) lattices to a precision (of the order 10^{-8}) slightly superior to that of the best available Monte Carlo simulations.