A generalized Beraha conjecture for non-planar graphs |
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Authors: | Jesper Lykke Jacobsen,Jesú s Salas |
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Affiliation: | 1. Laboratoire de Physique Théorique, École Normale Supérieure, 24 rue Lhomond, 75231 Paris, France;2. Université Pierre et Marie Curie, 4 place Jussieu, 75252 Paris, France;3. Grupo de Modelización, Simulación Numérica y Matemática Industrial, Universidad Carlos III de Madrid, Avda. de la Universidad, 30, 28911 Leganés, Spain;4. Grupo de Teorías de Campos y Física Estadística, Instituto Gregorio Millán, Universidad Carlos III de Madrid, Unidad Asociada al IEM–CSIC, Madrid, Spain |
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Abstract: | We study the partition function ZG(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 ZG(nk,k), namely the chromatic polynomial PG(nk,k)(Q) (corresponding to v=−1), and the flow polynomial ΦG(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→∞. 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. |
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Keywords: | Potts model Non-planar graphs Beraha conjecture Generalized Petersen graphs Transfer matrix Berker&ndash Kadanoff phase |
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