112 research outputs found
Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term
We consider the parametric representation of the amplitudes of Abelian models
in the so-called framework of rank Tensorial Group Field Theory. These
models are called Abelian because their fields live on . We concentrate
on the case when these models are endowed with particular kinetic terms
involving a linear power in momenta. New dimensional regularization and
renormalization schemes are introduced for particular models in this class: a
rank 3 tensor model, an infinite tower of matrix models over
, and a matrix model over . For all divergent amplitudes, we
identify a domain of meromorphicity in a strip determined by the real part of
the group dimension . From this point, the ordinary subtraction program is
applied and leads to convergent and analytic renormalized integrals.
Furthermore, we identify and study in depth the Symanzik polynomials provided
by the parametric amplitudes of generic rank Abelian models. We find that
these polynomials do not satisfy the ordinary Tutte's rules
(contraction/deletion). By scrutinizing the "face"-structure of these
polynomials, we find a generalized polynomial which turns out to be stable only
under contraction.Comment: 69 pages, 35 figure
The superconducting phase transition and gauge dependence
The gauge dependence of the renormalization group functions of the
Ginzburg-Landau model is investigated. The analysis is done by means of the
Ward-Takahashi identities. After defining the local superconducting order
parameter, it is shown that its exponent is in fact gauge independent.
This happens because in the Landau gauge is the only gauge having a
physical meaning, a property not shared by the four-dimensional model where any
gauge choice is possible. The analysis is done in both the context of the
-expansion and in the fixed dimension approach. It is pointed out the
differences that arise in both of these approaches concerning the gauge
dependence.Comment: RevTex, 3 pages, no figures; accepted for publication in PRB; this
paper is a short version of cond-mat/990527
Critical behaviour of the compactified theory
We investigate the critical behaviour of the -component Euclidean model at leading order in -expansion. We consider it in
three situations: confined between two parallel planes a distance apart
from one another, confined to an infinitely long cylinder having a square
cross-section of area and to a cubic box of volume . Taking the mass
term in the form , we retrieve Ginzburg-Landau
models which are supposed to describe samples of a material undergoing a phase
transition, respectively in the form of a film, a wire and of a grain, whose
bulk transition temperature () is known. We obtain equations for the
critical temperature as functions of (film), (wire), (grain) and of
, and determine the limiting sizes sustaining the transition.Comment: 12 pages, no figure
Multiplicative processes and power laws
[Takayasu et al., Phys. Rev.Lett. 79, 966 (1997)] revisited the question of
stochastic processes with multiplicative noise, which have been studied in
several different contexts over the past decades. We focus on the regime, found
for a generic set of control parameters, in which stochastic processes with
multiplicative noise produce intermittency of a special kind, characterized by
a power law probability density distribution. We briefly explain the physical
mechanism leading to a power law pdf and provide a list of references for these
results dating back from a quarter of century. We explain how the formulation
in terms of the characteristic function developed by Takayasu et al. can be
extended to exponents , which explains the ``reason of the lucky
coincidence''. The multidimensional generalization of (\ref{eq1}) and the
available results are briefly summarized. The discovery of stretched
exponential tails in the presence of the cut-off introduced in \cite{Taka} is
explained theoretically. We end by briefly listing applications.Comment: Extended version (7 pages). Phys. Rev. E (to appear April 1998
Partial survival and inelastic collapse for a randomly accelerated particle
We present an exact derivation of the survival probability of a randomly
accelerated particle subject to partial absorption at the origin. We determine
the persistence exponent and the amplitude associated to the decay of the
survival probability at large times. For the problem of inelastic reflection at
the origin, with coefficient of restitution , we give a new derivation of
the condition for inelastic collapse, , and determine
the persistence exponent exactly.Comment: 6 page
Scaling critical behavior of superconductors at zero magnetic field
We consider the scaling behavior in the critical domain of superconductors at
zero external magnetic field. The first part of the paper is concerned with the
Ginzburg-Landau model in the zero magnetic field Meissner phase. We discuss the
scaling behavior of the superfluid density and we give an alternative proof of
Josephson's relation for a charged superfluid. This proof is obtained as a
consequence of an exact renormalization group equation for the photon mass. We
obtain Josephson's relation directly in the form , that
is, we do not need to assume that the hyperscaling relation holds. Next, we
give an interpretation of a recent experiment performed in thin films of
. We argue that the measured mean field like
behavior of the penetration depth exponent is possibly associated with a
non-trivial critical behavior and we predict the exponents and
for the correlation lenght and specific heat, respectively. In the
second part of the paper we discuss the scaling behavior in the continuum dual
Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau
model, we discuss the continuum dual version by considering a family of
scalings characterized by a parameter introduced such that
, where is the bare mass of the magnetic
induction field. We discuss the difficulties in identifying the renormalized
magnetic induction mass with the photon mass. We show that the only way to have
a critical regime with is having , that
is, with having the scaling behavior of the renormalized photon mass.Comment: RevTex, 15 pages, no figures; the subsection III-C has been removed
due to a mistak
Even-visiting random walks: exact and asymptotic results in one dimension
We reconsider the problem of even-visiting random walks in one dimension.
This problem is mapped onto a non-Hermitian Anderson model with binary
disorder. We develop very efficient numerical tools to enumerate and
characterize even-visiting walks. The number of closed walks is obtained as an
exact integer up to 1828 steps, i.e., some walks. On the analytical
side, the concepts and techniques of one-dimensional disordered systems allow
to obtain explicit asymptotic estimates for the number of closed walks of
steps up to an absolute prefactor of order unity, which is determined
numerically. All the cumulants of the maximum height reached by such walks are
shown to grow as , with exactly known prefactors. These results
illustrate the tight relationship between even-visiting walks, trapping models,
and the Lifshitz tails of disordered electron or phonon spectra.Comment: 24 pages, 4 figures. To appear in J. Phys.
Large-N transition temperature for superconducting films in a magnetic field
We consider the -component Ginzburg-Landau model in the large limit,
the system being embedded in an external constant magnetic field and confined
between two parallel planes a distance apart from one another. On physical
grounds, this corresponds to a material in the form of a film in the presence
of an external magnetic field. Using techniques from dimensional and
-function regularization, modified by the external field and the
confinement conditions, we investigate the behavior of the system as a function
of the film thickness . This behavior suggests the existence of a minimal
critical thickness below which superconductivity is suppressed.Comment: Revtex, two column, 4 pages, 2 figure
Quenched Random Graphs
Spin models on quenched random graphs are related to many important
optimization problems. We give a new derivation of their mean-field equations
that elucidates the role of the natural order parameter in these models.Comment: 9 pages, report CPTH-A264.109
Random wetting transition on the Cayley tree : a disordered first-order transition with two correlation length exponents
We consider the random wetting transition on the Cayley tree, i.e. the
problem of a directed polymer on the Cayley tree in the presence of random
energies along the left-most bonds. In the pure case, there exists a
first-order transition between a localized phase and a delocalized phase, with
a correlation length exponent . In the disordered case, we find
that the transition remains first-order, but that there exists two diverging
length scales in the critical region : the typical correlation length diverges
with the exponent , whereas the averaged correlation length
diverges with the bigger exponent and governs the finite-size
scaling properties. We describe the relations with previously studied models
that are governed by the same "Infinite Disorder Fixed Point". For the present
model, where the order parameter is the contact density
(defined as the ratio of the number of contacts over the total length
), the notion of "infinite disorder fixed point" means that the thermal
fluctuations of within a given sample, become negligeable at large
scale with respect to sample-to-sample fluctuations. We characterize the
statistics over the samples of the free-energy and of the contact density. In
particular, exactly at criticality, we obtain that the contact density is not
self-averaging but remains distributed over the samples in the thermodynamic
limit, with the distribution .Comment: 15 pages, 1 figur
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