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Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial

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Abstract

We study the chromatic polynomials (= zero-temperature antiferromagnetic Potts-model partition functions) P G (q) for m×n rectangular subsets of the square lattice, with m≤8 (free or periodic transverse boundary conditions) and n arbitrary (free longitudinal boundary conditions), using a transfer matrix in the Fortuin–Kasteleyn representation. In particular, we extract the limiting curves of partition-function zeros when n→∞, which arise from the crossing in modulus of dominant eigenvalues (Beraha–Kahane–Weiss theorem). We also provide evidence that the Beraha numbers B 2,B 3,B 4,B 5 are limiting points of partition-function zeros as n→∞ whenever the strip width m is ≥7 (periodic transverse b.c.) or ≥8 (free transverse b.c.). Along the way, we prove that a noninteger Beraha number (except perhaps B 10) cannot be a chromatic root of any graph.

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REFERENCES

  1. R. B. Potts, Proc. Cambridge Philos. Soc. 48:106 (1952).

    Google Scholar 

  2. F. Y. Wu, Rev. Mod. Phys. 54:235 (1982); 55:315 (E) (1983).

    Google Scholar 

  3. F. Y. Wu, J. Appl. Phys. 55:2421 (1984).

    Google Scholar 

  4. R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, LondonûNew York, 1982).

    Google Scholar 

  5. C. Itzykson, H. Saleur, and J.-B. Zuber, eds., Conformal Invariance and Applications to Statistical Mechanics (World Scientific, Singapore, 1988).

    Google Scholar 

  6. P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory (Springer-Verlag, New York, 1997).

    Google Scholar 

  7. B. Nienhuis, J. Stat. Phys. 34:731 (1984).

    Google Scholar 

  8. C. N. Yang and T. D. Lee, Phys. Rev. 87:404 (1952).

    Google Scholar 

  9. P. W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Japan 26(Suppl.):11 (1969).

    Google Scholar 

  10. C. M. Fortuin and P. W. Kasteleyn, Physica 57:536 (1972).

    Google Scholar 

  11. R. Shrock and S.-H. Tsai, Phys. Rev. E 55:5165 (1997), cond-mat/9612249.

    Google Scholar 

  12. R. Shrock and S.-H. Tsai, Phys. Rev. E 56:1342 (1997), cond-mat/9703249.

    Google Scholar 

  13. R. Shrock and S.-H. Tsai, Phys. Rev. E 56:3935 (1997), cond-mat/9707096.

    Google Scholar 

  14. R. Shrock and S.-H. Tsai, Phys. Rev. E 56:4111 (1997), cond-mat/9707306.

    Google Scholar 

  15. H. Feldmann, R. Shrock, and S.-H. Tsai, Phys. Rev. E 57:1335 (1998), condmat/ 9711058.

    Google Scholar 

  16. H. Feldmann, A. J. Guttmann, I. Jensen, R. Shrock, and S.-H. Tsai, J. Phys. A 31:2287 (1998), cond-mat/9801305.

    Google Scholar 

  17. M. Roček, R. Shrock, and S.-H. Tsai, Physica A 252:505 (1998), cond-mat/9712148.

    Google Scholar 

  18. R. Shrock and S.-H. Tsai, Physica A 259:315 (1998), cond-mat/9807105.

    Google Scholar 

  19. M. Roček, R. Shrock, and S.-H. Tsai, Physica A 259:367 (1998), cond-mat/9807106.

    Google Scholar 

  20. S.-H. Tsai, Physica A 259:349 (1998), cond-mat/9807107.

    Google Scholar 

  21. R. Shrock and S.-H. Tsai, J. Phys. A 31:9641 (1998), cond-mat/9810057.

    Google Scholar 

  22. R. Shrock and S.-H. Tsai, Physica A 265:186 (1999), cond-mat/9811410.

    Google Scholar 

  23. R. Shrock and S.-H. Tsai, J. Phys. A 32:L195 (1999), cond-mat/9903233.

    Google Scholar 

  24. R. Shrock and S.-H. Tsai, Phys. Rev. E 60:3512 (1999), cond-mat/9910377.

    Google Scholar 

  25. R. Shrock and S.-H. Tsai, J. Phys. A 32:5053 (1999), cond-mat/9905431.

    Google Scholar 

  26. N. Biggs and R. Shrock, J. Phys. A 32:L489 (1999), cond-mat/0001407.

    Google Scholar 

  27. R. Shrock, Phys. Lett. A 261:57 (1999), cond-mat/9908323.

    Google Scholar 

  28. R. Shrock, Discrete Math. 231:421 (2001), cond-mat/9908387.

    Google Scholar 

  29. R. Shrock and S.-H. Tsai, Physica A 275:429 (2000), cond-mat/9907403.

    Google Scholar 

  30. R. Shrock, Physica A 283:388 (2000), cond-mat/0001389.

    Google Scholar 

  31. S.-C. Chang and R. Shrock, Ground state entropy of the Potts antiferromagnet on triangular lattice strips, cond-mat/0004129.

  32. S.-C. Chang and R. Shrock, Physica A 290:402 (2001), cond-mat/0004161.

    Google Scholar 

  33. S.-C. Chang and R. Shrock, Physica A 286:189 (2000), cond-mat/0004181.

    Google Scholar 

  34. N. Biggs, J. Combin. Theory B 82:19 (2001).

    Google Scholar 

  35. N. Biggs, The Chromatic Polynomial of the 3×n Toroidal Lattice, London School of Economics CDAM Research Report LSE-CDAM-99-05 (1999).

  36. S. Beraha, J. Kahane, and N. J. Weiss, Proc. Nat. Acad. Sci. USA 72:4209 (1975).

    Google Scholar 

  37. S. Beraha, J. Kahane, and N. J. Weiss, in Studies in Foundations and Combinatorics (Advances in Mathematics Supplementary Studies, Vol. 1), G.-C. Rota, ed. (Academic Press, New York, 1978).

    Google Scholar 

  38. S. Beraha and J. Kahane, J. Combin. Theory B 27:1 (1979).

    Google Scholar 

  39. S. Beraha, J. Kahane, and N. J. Weiss, J. Combin. Theory B 28:52 (1980).

    Google Scholar 

  40. A. D. Sokal, Chromatic roots are dense in the whole complex plane, cond-mat/0012369.

  41. J. Salas and A. D. Sokal, J. Stat. Phys. 86:551 (1997), cond-mat/9603068.

    Google Scholar 

  42. R. J. Baxter, Proc. Roy. Soc. London A 383:43 (1982).

    Google Scholar 

  43. L. Onsager, Phys. Rev. 65:117 (1944).

    Google Scholar 

  44. A. Lenard, cited in E. H. Lieb, Phys. Rev. 162:162 (1967) at pp. 169û170.

    Google Scholar 

  45. R. J. Baxter, J. Math. Phys. 11:3116 (1970).

    Google Scholar 

  46. M. den Nijs, M. P. Nightingale, and M. Schick, Phys. Rev. B 26:2490 (1982).

    Google Scholar 

  47. J. K. Burton Jr. and C. L. Henley, J. Phys. A 30:8385 (1997), cond-mat/9708171.

    Google Scholar 

  48. J. Salas and A. D. Sokal, J. Stat. Phys. 92:729 (1998), cond-mat/9801079.

    Google Scholar 

  49. S. L. A. de Queiroz, T. Paiva, J. S. de Sa Martins, and R. R. dos Santos, Phys. Rev. E 59:2772 (1999), cond-mat/9812341.

    Google Scholar 

  50. S. J. Ferreira and A. D. Sokal, J. Stat. Phys. 96:461 (1999), cond-mat/9811345.

    Google Scholar 

  51. J. Cardy, J.-L. Jacobsen, and A. D. Sokal, Unusual corrections to scaling in the 3-state Potts antiferromagnet on a square lattice, cond-mat/0101197, to appear in J. Stat. Phys. 105(1/2) (Oct. 2001).

  52. H. Saleur, Commun. Math. Phys. 132:657 (1990).

    Google Scholar 

  53. H. Saleur, Nucl. Phys. B 360:219 (1991).

    Google Scholar 

  54. R. J. Baxter, H. N. V. Temperley, and S. E. Ashley, Proc. Roy. Soc. London A 358:535 (1978).

    Google Scholar 

  55. R. J. Baxter, J. Phys. A 19:2821 (1986).

    Google Scholar 

  56. R. J. Baxter, J. Phys. A 20:5241 (1987).

    Google Scholar 

  57. B. Nienhuis, Phys. Rev. Lett. 49:1062 (1982).

    Google Scholar 

  58. J. Stephenson, J. Math. Phys. 5:1009 (1964).

    Google Scholar 

  59. H. W. J. Blöte and H. J. Hilhorst, J. Phys. A 15:L631 (1982).

    Google Scholar 

  60. B. Nienhuis, H. J. Hilhorst, and H. W. J. Blöte, J. Phys. A 17:3559 (1984).

    Google Scholar 

  61. R. J. Baxter, J. Math. Phys. 11:784 (1970).

    Google Scholar 

  62. C. L. Henley, private communications.

  63. J. Salas and A. D. Sokal, unpublished.

  64. A. C. D. van Enter, R. Fernández, and A. D. Sokal, unpublished (1996).

  65. J. Adler, A. Brandt, W. Janke, and S. Shmulyan, J. Phys. A 28:5117 (1995).

    Google Scholar 

  66. J. L. Jacobsen, J. Salas, and A. D. Sokal, Transfer matrices and partition-function zeros for antiferromagnetic Potts models. V. Nonzero temperature, work in progress.

  67. H. W. J. Blöte and M. P. Nightingale, Physica 112A:405 (1982).

    Google Scholar 

  68. S. Beraha, unpublished, circa 1974.

  69. G. Berman and W. T. Tutte, J. Combin. Theory 6:301 (1969).

    Google Scholar 

  70. W. T. Tutte, J. Combin. Theory 9:289 (1970).

    Google Scholar 

  71. W. T. Tutte, Ann. N.Y. Acad. Sci. 175:391 (1970).

    Google Scholar 

  72. W. T. Tutte, Canad. J. Math. 26:893 (1974).

    Google Scholar 

  73. W. T. Tutte, in Studies in Graph Theory, Part II, D. R. Fulkerson ed., Studies in Mathematics #12 (Mathematical Association of America, Washington, 1975), pp. 361–377.

    Google Scholar 

  74. W. T. Tutte, Canad. J. Math. 34:741 (1982).

    Google Scholar 

  75. W. T. Tutte, Canad. J. Math. 34:952 (1982).

    Google Scholar 

  76. P. P. Martin, J. Phys. A 20:L399 (1987).

    Google Scholar 

  77. P. P. Martin, J. Phys. A 22:3991 (1989).

    Google Scholar 

  78. P. P. Martin, Potts Models and Related Problems in Statistical Mechanics (World Scientific, Singapore, 1991).

    Google Scholar 

  79. F. Jaeger, J. Combin. Theory B 52:259 (1991).

    Google Scholar 

  80. L. H. Kauffmann and H. Saleur, Commun. Math. Phys. 152:565 (1993).

    Google Scholar 

  81. H. N. V. Temperley, in Low-Dimensional Topology and Quantum Field Theory (Cambridge, 1992), H. Osborn, ed., NATO Advanced Science Institutes Series B: Physics #315 (Plenum, New York, 1993), pp. 203–212.

    Google Scholar 

  82. W. T. Tutte, J. Combin. Theory. B 57:269 (1993).

    Google Scholar 

  83. D. M. Jackson, European J. Combin. 15:245 (1994).

    Google Scholar 

  84. J.-M. Maillard, G. Rollet, and F. Y. Wu, J. Phys. A 27:3373 (1994).

    Google Scholar 

  85. J.-M. Maillard and G. Rollet, J. Phys. A 27:6963 (1994).

    Google Scholar 

  86. J.-M. Maillard, Math. Comput. Modelling 26:169 (1997).

    Google Scholar 

  87. J. L. Jacobsen and J. Salas, Transfer matrices and partition-function zeros for antiferromagnetic Potts models. II. Extended results for square-lattice chromatic polynomial, cond-mat/0011456, J. Stat. Phys. 104:703 (2001), this issue.

    Google Scholar 

  88. J. L. Jacobsen, J. Salas, and A. D. Sokal, Transfer matrices and partition-function zeros for antiferromagneticPottsmodels. III. Triangular-latticechromaticpolynomial, in preparation.

  89. J. L. Jacobsen, J. Salas, and A. D. Sokal, Transfer matrices and partition-function zeros for antiferromagnetic Potts models. IV. Periodic boundary conditions in the longitudinal direction, work in progress.

  90. G. D. Birkhoff, Ann. Math. 14:42 (1912).

    Google Scholar 

  91. H. Whitney, Bull. Amer. Math. Soc. 38:572 (1932).

    Google Scholar 

  92. W. T. Tutte, Proc. Cambridge Philos. Soc. 43:26 (1947).

    Google Scholar 

  93. W. T. Tutte, Canad. J. Math. 6:80 (1954).

    Google Scholar 

  94. R. G. Edwards and A. D. Sokal, Phys. Rev. D 38:2009 (1988).

    Google Scholar 

  95. R. C. Read, J. Combin. Theory 4:52 (1968).

    Google Scholar 

  96. R. C. Read and W. T. Tutte, in Selected Topics in Graph Theory 3, L. W. Beineke and R. J. Wilson, eds. (Academic Press, London, 1988).

    Google Scholar 

  97. G. L. Chia, Discrete Math. 172:175 (1997).

    Google Scholar 

  98. I. Stewart and D. Tall, Algebraic Number Theory, 2nd ed. (Chapman and Hall, LondonûNew York, 1987).

    Google Scholar 

  99. P. M. Cohn, Algebra, 2nd ed., vol. 2 (Wiley, Chichester, 1989).

    Google Scholar 

  100. I. Stewart, Galois Theory, 2nd ed. (Chapman and Hall, LondonûNew York, 1989).

    Google Scholar 

  101. B. Jackson, Combin. Probab. Comput. 2:325 (1993).

    Google Scholar 

  102. G. D. Birkhoff and D. C. Lewis, Trans. Amer. Math. Soc. 60:355 (1946).

    Google Scholar 

  103. D. R. Woodall, Discrete Math. 172:141 (1997).

    Google Scholar 

  104. C. Thomassen, Combin. Probab. Comput. 6:497 (1997).

    Google Scholar 

  105. D. R. Woodall, in Combinatorial Surveys: Proceedings of the Sixth British Combinatorial Conference, P. J. Cameron, ed. (Academic Press, London, 1977).

    Google Scholar 

  106. D. R. Woodall, Discrete Math. 101:327 (1992).

    Google Scholar 

  107. D. R. Woodall, Discrete Math. 101:333 (1992).

    Google Scholar 

  108. H. N. V. Temperley and E. H. Lieb, Proc. Roy. Soc. London A 322:251 (1971).

    Google Scholar 

  109. E. H. Lieb and W. A. Beyer, Stud. Appl. Math. 48:77 (1969).

    Google Scholar 

  110. N. L. Biggs, R. M. Damerell, and D. A. Sands, J. Combin. Theory B 12:123 (1972).

    Google Scholar 

  111. D. A. Sands, Dichromatic polynomials of linear graphs, Ph.D. thesis (University of London, 1972).

  112. N. L. Biggs and G. H. J. Meredith, J. Combin. Theory B 20:5 (1976).

    Google Scholar 

  113. B. Derrida and J. Vannimenus, J. Physique Lett. 41:L-473 (1980).

    Google Scholar 

  114. H. W. J. Blöte and M. P. Nightingale, Physica 129A:1 (1984).

    Google Scholar 

  115. H. W. J. Blöte and B. Nienhuis, J. Phys. A 22:1415 (1989).

    Google Scholar 

  116. J. L. Jacobsen and J. Cardy, Nucl. Phys. B 515[FS]:701 (1998), cond-mat/9711279.

    Google Scholar 

  117. V. Dotsenko, J. L. Jacobsen, M.-A. Lewis, and M. Picco, Nucl. Phys. B 546[FS]:505 (1999), cond-mat/9812227.

    Google Scholar 

  118. R. P. Stanley, Enumerative Combinatorics, vol. 1 (Wadsworth & Brooks/Cole, Monterey, CA, 1986). Reprinted by Cambridge University Press, 1999.

    Google Scholar 

  119. R. P. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge University Press, CambridgeûNew York, 1999).

    Google Scholar 

  120. N. J. A. Sloane, Sloane's On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/index.html.

  121. N. G. De Bruijn, Asymptotic Methods in Analysis, 2nd ed. (North-Holland, Amsterdam, 1961).

    Google Scholar 

  122. V. N. Sachkov, Combinatorial Methods in Discrete Mathematics, Encyclopedia of Mathematics and its Applications #55 (Cambridge University Press, Cambridge, 1996).

    Google Scholar 

  123. A. M. Odlyzko, in Handbook of Combinatorics, vol. 2, R. L. Graham, M. Grötschel, and L. Lovász, eds. (Elsevier/MIT Press, Amsterdam/Cambridge, 1995), pp. 1063–1229.

    Google Scholar 

  124. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, Adv. Comput. Math. 5:329 (1996).

    Google Scholar 

  125. R. Simion and D. Ullman, Discrete Math. 98:193 (1991).

    Google Scholar 

  126. M. Klazar, Discrete Appl. Math. 82:263 (1998).

    Google Scholar 

  127. R. Donaghey and L. W. Shapiro, J. Combin. Theory A 23:291 (1977).

    Google Scholar 

  128. J. Riordan, J. Combin. Theory A 19:214 (1975).

    Google Scholar 

  129. D. Gouyou-Beauchamps and G. Viennot, Adv. Appl. Math. 9:334 (1988).

    Google Scholar 

  130. F. R. Bernhart, Discrete Math. 204:73 (1999).

    Google Scholar 

  131. S. Lang, Algebra, 3rd ed. (AddisonûWesley, Reading, Mass., 1993).

    Google Scholar 

  132. D. Cox, J. Little, and D. O'shea, Using Algebraic Geometry (Springer-Verlag, New York, 1998).

    Google Scholar 

  133. D. A. Bini and G. Fiorentino, Numerical computation of polynomial roots: MPSolve û Version 2.0. FRISCO report (1998). Available at http://www.dm.unipi.it/pages/bini/public_html/papers/mpsolve.ps.Z. Software package available at http://www.dm.unipi.it/pages/bini/public_html/software/mps2.tar.gz.

  134. D. A. Bini and G. Fiorentino, Numer. Algorithms 23:127 (2000).

    Google Scholar 

  135. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., corrected printing (Springer-Verlag, BerlinûNew York, 1980).

    Google Scholar 

  136. C. G. Gibson, Elementary Geometry of Algebraic Curves (Cambridge University Press, Cambridge, 1998).

    Google Scholar 

  137. V. Matveev and R. Shrock, J. Phys. A 28:L533 (1995).

    Google Scholar 

  138. E. J. Farrell, Discrete Math. 29:161 (1980).

    Google Scholar 

  139. R. C. Read and G. F. Royle, in Graph Theory, Combinatorics, and Applications (Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Western Michigan University, 1988), Volume 2, Y. Alavi et al., eds. (Wiley, New York, 1991).

    Google Scholar 

  140. A. V. Bakaev and V. I. Kabanovich, J. Phys. A 27:6731 (1994).

    Google Scholar 

  141. B. Bollobas, Modern Graph Theory (Springer-Verlag, New YorkûBerlinûHeidelberg, 1998).

    Google Scholar 

  142. K. Appel and W. Haken, Illinois J. Math. 21:429 (1977).

    Google Scholar 

  143. K. Appel, W. Haken, and J. Koch, Illinois J. Math. 21:491 (1977).

    Google Scholar 

  144. K. Appel and W. Haken, Every Planar Mapis Four Colorable, Contemporary Mathematics #98 (American Mathematical Society, Providence RI, 1989).

    Google Scholar 

  145. N. Robertson, D. P. Sanders, P. D. Seymour, and R. Thomas, J. Combin. Theory B 70:2 (1997).

    Google Scholar 

  146. R. Thomas, Notices Amer. Math. Soc. 45:848 (1998)

    Google Scholar 

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  1. Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, Zaragoza, 50009, Spain

    Jesús Salas

  2. Department of Physics, New York University, 4 Washington Place, New York, New York, 10003

    Alan D. Sokal

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Salas, J., Sokal, A.D. Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. I. General Theory and Square-Lattice Chromatic Polynomial. Journal of Statistical Physics 104, 609–699 (2001). https://doi.org/10.1023/A:1010376605067

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