## Abstract

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width L_{y} and arbitrary length L_{x} has the form Z(G,q,v) = ∑_{j=1}^{NZ,G,λ} c_{Z,G,j}(λ_{Z,G,j})^{Lx}, where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet (v=-1) is the chromatic polynomial P(G,q). Using coloring and transfer matrix methods, we give general formulas for C_{X,G} = ∑_{j=1}^{NX,G,λ} c_{X,G,j} for X = Z, P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient c_{Z,G,j} of degree d in q: c^{(d)} = U_{2d}(√q/2), where U_{n}(x) is the Chebyshev polynomial of the second kind, we determine the number of λ_{Z,G,j}'s with coefficient c^{(d)} in Z(G,q,v) for these cyclic strips of width L_{y} to be n_{Z}(L_{y},d)=(2d+1) (L_{y}+d+1)^{-1}(_{Ly-d}^{2Ly}) for 0 ≤ d ≤ L_{y} and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λ_{Z,G,j} is calculated to be N_{Z,Ly,λ}=(_{Ly}^{2Ly}). Results are also presented for the analogous numbers n_{P}(L_{y},d) and N_{P,Ly,λ} for P(G,q). We find that n_{P}(L_{y},0)=n_{P}(L_{y}-1,1)= M_{Ly-1} (Motzkin number), n_{Z}(L_{y},0)=C_{Ly} (the Catalan number), and give an exact expression for N_{P,Ly,λ}. Our results for N_{Z,Ly,λ} and N_{P,Ly,λ} apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations N_{Z,Ly,λ}=2N_{DA,tri,Ly} and N_{P,Ly,λ}=2N_{DA,sq,Ly}, where N_{DA,Λ,n} denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths N_{Z,Ly,λ}∼L_{y}^{-1/2}4^{Ly} and N_{P,Ly,λ}∼L_{y}^{-1/2}3^{Ly} as L_{y}→∞. Some general geometric identities for Potts model partition functions are also presented.

Original language | English |
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Pages (from-to) | 131-182 |

Number of pages | 52 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 296 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2001 Jul 1 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics