A generalized Beraha conjecture for non-planar graphs

Abstract

We study the partition function Z_{G(nk,k)}(Q,v) of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k). We study its zeros in the plane (Q,v) for 1<= k <= 7. We also consider two specializations of Z_{G(nk,k)}, namely the chromatic polynomial P_{G(nk,k)}(Q) (corresponding to v=-1), and the flow polynomial Phi_{G(nk,k)}(Q) (corresponding to v=-Q). In these two cases, we study their zeros in the complex Q-plane for 1 <= k <= 7. We pay special attention to the accumulation loci of the corresponding zeros when n -> infinity. We observe that the Berker-Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.

47 pages (LaTeX2e). Includes tex file, three sty files, and 40 Postscript figures. Minor changes from version 1. Final version published in Nucl. Phys. B

References in corpus (5)

• Phase diagram of the chromatic polynomial on a torus
• Character decomposition of Potts model partition functions. I. Cyclic geometry
• Critical manifold of the kagome-lattice Potts model
• Eigenvalue amplitudes of the Potts model on a torus
• Transfer matrix computation of critical polynomials for two-dimensional Potts models

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• Devil's Staircase Continuum in the Chiral Clock Spin Glass with Competing Ferromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
• Phase diagram of the triangular-lattice Potts antiferromagnet
• Cylinder partition function of the 6-vertex model from algebraic geometry
• Planar diagrams for local invariants of graphs in surfaces
• Torus partition function of the six-vertex model from algebraic geometry