50 research outputs found
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics
The scaling limit of the energy correlations in non integrable Ising models
We obtain an explicit expression for the multipoint energy correlations of a
non solvable two-dimensional Ising models with nearest neighbor ferromagnetic
interactions plus a weak finite range interaction of strength , in a
scaling limit in which we send the lattice spacing to zero and the temperature
to the critical one. Our analysis is based on an exact mapping of the model
into an interacting lattice fermionic theory, which generalizes the one
originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising
model. The interacting model is then analyzed by a multiscale method first
proposed by Pinson and Spencer. If the lattice spacing is finite, then the
correlations cannot be computed in closed form: rather, they are expressed in
terms of infinite, convergent, power series in . In the scaling limit,
these infinite expansions radically simplify and reduce to the limiting energy
correlations of the integrable Ising model, up to a finite renormalization of
the parameters. Explicit bounds on the speed of convergence to the scaling
limit are derived.Comment: 75 pages, 11 figure
The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma
We elucidate the close connection between the repulsive lattice gas in
equilibrium statistical mechanics and the Lovasz local lemma in probabilistic
combinatorics. We show that the conclusion of the Lovasz local lemma holds for
dependency graph G and probabilities {p_x} if and only if the independent-set
polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore,
we show that the usual proof of the Lovasz local lemma -- which provides a
sufficient condition for this to occur -- corresponds to a simple inductive
argument for the nonvanishing of the independent-set polynomial in a polydisc,
which was discovered implicitly by Shearer and explicitly by Dobrushin. We also
present some refinements and extensions of both arguments, including a
generalization of the Lovasz local lemma that allows for "soft" dependencies.
In addition, we prove some general properties of the partition function of a
repulsive lattice gas, most of which are consequences of the alternating-sign
property for the Mayer coefficients. We conclude with a brief discussion of the
repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity.
To be published in J. Stat. Phy
Advances in honeycomb layered oxides: Part II -- Theoretical advances in the characterisation of honeycomb layered oxides with optimised lattices of cations
The quest for a successful condensed matter theory that incorporates
diffusion of cations, whose trajectories are restricted to a
honeycomb/hexagonal pattern prevalent in honeycomb layered materials is
ongoing, with the recent progress discussed herein focusing on symmetries,
topological aspects and phase transition descriptions of the theory. Such a
theory is expected to differ both qualitatively and quantitatively from 2D
electron theory on static carbon lattices, by virtue of the dynamical nature of
diffusing cations within lattices in honeycomb layered materials. Herein, we
have focused on recent theoretical progress in the characterisation of
pnictogen- and chalcogen-based honeycomb layered oxides with emphasis on
hexagonal/honeycomb lattices of cations. Particularly, we discuss the link
between Liouville conformal field theory to expected experimental results
characterising the optimal nature of the honeycomb/hexagonal lattices in
congruent sphere packing problems. The diffusion and topological aspects are
captured by an idealised model, which successfully incorporates the duality
between the theory of cations and their vacancies. Moreover, the rather
intriguing experimental result that a wide class of silver-based layered
materials form stable Ag bilayers, each comprising a pair of triangular
sub-lattices, suggests a bifurcation mechanism for the Ag triangular
sub-lattices, which ultimately requires conformal symmetry breaking within the
context of the idealised model, resulting in a cation monolayer-bilayer phase
transition. Other relevant experimental, theoretical and computational
techniques applicable to the characterisation of honeycomb layered materials
have been availed for completeness.Comment: 93 pages, 21 figures, 4 tables, title updated, table of contents
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