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 $0\leq a <\infty$ 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 $\xi$ near the critical coupling $K=K_c$ is modulated by a function which is periodic in $\ln|\ln(K_c-K)|$. We have also shown that the common finite-size scaling hypothesis, according to which for a finite system at criticality $\xi$ 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 $\xi$ 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 nn-simplex fractals

The decimation map $\mathcal{D}$ for a network of admittances on an $n$-simplex lattice fractal is studied. The asymptotic behaviour of $\mathcal{D}$ 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 $n \geq 4$; 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 $\ln[(n+2)/3]/\ln 2$ to $\ln[(n+2)^3/n(n+1)^2]/\ln 2$.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