152 research outputs found
Renormalization group approach to interacting polymerised manifolds
We propose to study the infrared behaviour of polymerised (or tethered)
random manifolds of dimension D interacting via an exclusion condition with a
fixed impurity in d-dimensional Euclidean space in which the manifold is
embedded. We prove rigorously, via methods of Wilson's renormalization group,
the convergence to a non Gaussian fixed point for suitably chosen physical
parameters.Comment: 90 pages, Plain tex file. Updated version with more detailed
introduction and added reference
Shaken Dynamics: An Easy Way to Parallel Markov Chain Monte Carlo
We define a class of Markovian parallel dynamics for spin systems on arbitrary graphs with nearest neighbor interaction described by a Hamiltonian function H(sigma). These dynamics turn out to be reversible and their stationary measure is explicitly determined. Convergence to equilibrium and relation of the stationary measure to the usual Gibbs measure are discussed when the dynamics is defined on Z(2). Further it is shown how these dynamics can be used to define natively parallel algorithms to face problems in the context of combinatorial optimization
Phase transitions for the cavity approach to the clique problem on random graphs
We give a rigorous proof of two phase transitions for a disordered system
designed to find large cliques inside Erdos random graphs. Such a system is
associated with a conservative probabilistic cellular automaton inspired by the
cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure
CRITICAL (Phi^{4}_{3,\epsilon})
The Euclidean (\phi^{4})_{3,\epsilon model in corresponds to a
perturbation by a interaction of a Gaussian measure on scalar fields
with a covariance depending on a real parameter in the range . For one recovers the covariance of a massless
scalar field in . For is a marginal interaction.
For the covariance continues to be Osterwalder-Schrader and
pointwise positive. After introducing cutoffs we prove that for ,
sufficiently small, there exists a non-gaussian fixed point (with one unstable
direction) of the Renormalization Group iterations. These iterations converge
to the fixed point on its stable (critical) manifold which is constructed.Comment: 49 pages, plain tex, macros include
Metastability and small eigenvalues in Markov chains
In this letter we announce rigorous results that elucidate the relation
between metastable states and low-lying eigenvalues in Markov chains in a much
more general setting and with considerable greater precision as was so far
available. This includes a sharp uncertainty principle relating all low-lying
eigenvalues to mean times of metastable transitions, a relation between the
support of eigenfunctions and the attractor of a metastable state, and sharp
estimates on the convergence of probability distribution of the metastable
transition times to the exponential distribution.Comment: 5pp, AMSTe
The Global Renormalization Group Trajectory in a Critical Supersymmetric Field Theory on the Lattice Z^3
We consider an Euclidean supersymmetric field theory in given by a
supersymmetric perturbation of an underlying massless Gaussian measure
on scalar bosonic and Grassmann fields with covariance the Green's function of
a (stable) L\'evy random walk in . The Green's function depends on the
L\'evy-Khintchine parameter with . For
the interaction is marginal. We prove for
sufficiently small and initial
parameters held in an appropriate domain the existence of a global
renormalization group trajectory uniformly bounded on all renormalization group
scales and therefore on lattices which become arbitrarily fine. At the same
time we establish the existence of the critical (stable) manifold. The
interactions are uniformly bounded away from zero on all scales and therefore
we are constructing a non-Gaussian supersymmetric field theory on all scales.
The interest of this theory comes from the easily established fact that the
Green's function of a (weakly) self-avoiding L\'evy walk in is a second
moment (two point correlation function) of the supersymmetric measure governing
this model. The control of the renormalization group trajectory is a
preparation for the study of the asymptotics of this Green's function. The
rigorous control of the critical renormalization group trajectory is a
preparation for the study of the critical exponents of the (weakly)
self-avoiding L\'evy walk in .Comment: 82 pages, Tex with macros supplied. Revision includes 1. redefinition
of norms involving fermions to ensure uniqueness. 2. change in the definition
of lattice blocks and lattice polymer activities. 3. Some proofs have been
reworked. 4. New lemmas 5.4A, 5.14A, and new Theorem 6.6. 5.Typos
corrected.This is the version to appear in Journal of Statistical Physic
A Method to Study Relaxation of Metastable Phases: Macroscopic Mean-Field Dynamics
We propose two different macroscopic dynamics to describe the decay of
metastable phases in many-particle systems with local interactions. These
dynamics depend on the macroscopic order parameter through the restricted
free energy and are designed to give the correct equilibrium
distribution for . The connection between macroscopic dynamics and the
underlying microscopic dynamic are considered in the context of a projection-
operator formalism. Application to the square-lattice nearest-neighbor Ising
ferromagnet gives good agreement with droplet theory and Monte Carlo
simulations of the underlying microscopic dynamic. This includes quantitative
agreement for the exponential dependence of the lifetime on the inverse of the
applied field , and the observation of distinct field regions in which the
derivative of the lifetime with respect to depends differently on . In
addition, at very low temperatures we observe oscillatory behavior of this
derivative with respect to , due to the discreteness of the lattice and in
agreement with rigorous results. Similarities and differences between this work
and earlier works on finite Ising models in the fixed-magnetization ensemble
are discussed.Comment: 44 pages RevTeX3, 11 uuencoded Postscript figs. in separate file
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