111 research outputs found
New technique for replica symmetry breaking with application to the SK-model at and near T=0
We describe a novel method which allows the treatment of high orders of
replica-symmetry-breaking (RSB) at low temperatures as well as at T=0 directly,
without a need for approximations or scaling assumptions. It yields the low
temperature order function q(a,T) in the full range and is
complete in the sense that all observables can be calculated from it. The
behavior of some observables and the finite RSB theory itself is analyzed as
one approaches continuous RSB. The validity and applicability of the
traditional continuous formulation is then scrutinized and a new continuous RSB
formulation is proposed
Double Criticality of the Sherrington-Kirkpatrick Model at T=0
Numerical results up to 42nd order of replica symmetry breaking (RSB) are
used to predict the singular structure of the SK spin glass at T=0. We confirm
predominant single parameter scaling and derive corrections for the T=0 order
function q(a), related to a Langevin equation with pseudotime 1/a. a=0 and
a=\infty are shown to be two critical points for \infty-RSB, associated with
two discrete spectra of Parisi block size ratios, attached to a continuous
spectrum. Finite-RSB-size scaling, associated exponents, and T=0-energy are
obtained with unprecedented accuracy.Comment: 4 pages, 5 figure
The branching structure of diffusion-limited aggregates
I analyze the topological structures generated by diffusion-limited
aggregation (DLA), using the recently developed "branched growth model". The
computed bifurcation number B for DLA in two dimensions is B ~ 4.9, in good
agreement with the numerically obtained result of B ~ 5.2. In high dimensions,
B -> 3.12; the bifurcation ratio is thus a decreasing function of
dimensionality. This analysis also determines the scaling properties of the
ramification matrix, which describes the hierarchy of branches.Comment: 6 pages, 1 figure, Euro-LaTeX styl
The statistical mechanics of combinatorial optimization problems with site disorder
We study the statistical mechanics of a class of problems whose phase space
is the set of permutations of an ensemble of quenched random positions.
Specific examples analyzed are the finite temperature traveling salesman
problem on several different domains and various problems in one dimension such
as the so called descent problem. We first motivate our method by analyzing
these problems using the annealed approximation, then the limit of a large
number of points we develop a formalism to carry out the quenched calculation.
This formalism does not require the replica method and its predictions are
found to agree with Monte Carlo simulations. In addition our method reproduces
an exact mathematical result for the Maximum traveling salesman problem in two
dimensions and suggests its generalization to higher dimensions. The general
approach may provide an alternative method to study certain systems with
quenched disorder.Comment: 21 pages RevTex, 8 figure
Oscillatory Behavior of Critical Amplitudes of the Gaussian Model on a Hierarchical Structure
We studied oscillatory behavior of critical amplitudes for the Gaussian model
on a hierarchical structure presented by a modified Sierpinski gasket lattice.
This model is known to display non-standard critical behavior on the lattice
under study. The leading singular behavior of the correlation length near
the critical coupling is modulated by a function which is periodic in
. We have also shown that the common finite-size scaling
hypothesis, according to which for a finite system at criticality should
be of the order of the size of system, is not applicable in this case. As a
consequence of this, the exact form of the leading singular behavior of
differs from the one described earlier (which was based on the finite-size
scaling assumption).Comment: 9 pages (REVTEX), 2 figures (EPS), Phys. Rev. E (accepted
Electrical networks on -simplex fractals
The decimation map for a network of admittances on an
-simplex lattice fractal is studied. The asymptotic behaviour of
for large-size fractals is examined. It is found that in the
vicinity of the isotropic point the eigenspaces of the linearized map are
always three for ; they are given a characterization in terms of
graph theory. A new anisotropy exponent, related to the third eigenspace, is
found, with a value crossing over from to
.Comment: 14 pages, 8 figure
Universal interface width distributions at the depinning threshold
We compute the probability distribution of the interface width at the
depinning threshold, using recent powerful algorithms. It confirms the
universality classes found previously. In all cases, the distribution is
surprisingly well approximated by a generalized Gaussian theory of independant
modes which decay with a characteristic propagator G(q)=1/q^(d+2 zeta); zeta,
the roughness exponent, is computed independently. A functional renormalization
analysis explains this result and allows to compute the small deviations, i.e.
a universal kurtosis ratio, in agreement with numerics. We stress the
importance of the Gaussian theory to interpret numerical data and experiments.Comment: 4 pages revtex4. See also the following article cond-mat/030146
Modified Thouless-Anderson-Palmer equations for the Sherrington-Kirkpatrick spin glass: Numerical solutions
For large but finite systems the static properties of the infinite ranged
Sherrington-Kirkpatrick model are numerically investigated in the entire the
glass regime. The approach is based on the modified Thouless-Anderson-Palmer
equations in combination with a phenomenological relaxational dynamics used as
a numerical tool. For all temperatures and all bond configurations stable and
meta stable states are found. Following a discussion of the finite size
effects, the static properties of the state of lowest free energy are presented
in the presence of a homogeneous magnetic field for all temperatures below the
spin glass temperature. Moreover some characteristic features of the meta
stable states are presented. These states exist in finite temperature intervals
and disappear via local saddle node bifurcations. Numerical evidence is found
that the excess free energy of the meta stable states remains finite in the
thermodynamic limit. This implies a the `multi-valley' structure of the free
energy on a sub-extensive scale.Comment: Revtex 10 pages 13 figures included, submitted to Phys.Rev.B.
Shortend and improved version with additional numerical dat
Avalanches in mean-field models and the Barkhausen noise in spin-glasses
We obtain a general formula for the distribution of sizes of "static
avalanches", or shocks, in generic mean-field glasses with
replica-symmetry-breaking saddle points. For the Sherrington-Kirkpatrick (SK)
spin-glass it yields the density rho(S) of the sizes of magnetization jumps S
along the equilibrium magnetization curve at zero temperature. Continuous
replica-symmetry breaking allows for a power-law behavior rho(S) ~ 1/(S)^tau
with exponent tau=1 for SK, related to the criticality (marginal stability) of
the spin-glass phase. All scales of the ultrametric phase space are implicated
in jump events. Similar results are obtained for the sizes S of static jumps of
pinned elastic systems, or of shocks in Burgers turbulence in large dimension.
In all cases with a one-step solution, rho(S) ~ S exp(-A S^2). A simple
interpretation relating droplets to shocks, and a scaling theory for the
equilibrium analog of Barkhausen noise in finite-dimensional spin glasses are
discussed.Comment: 6 pages, 1 figur
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