1,103 research outputs found
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. IV. Chromatic polynomial with cyclic boundary conditions
We study the chromatic polynomial P_G(q) for m \times n square- and
triangular-lattice strips of widths 2\leq m \leq 8 with cyclic boundary
conditions. This polynomial gives the zero-temperature limit of the partition
function for the antiferromagnetic q-state Potts model defined on the lattice
G. We show how to construct the transfer matrix in the Fortuin--Kasteleyn
representation for such lattices and obtain the accumulation sets of chromatic
zeros in the complex q-plane in the limit n\to\infty. We find that the
different phases that appear in this model can be characterized by a
topological parameter. We also compute the bulk and surface free energies and
the central charge.Comment: 55 pages (LaTeX2e). Includes tex file, three sty files, and 22
Postscript figures. Also included are Mathematica files transfer4_sq.m and
transfer4_tri.m. Journal versio
Phase diagram of the chromatic polynomial on a torus
We study the zero-temperature partition function of the Potts antiferromagnet
(i.e., the chromatic polynomial) on a torus using a transfer-matrix approach.
We consider square- and triangular-lattice strips with fixed width L, arbitrary
length N, and fully periodic boundary conditions. On the mathematical side, we
obtain exact expressions for the chromatic polynomial of widths L=5,6,7 for the
square and triangular lattices. On the physical side, we obtain the exact
``phase diagrams'' for these strips of width L and infinite length, and from
these results we extract useful information about the infinite-volume phase
diagram of this model: in particular, the number and position of the different
phases.Comment: 72 pages (LaTeX2e). Includes tex file, three sty files, and 26
Postscript figures. Also included are Mathematica files transfer6_sq.m and
transfer6_tri.m. Final version to appear in Nucl. Phys.
Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial
We derive some new structural results for the transfer matrix of
square-lattice Potts models with free and cylindrical boundary conditions. In
particular, we obtain explicit closed-form expressions for the dominant (at
large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as
the solution of a special one-dimensional polymer model. We also obtain the
large-q expansion of the bulk and surface (resp. corner) free energies for the
zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47}
(resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <=
m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19
Postscript figures. Also included are Mathematica files data_CYL.m and
data_FREE.m. Many changes from version 1: new material on series expansions
and their analysis, and several proofs of previously conjectured results.
Final version to be published in J. Stat. Phy
Chromatic Zeros On Hierarchical Lattices and Equidistribution on Parameter Space
Associated to any finite simple graph is the chromatic polynomial
whose complex zeroes are called the chromatic zeros of .
A hierarchical lattice is a sequence of finite simple graphs
built recursively using a substitution rule
expressed in terms of a generating graph. For each , let denote the
probability measure that assigns a Dirac measure to each chromatic zero of
. Under a mild hypothesis on the generating graph, we prove that the
sequence converges to some measure as tends to infinity. We
call the limiting measure of chromatic zeros associated to
. In the case of the Diamond Hierarchical Lattice we
prove that the support of has Hausdorff dimension two.
The main techniques used come from holomorphic dynamics and more specifically
the theories of activity/bifurcation currents and arithmetic dynamics. We prove
a new equidistribution theorem that can be used to relate the chromatic zeros
of a hierarchical lattice to the activity current of a particular marked point.
We expect that this equidistribution theorem will have several other
applications.Comment: To appear in Annales de l'Institut Henri Poincar\'e D. We have added
considerably more background on activity currents and especially on the
Dujardin-Favre classification of the passive locus. Exposition in the proof
of the main theorem was improved. Comments welcome
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