44 research outputs found
Non-degenerated groundstates in the antiferromagnetic Ising model on triangulations
A triangulation is an embedding of a graph into a closed Riemann surface so
that each face boundary is a 3-cycle of the graph. In this work, groundstate
degeneracy in the antiferromagnetic Ising model on triangulations is studied.
We show that for every fixed closed Riemann surface S, there are
vertex-increasing sequences of triangulations of S with a non-degenerated
groundstate. In particular, we exhibit geometrically frustrated systems with a
non-degenerated groundstate.Comment: 11 pages, 9 figure
A statistical formalism of Causal Dynamical Triangulations
We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system and
thus provide a new method of solution of the model.Comment: 21 pages, 19 pictures, 1 graph, Published in section: Field Theory
And Statistical System
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P-G(q) for the generalized theta graphs Theta((s.p)) are taken together, dense in the whole complex plane with the possible exception of the disc \q - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof
Chromatic roots are dense in the whole complex plane
I show that the zeros of the chromatic polynomials P_G(q) for the generalized
theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex
plane with the possible exception of the disc |q-1| < 1. The same holds for
their dichromatic polynomials (alias Tutte polynomials, alias Potts-model
partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate
corollary is that the chromatic zeros of not-necessarily-planar graphs are
dense in the whole complex plane. The main technical tool in the proof of these
results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for
certain sequences of analytic functions, for which I give a new and simpler
proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3
adds a new Theorem 1.4 and a new Section 5, and makes several small
improvements. To appear in Combinatorics, Probability & Computin
Folding of the Triangular Lattice with Quenched Random Bending Rigidity
We study the problem of folding of the regular triangular lattice in the
presence of a quenched random bending rigidity + or - K and a magnetic field h
(conjugate to the local normal vectors to the triangles). The randomness in the
bending energy can be understood as arising from a prior marking of the lattice
with quenched creases on which folds are favored. We consider three types of
quenched randomness: (1) a ``physical'' randomness where the creases arise from
some prior random folding; (2) a Mattis-like randomness where creases are
domain walls of some quenched spin system; (3) an Edwards-Anderson-like
randomness where the bending energy is + or - K at random independently on each
bond. The corresponding (K,h) phase diagrams are determined in the hexagon
approximation of the cluster variation method. Depending on the type of
randomness, the system shows essentially different behaviors.Comment: uses harvmac (l), epsf, 17 figs included, uuencoded, tar compresse
Ergodicity of the Wang--Swendsen--Koteck\'y algorithm on several classes of lattices on the torus
We prove the ergodicity of the Wang--Swendsen--Koteck\'y (WSK) algorithm for
the zero-temperature -state Potts antiferromagnet on several classes of
lattices on the torus. In particular, the WSK algorithm is ergodic for
on any quadrangulation of the torus of girth . It is also ergodic for (resp. ) on any Eulerian triangulation of the torus such that
one sublattice consists of degree-4 vertices while the other two sublattices
induce a quadrangulation of girth (resp.~a bipartite quadrangulation)
of the torus. These classes include many lattices of interest in statistical
mechanics.Comment: 27 pages, pdflatex, and 22 pdf figures. Corrected an error in Remark
4 after Theorem 4.4. Final versio
The Gonihedric Ising Model and Glassiness
The Gonihedric 3D Ising model is a lattice spin model in which planar Peierls
boundaries between + and - spins can be created at zero energy cost. Instead of
weighting the area of Peierls boundaries as the case for the usual 3D Ising
model with nearest neighbour interactions, the edges, or "bends" in an
interface are weighted, a concept which is related to the intrinsic curvature
of the boundaries in the continuum.
In these notes we follow a roughly chronological order by first reviewing the
background to the formulation of the model, before moving on to the elucidation
of the equilibrium phase diagram by various means and then to investigation of
the non-equilibrium, glassy behaviour of the model.Comment: To appear as Chapter 7 in Rugged Free-Energy Landscapes - An
Introduction, Springer Lecture Notes in Physics, 736, ed. W. Janke, (2008
A Little Statistical Mechanics for the Graph Theorist
In this survey, we give a friendly introduction from a graph theory
perspective to the q-state Potts model, an important statistical mechanics tool
for analyzing complex systems in which nearest neighbor interactions determine
the aggregate behavior of the system. We present the surprising equivalence of
the Potts model partition function and one of the most renowned graph
invariants, the Tutte polynomial, a relationship that has resulted in a
remarkable synergy between the two fields of study. We highlight some of these
interconnections, such as computational complexity results that have alternated
between the two fields. The Potts model captures the effect of temperature on
the system and plays an important role in the study of thermodynamic phase
transitions. We discuss the equivalence of the chromatic polynomial and the
zero-temperature antiferromagnetic partition function, and how this has led to
the study of the complex zeros of these functions. We also briefly describe
Monte Carlo simulations commonly used for Potts model analysis of complex
systems. The Potts model has applications as widely varied as magnetism, tumor
migration, foam behaviors, and social demographics, and we provide a sampling
of these that also demonstrates some variations of the Potts model. We conclude
with some current areas of investigation that emphasize graph theoretic
approaches.
This paper is an elementary general audience survey, intended to popularize
the area and provide an accessible first point of entry for further
exploration.Comment: 30 pages, 3 figure
A polynomial version of Cereceda's conjecture
Let k and d be such that k ≥ d + 2. Consider two k-colourings of a d-degenerate graph G. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The k-reconfiguration graph of G is the graph whose vertices are the proper k-colourings of G, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the (d + 2)-reconfiguration graph of any d-degenerate graph on n vertices is O(n 2). So far, the existence of a polynomial diameter is open even for d = 2. In this paper, we prove that the diameter of the k-reconfiguration graph of a d-degenerate graph is O(n d+1) for k ≥ d + 2. Moreover, we prove that if k ≥ 3 2 (d + 1) then the diameter of the k-reconfiguration graph is quadratic, improving the previous bound of k ≥ 2d + 1. We also show that the 5-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs