A generalized Beraha conjecture for non-planar graphs

作者：Jacobsen Jesper Lykke; Salas Jesus^*

来源：Nuclear Physics B, 2013, 875(3): 678-718.

DOI：10.1016/j.nuclphysb.2013.07.012

• Potts model
• Non-planar graphs
• Beraha conjecture
• Generalized Petersen graphs
• Transfer matrix
• Berker-Kadanoff phase

We study the partition function Z(G( k,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 %26lt;= k %26lt;= 7. We also consider two specializations of Z(G( k,k)), namely the chromatic polynomial P-G(nk,P-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 %26lt;= k %26lt;= 7. We pay special attention to the accumulation loci of the corresponding zeros when n -%26gt; infinity. We observe that the Berker Kadanoff phase that is present in twodimensional 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.

• 出版日期2013-10-21