TY - JOUR

T1 - Flow Polynomials and their Asymptotic Limits for Lattice Strip Graphs

AU - Chang, Shu Chiuan

AU - Shrock, Robert

N1 - Funding Information:
It is a pleasure to dedicate this paper to Professor F. Y. Wu, who has contributed so much to statistical mechanics and, in particular, to the understanding of the Potts model, on the occasion of his 70’th birthday. This research was partially supported by the NSF Grant PHY-97-22101.

PY - 2003/8

Y1 - 2003/8

N2 - We present exact calculations of flow polynomials F(G, q) for lattice strips of various fixed widths L y ≤ 4 and arbitrarily great lengths L x, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G, q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros ℬ in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case L y = 1, the maximal point, q cf, where ℬ crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf = 3 for the infinite square lattice.

AB - We present exact calculations of flow polynomials F(G, q) for lattice strips of various fixed widths L y ≤ 4 and arbitrarily great lengths L x, with several different boundary conditions. Square, honeycomb, and triangular lattice strips are considered. We introduce the notion of flows per face fl in the infinite-length limit. We study the zeros of F(G, q) in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros ℬ in the infinite-length limit for the various families of strips. The function fl is nonanalytic on this locus. The loci are found to be noncompact for many strip graphs with periodic (or twisted periodic) longitudinal boundary conditions, and compact for strips with free longitudinal boundary conditions. We also find the interesting feature that, aside from the trivial case L y = 1, the maximal point, q cf, where ℬ crosses the real axis, is universal on cyclic and Möbius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf = 3 for the infinite square lattice.

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U2 - 10.1023/A:1023836311251

DO - 10.1023/A:1023836311251

M3 - Article

AN - SCOPUS:0037500396

VL - 112

SP - 815

EP - 879

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -