Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

Abstract

We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s=\{1,...,s\}$ of the total set of possible spin values, $Z(G,q,s,v,w)$, where $v$ and $w$ are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial $Ph(G,q,s,w)$ that counts the number of colorings of the vertices of $G$ subject to the condition that colors of adjacent vertices are different, with a weighting $w$ that favors or disfavors colors in the interval $I_s$. We derive powerful new upper and lower bounds on $Z(G,q,s,v,w)$ for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for $Z(G,q,s,v,w)$ on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.

39 pages, 1 figure

References in corpus (8)

• A Little Statistical Mechanics for the Graph Theorist
• Families of Graphs With Chromatic Zeros Lying on Circles
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• A Mapping Relating Complex and Physical Temperatures in the 2D $q$-state Potts Model and Some Applications
• Some Exact Results on the Potts Model Partition Function in a Magnetic Field
• Structure of the Partition Function and Transfer Matrices for the Potts Model in a Magnetic Field on Lattice Strips
• Weighted Graph Colorings
• Weighted-Set Graph Colorings

Cited by in corpus (5)

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• Measures of Spin Ordering in the Potts Model with a Generalized External Magnetic Field
• Exact Partition Functions for the $q$-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs
• A note on recognizing an old friend in a new place: list coloring and the zero-temperature Potts model
• Some Exact Results on Bond Percolation