26 research outputs found
A Potts/Ising Correspondence on Thin Graphs
We note that it is possible to construct a bond vertex model that displays
q-state Potts criticality on an ensemble of phi3 random graphs of arbitrary
topology, which we denote as ``thin'' random graphs in contrast to the fat
graphs of the planar diagram expansion.
Since the four vertex model in question also serves to describe the critical
behaviour of the Ising model in field, the formulation reveals an isomorphism
between the Potts and Ising models on thin random graphs. On planar graphs a
similar correspondence is present only for q=1, the value associated with
percolation.Comment: 6 pages, 5 figure
Thin Animals
Lattice animals provide a discretized model for the theta transition
displayed by branched polymers in solvent. Exact graph enumeration studies have
given some indications that the phase diagram of such lattice animals may
contain two collapsed phases as well as an extended phase. This has not been
confirmed by studies using other means. We use the exact correspondence between
the q --> 1 limit of an extended Potts model and lattice animals to investigate
the phase diagram of lattice animals on phi-cubed random graphs of arbitrary
topology (``thin'' random graphs). We find that only a two phase structure
exists -- there is no sign of a second collapsed phase.
The random graph model is solved in the thermodynamic limit by saddle point
methods. We observe that the ratio of these saddle point equations give
precisely the fixed points of the recursion relations that appear in the
solution of the model on the Bethe lattice by Henkel and Seno. This explains
the equality of non-universal quantities such as the critical lines for the
Bethe lattice and random graph ensembles.Comment: Latex, 10 pages plus 6 ps/eps figure
Global Bethe lattice consideration of the spin-1 Ising model
The spin-1 Ising model with bilinear and biquadratic exchange interactions
and single-ion crystal field is solved on the Bethe lattice using exact
recursion equations. The general procedure of critical properties investigation
is discussed and full set of phase diagrams are constructed for both positive
and negative biquadratic couplings. In latter case we observe all remarkable
features of the model, uncluding doubly-reentrant behavior and ferrimagnetic
phase. A comparison with the results of other approximation schemes is done.Comment: Latex, 11 pages, 13 ps figures available upon reques
New order parameters in the Potts model on a Cayley tree
For the state Potts model new order parameters projecting on a group of
spins instead of a single spin are introduced. On a Cayley tree this allows the
physical interpretation of the Potts model at noninteger values q of the number
of states. The model can be solved recursively. This recursion exhibits chaotic
behaviour changing qualitatively at critical values of . Using an
additional order parameter belonging to a group of zero extrapolated size the
additional ordering is related to a percolation problem. This percolation
distinguishes different phases and explains the critical indices of percolation
class occuring at the Peierls temperature.Comment: 16 pages TeX, 5 figures PostScrip
Phase transitions for -adic Potts model on the Cayley tree of order three
In the present paper, we study a phase transition problem for the -state
-adic Potts model over the Cayley tree of order three. We consider a more
general notion of -adic Gibbs measure which depends on parameter
\rho\in\bq_p. Such a measure is called {\it generalized -adic quasi Gibbs
measure}. When equals to -adic exponent, then it coincides with the
-adic Gibbs measure. When , then it coincides with -adic quasi
Gibbs measure. Therefore, we investigate two regimes with respect to the value
of . Namely, in the first regime, one takes for some
J\in\bq_p, in the second one . In each regime, we first find
conditions for the existence of generalized -adic quasi Gibbs measures.
Furthermore, in the first regime, we establish the existence of the phase
transition under some conditions. In the second regime, when we prove the existence of a quasi phase transition. It turns out that
if and \sqrt{-3}\in\bq_p, then one finds the existence
of the strong phase transition.Comment: 27 page
Study of Percolative Transitions with First-Order Characteristics in the Context of CMR Manganites
The unusual magneto-transport properties of manganites are widely believed to
be caused by mixed-phase tendencies and concomitant percolative processes.
However, dramatic deviations from "standard" percolation have been unveiled
experimentally. Here, a semi-phenomenological description of Mn oxides is
proposed based on coexisting clusters with smooth surfaces, as suggested by
Monte Carlo simulations of realistic models for manganites, also briefly
discussed here. The present approach produces fairly abrupt percolative
transitions and even first-order discontinuities, in agreement with
experiments. These transitions may describe the percolation that occurs after
magnetic fields align the randomly oriented ferromagnetic clusters believed to
exist above the Curie temperature in Mn oxides. In this respect, part of the
manganite phenomenology could belong to a new class of percolative processes
triggered by phase competition and correlations.Comment: 4 pages, 4 eps figure
On Quantum Markov Chains on Cayley tree II: Phase transitions for the associated chain with XY-model on the Cayley tree of order three
In the present paper we study forward Quantum Markov Chains (QMC) defined on
a Cayley tree. Using the tree structure of graphs, we give a construction of
quantum Markov chains on a Cayley tree. By means of such constructions we prove
the existence of a phase transition for the XY-model on a Cayley tree of order
three in QMC scheme. By the phase transition we mean the existence of two now
quasi equivalent QMC for the given family of interaction operators
.Comment: 34 pages, 1 figur
Probability measures and Hamiltonian models on Bethe lattices. I. Properties and construction of MRT probability measures
The properties of one???step Markov, rotationally and m???step (m=1 or 2) translationally invariant (MRT) probability measures on q???state???site (qSS) Bethe lattices are studied. A theorem is proven, which completely defines such measures in terms of m(q2+q) fundamental probabilities. These are explicitly calculated for any MRT???qSS Hamiltonian model. As a consequence of our approach, the dychotomy between alternative solutions of Hamiltonian models on Bethe lattices is solved
Clusters and Ising droplets in the antiferromagnetic lattice gas
A definition of clusters of particles and holes with antiferromagnetic order is given for a lattice gas with coupling constant K < 0. In two dimensions it is shown that the Ising antiferromagnetic critical line is also a percolation line if Pb = 1 - exp(-K/2). Along this line these clusters called ``droplets'' diverge with Ising exponents