267 research outputs found
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
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]
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
Successful network inference from time-series data using Mutual Information Rate
This work uses an information-based methodology to infer the connectivity of complex systems from observed time-series data. We first derive analytically an expression for the Mutual Information Rate (MIR), namely, the amount of information exchanged per unit of time, that can be used to estimate the MIR between two finite-length low-resolution noisy time-series, and then apply it after a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems. In particular, we show that our methodology successfully infers the connectivity for heterogeneous networks, different time-series lengths or coupling strengths, and even in the presence of additive noise. Finally, we show that our methodology based on MIR successfully infers the connectivity of networks composed of nodes with different time-scale dynamics, where inference based on Mutual Information fails
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
Entropy-driven phase transition in a polydisperse hard-rods lattice system
We study a system of rods on the 2d square lattice, with hard-core exclusion.
Each rod has a length between 2 and N. We show that, when N is sufficiently
large, and for suitable fugacity, there are several distinct Gibbs states, with
orientational long-range order. This is in sharp contrast with the case N=2
(the monomer-dimer model), for which Heilmann and Lieb proved absence of phase
transition at any fugacity. This is the first example of a pure hard-core
system with phases displaying orientational order, but not translational order;
this is a fundamental characteristic feature of liquid crystals
Pattern densities in fluid dimer models
In this paper, we introduce a family of observables for the dimer model on a
bi-periodic bipartite planar graph, called pattern density fields. We study the
scaling limit of these objects for liquid and gaseous Gibbs measures of the
dimer model, and prove that they converge to a linear combination of a
derivative of the Gaussian massless free field and an independent white noise.Comment: 38 pages, 3 figure
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