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Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs

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  • Chang, Shu-Chiuan
  • Shrock, Robert

Abstract

We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs G of finite width and arbitrarily great length, in the infinite-length limit, denoted {G}. Specifically, we calculate the exponential growth constants for (i) acyclic orientations, α({G}), (ii) acyclic orientations with a single source vertex, α0({G}), and (iii) totally cyclic orientations, β({G}). We consider several lattices, including square (sq), triangular (tri), and honeycomb (hc). From our calculations, we infer lower and upper bounds on these exponential growth constants for the respective infinite lattices. To our knowledge, these are the best current bounds on these quantities. Since our lower and upper bounds are quite close to each other, we can infer very accurate approximate values for the exponential growth constants, with fractional uncertainties ranging from O(10−4) to O(10−2). Further, we present exact values of α(tri), α0(tri), and β(hc) and use them to show that our lower and upper bounds on these quantities are very close to these exact values, even for modest strip widths. Results are also given for a nonplanar lattice denoted sqd. We show that α({G}), α0({G}), and β({G}) are monotonically increasing functions of vertex degree for these lattices. A comparison is given of these exponential growth constants with the corresponding exponential growth constant τ({G}) for spanning trees. Our results are in agreement with inequalities following from the Merino–Welsh and Conde–Merino conjectures.

Suggested Citation

  • Chang, Shu-Chiuan & Shrock, Robert, 2020. "Asymptotic behavior of acyclic and cyclic orientations of directed lattice graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
  • Handle: RePEc:eee:phsmap:v:540:y:2020:i:c:s0378437119317285
    DOI: 10.1016/j.physa.2019.123059
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    References listed on IDEAS

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    1. Roček, Martin & Shrock, Robert & Tsai, Shan-Ho, 1998. "Chromatic polynomials for families of strip graphs and their asymptotic limits," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 252(3), pages 505-546.
    2. Chang, Shu-Chiuan & Shrock, Robert, 2001. "T=0 partition functions for Potts antiferromagnets on lattice strips with fully periodic boundary conditions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 292(1), pages 307-345.
    3. Chang, Shu-Chiuan & Shrock, Robert, 2001. "Exact Potts model partition functions on wider arbitrary-length strips of the square lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(1), pages 234-288.
    4. Chang, Shu-Chiuan & Shrock, Robert, 2001. "Potts model partition functions for self-dual families of strip graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 301(1), pages 301-329.
    5. Chang, Shu-Chiuan & Shrock, Robert, 2000. "Exact Potts model partition function on strips of the triangular lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 286(1), pages 189-238.
    6. Shrock, Robert & Tsai, Shan-Ho, 2000. "Exact partition functions for Potts antiferromagnets on cyclic lattice strips," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 275(3), pages 429-449.
    7. Shrock, Robert & Tsai, Shan-Ho, 1998. "Ground state entropy of Potts antiferromagnets on homeomorphic families of strip graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 259(3), pages 315-348.
    8. Shrock, Robert, 2000. "Exact Potts model partition functions on ladder graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 388-446.
    9. Chang, Shu-Chiuan & Shrock, Robert, 2001. "Exact Potts model partition functions on strips of the honeycomb lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 296(1), pages 183-233.
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