261 research outputs found
On the Convergence to the Continuum of Finite Range Lattice Covariances
In J. Stat. Phys. 115, 415-449 (2004) Brydges, Guadagni and Mitter proved the
existence of multiscale expansions of a class of lattice Green's functions as
sums of positive definite finite range functions (called fluctuation
covariances). The lattice Green's functions in the class considered are
integral kernels of inverses of second order positive self adjoint operators
with constant coefficients and fractional powers thereof. The fluctuation
coefficients satisfy uniform bounds and the sequence converges in appropriate
norms to a smooth, positive definite, finite range continuum function. In this
note we prove that the convergence is actually exponentially fast.Comment: 14 pages. We have added further references as well as a proof of
Corollary 2.2. This version submitted for publicatio
Abstract polymer models with general pair interactions
A convergence criterion of cluster expansion is presented in the case of an
abstract polymer system with general pair interactions (i.e. not necessarily
hard core or repulsive). As a concrete example, the low temperature disordered
phase of the BEG model with infinite range interactions, decaying polynomially
as with , is studied.Comment: 19 pages. Corrected statement for the stability condition (2.3) and
modified section 3.1 of the proof of theorem 1 consistently with (2.3). Added
a reference and modified a sentence at the end of sec. 2.
Abstract cluster expansion with applications to statistical mechanical systems
We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions
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
The scaling limit of the energy correlations in non integrable Ising models
We obtain an explicit expression for the multipoint energy correlations of a
non solvable two-dimensional Ising models with nearest neighbor ferromagnetic
interactions plus a weak finite range interaction of strength , in a
scaling limit in which we send the lattice spacing to zero and the temperature
to the critical one. Our analysis is based on an exact mapping of the model
into an interacting lattice fermionic theory, which generalizes the one
originally used by Schultz, Mattis and Lieb for the nearest neighbor Ising
model. The interacting model is then analyzed by a multiscale method first
proposed by Pinson and Spencer. If the lattice spacing is finite, then the
correlations cannot be computed in closed form: rather, they are expressed in
terms of infinite, convergent, power series in . In the scaling limit,
these infinite expansions radically simplify and reduce to the limiting energy
correlations of the integrable Ising model, up to a finite renormalization of
the parameters. Explicit bounds on the speed of convergence to the scaling
limit are derived.Comment: 75 pages, 11 figure
Debye screening
The existence and exponential clustering of correlation functions for a classical coulomb system at low density or high temperature are proven using methods from constructive quantum field theory, the sine gordon transformation and the Glimm, Jaffe, Spencer expansion about mean field theory. This is a vindication of a belief of long standing among physicists, known as Debye screening. That is, because of special properties of the coulomb potential, the configurations of significant probability are those in which the long range parts of r −1 are mostly cancelled, leaving an effective exponentially decaying potential acting between charge clouds. This paper generalizes a previous paper of one of the authors in which these results were obtained for a special lattice system. The present treatment covers the continuous mechanics situation, with essentially arbitrary short range forces and charge species. Charge symmetry is not assumed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46519/1/220_2005_Article_BF01197700.pd
Hard squares with negative activity
We show that the hard-square lattice gas with activity z= -1 has a number of
remarkable properties. We conjecture that all the eigenvalues of the transfer
matrix are roots of unity. They fall into groups (``strings'') evenly spaced
around the unit circle, which have interesting number-theoretic properties. For
example, the partition function on an M by N lattice with periodic boundary
condition is identically 1 when M and N are coprime. We provide evidence for
these conjectures from analytical and numerical arguments.Comment: 8 page
Scaling behaviour of lattice animals at the upper critical dimension
We perform numerical simulations of the lattice-animal problem at the upper
critical dimension d=8 on hypercubic lattices in order to investigate
logarithmic corrections to scaling there. Our stochastic sampling method is
based on the pruned-enriched Rosenbluth method (PERM), appropriate to linear
polymers, and yields high statistics with animals comprised of up to 8000
sites. We estimate both the partition sums (number of different animals) and
the radii of gyration. We re-verify the Parisi-Sourlas prediction for the
leading exponents and compare the logarithmic-correction exponents to two
partially differing sets of predictions from the literature. Finally, we
propose, and test, a new Parisi-Sourlas-type scaling relation appropriate for
the logarithmic-correction exponents.Comment: 10 pages, 5 figure
Mayer and virial series at low temperature
We analyze the Mayer pressure-activity and virial pressure-density series for
a classical system of particles in continuous configuration space at low
temperature. Particles interact via a finite range potential with an attractive
tail. We propose physical interpretations of the Mayer and virial series'
radius of convergence, valid independently of the question of phase transition:
the Mayer radius corresponds to a fast increase from very small to finite
density, and the virial radius corresponds to a cross-over from monatomic to
polyatomic gas. Our results have consequences for the search of a low density,
low temperature solid-gas phase transition, consistent with the Lee-Yang
theorem for lattice gases and with the continuum Widom-Rowlinson model.Comment: 36 pages, 1 figur
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