15 research outputs found
Critical and tricritical singularities of the three-dimensional random-bond Potts model for large
We study the effect of varying strength, , of bond randomness on the
phase transition of the three-dimensional Potts model for large . The
cooperative behavior of the system is determined by large correlated domains in
which the spins points into the same direction. These domains have a finite
extent in the disordered phase. In the ordered phase there is a percolating
cluster of correlated spins. For a sufficiently large disorder
this percolating cluster coexists with a percolating cluster
of non-correlated spins. Such a co-existence is only possible in more than two
dimensions. We argue and check numerically that is the tricritical
disorder, which separates the first- and second-order transition regimes. The
tricritical exponents are estimated as and
. We claim these exponents are independent, for sufficiently
large . In the second-order transition regime the critical exponents
and are independent of the strength of
disorder.Comment: 12 pages, 11 figure
Complexity spectrum of some discrete dynamical systems
We first study birational mappings generated by the composition of the matrix
inversion and of a permutation of the entries of matrices. We
introduce a semi-numerical analysis which enables to compute the Arnold
complexities for all the possible birational transformations. These
complexities correspond to a spectrum of eighteen algebraic values. We then
drastically generalize these results, replacing permutations of the entries by
homogeneous polynomial transformations of the entries possibly depending on
many parameters. Again it is shown that the associated birational, or even
rational, transformations yield algebraic values for their complexities.Comment: 1 LaTex fil
Disorder driven phase transitions of the large q-state Potts model in 3d
Phase transitions induced by varying the strength of disorder in the large-q
state Potts model in 3d are studied by analytical and numerical methods. By
switching on the disorder the transition stays of first order, but different
thermodynamical quantities display essential singularities. Only for strong
enough disorder the transition will be soften into a second-order one, in which
case the ordered phase becomes non-homogeneous at large scales, while the
non-correlated sites percolate the sample. In the critical regime the critical
exponents are found universal: \beta/\nu=0.60(2) and \nu=0.73(1).Comment: 4 pages; 3 figure
Integrability of the critical point of the Kagom\'e three-state Potts mode
The vicinity of the critical point of the three-state Potts model on a
Kagom\'e lattice is studied by mean of Random Matrix Theory. Strong evidence
that the critical point is integrable is given.Comment: 1 LaTex file + 3 eps files 7 page
Symmetry, complexity and multicritical point of the two-dimensional spin glass
We analyze models of spin glasses on the two-dimensional square lattice by
exploiting symmetry arguments. The replicated partition functions of the Ising
and related spin glasses are shown to have many remarkable symmetry properties
as functions of the edge Boltzmann factors. It is shown that the applications
of homogeneous and Hadamard inverses to the edge Boltzmann matrix indicate
reduced complexities when the elements of the matrix satisfy certain
conditions, suggesting that the system has special simplicities under such
conditions. Using these duality and symmetry arguments we present a conjecture
on the exact location of the multicritical point in the phase diagram.Comment: 32 pages, 6 figures; a few typos corrected. To be published in J.
Phys.
On the complexity of some birational transformations
Using three different approaches, we analyze the complexity of various
birational maps constructed from simple operations (inversions) on square
matrices of arbitrary size. The first approach consists in the study of the
images of lines, and relies mainly on univariate polynomial algebra, the second
approach is a singularity analysis, and the third method is more numerical,
using integer arithmetics. Each method has its own domain of application, but
they give corroborating results, and lead us to a conjecture on the complexity
of a class of maps constructed from matrix inversions
A classification of four-state spin edge Potts models
We classify four-state spin models with interactions along the edges
according to their behavior under a specific group of symmetry transformations.
This analysis uses the measure of complexity of the action of the symmetries,
in the spirit of the study of discrete dynamical systems on the space of
parameters of the models, and aims at uncovering solvable ones. We find that
the action of these symmetries has low complexity (polynomial growth, zero
entropy). We obtain natural parametrizations of various models, among which an
unexpected elliptic parametrization of the four-state chiral Potts model, which
we use to localize possible integrability conditions associated with high genus
curves.Comment: 5 figure
Random Matrix Theory and higher genus integrability: the quantum chiral Potts model
We perform a Random Matrix Theory (RMT) analysis of the quantum four-state
chiral Potts chain for different sizes of the chain up to size L=8. Our
analysis gives clear evidence of a Gaussian Orthogonal Ensemble statistics,
suggesting the existence of a generalized time-reversal invariance.
Furthermore a change from the (generic) GOE distribution to a Poisson
distribution occurs when the integrability conditions are met. The chiral Potts
model is known to correspond to a (star-triangle) integrability associated with
curves of genus higher than zero or one. Therefore, the RMT analysis can also
be seen as a detector of ``higher genus integrability''.Comment: 23 pages and 10 figure
Functional relations in lattice statistical mechanics, enumerative combinatorics, and discrete dynamical systems
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