384 research outputs found
Phase Transition in the 1d Random Field ising model with long range interaction
We study the one dimensional Ising model with ferromagnetic, long range
interaction which decays as |i-j|^{-2+a}, 1/2< a<1, in the presence of an
external random filed. we assume that the random field is given by a collection
of independent identically distributed random variables, subgaussian with mean
zero. We show that for temperature and strength of the randomness (variance)
small enough with P=1 with respect to the distribution of the random fields
there are at least two distinct extremal Gibbs measures
On the formation/dissolution of equilibrium droplets
We consider liquid-vapor systems in finite volume at parameter
values corresponding to phase coexistence and study droplet formation due to a
fixed excess of particles above the ambient gas density. We identify
a dimensionless parameter and a
\textrm{universal} value \Deltac=\Deltac(d), and show that a droplet of the
dense phase occurs whenever \Delta>\Deltac, while, for \Delta<\Deltac, the
excess is entirely absorbed into the gaseous background. When the droplet first
forms, it comprises a non-trivial, \textrm{universal} fraction of excess
particles. Similar reasoning applies to generic two-phase systems at phase
coexistence including solid/gas--where the ``droplet'' is crystalline--and
polymorphic systems. A sketch of a rigorous proof for the 2D Ising lattice gas
is presented; generalizations are discussed heuristically.Comment: An announcement of a forthcoming rigorous work on the 2D Ising model;
to appear in Europhys. Let
High Temperature Expansions and Dynamical Systems
We develop a resummed high-temperature expansion for lattice spin systems
with long range interactions, in models where the free energy is not, in
general, analytic. We establish uniqueness of the Gibbs state and exponential
decay of the correlation functions. Then, we apply this expansion to the
Perron-Frobenius operator of weakly coupled map lattices.Comment: 33 pages, Latex; [email protected]; [email protected]
Fluctuations of the Phase Boundary in the Ising Ferromagnet
We discuss statistical properties of phase boundary in the 2D low-temperature Ising ferromagnet in a box with the two-component boundary conditions. We prove the weak convergence in C [O, 1] of measures describing the fluctuations of phase boundaries in the canonical ensemble of interfaces with fixed endpoints and area enclosed below them. The limiting Gaussian measure coincides with the conditional distribution of certain Gaussian process obtained by the integral transformation of the white noise
Quantum Markov fields on graphs
We introduce generalized quantum Markov states and generalized d-Markov
chains which extend the notion quantum Markov chains on spin systems to that on
-algebras defined by general graphs. As examples of generalized d-Markov
chains, we construct the entangled Markov fields on tree graphs. The concrete
examples of generalized d-Markov chains on Cayley trees are also investigated.Comment: 23 pages, 1 figure. accepted to "Infinite Dimensional Anal. Quantum
Probability & Related Topics
The low-temperature phase of Kac-Ising models
We analyse the low temperature phase of ferromagnetic Kac-Ising models in
dimensions . We show that if the range of interactions is \g^{-1},
then two disjoint translation invariant Gibbs states exist, if the inverse
temperature \b satisfies \b -1\geq \g^\k where \k=\frac
{d(1-\e)}{(2d+1)(d+1)}, for any \e>0. The prove involves the blocking
procedure usual for Kac models and also a contour representation for the
resulting long-range (almost) continuous spin system which is suitable for the
use of a variant of the Peierls argument.Comment: 19pp, Plain Te
Absence of Phase Transition for Antiferromagnetic Potts Models via the Dobrushin Uniqueness Theorem
We prove that the -state Potts antiferromagnet on a lattice of maximum
coordination number exhibits exponential decay of correlations uniformly at
all temperatures (including zero temperature) whenever . We also prove
slightly better bounds for several two-dimensional lattices: square lattice
(exponential decay for ), triangular lattice (), hexagonal
lattice (), and Kagom\'e lattice (). The proofs are based on
the Dobrushin uniqueness theorem.Comment: 32 pages including 3 figures. Self-unpacking file containing the tex
file, the needed macros (epsf.sty, indent.sty, subeqnarray.sty, and
eqsection.sty) and the 3 ps file
One-Dimensional Hard-Rod Caricature of Hydrodynamics: Navier-Stokes Correction
One-dimensional system of hard-rod particles of length a is studied in the hydrodynamical limit. The Navier-Stokes correction to Euler's equation is found for an initial locally-equilibrium family of states of constant density ρ ϵ [0,a^(-1)). The correction is given, at t~0, by the non-linear second-order differential operator (Bf)(q,v) = (a^2/2)(∂/∂q)[∫dw|v-w|f(q,w)(∂/∂q)f(q,v) - f(q,v)∫dw|v-w|(∂/∂q)f(q,w)](1-ρa)^(-1) where f(q,v) is the (hydrodynamical) density at a point q ϵ R^1 of the species of particles with velocity v ϵ R^1
On the convergence of cluster expansions for polymer gases
We compare the different convergence criteria available for cluster
expansions of polymer gases subjected to hard-core exclusions, with emphasis on
polymers defined as finite subsets of a countable set (e.g. contour expansions
and more generally high- and low-temperature expansions). In order of
increasing strength, these criteria are: (i) Dobrushin criterion, obtained by a
simple inductive argument; (ii) Gruber-Kunz criterion obtained through the use
of Kirkwood-Salzburg equations, and (iii) a criterion obtained by two of us via
a direct combinatorial handling of the terms of the expansion. We show that for
subset polymers our sharper criterion can be proven both by a suitable
adaptation of Dobrushin inductive argument and by an alternative --in fact,
more elementary-- handling of the Kirkwood-Salzburg equations. In addition we
show that for general abstract polymers this alternative treatment leads to the
same convergence region as the inductive Dobrushin argument and, furthermore,
to a systematic way to improve bounds on correlations
Lattice Dynamics in the Half-Space, II. Energy Transport Equation
We consider the lattice dynamics in the half-space. The initial data are
random according to a probability measure which enforces slow spatial variation
on the linear scale . We establish two time regimes. For
times of order , , locally the measure
converges to a Gaussian measure which is time stationary with a covariance
inherited from the initial measure (non-Gaussian, in general). For times of
order , this covariance changes in time and is governed by a
semiclassical transport equation.Comment: 35 page
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