Dynamics of dilute disordered models: a solvable case

We study the dynamics of a dilute spherical model with two body interactions and random exchanges. We analyze the Langevin equations and we introduce a functional variational method to study generic dilute disordered models. A crossover temperature replaces the dynamic transition of the fully-connected limit. There are two asymptotic regimes, one determined by the central band of the spectral density of the interactions and a slower one determined by localized configurations on sites with high connectivity. We confront the behavior of this model to the one of real glasses.Comment: 7 pages, 4 figures. Clarified, final versio

Systematic perturbation approach for a dynamical scaling law in a kinetically constrained spin model

The dynamical behaviours of a kinetically constrained spin model (Fredrickson-Andersen model) on a Bethe lattice are investigated by a perturbation analysis that provides exact final states above the nonergodic transition point. It is observed that the time-dependent solutions of the derived dynamical systems obtained by the perturbation analysis become systematically closer to the results obtained by Monte Carlo simulations as the order of a perturbation series is increased. This systematic perturbation analysis also clarifies the existence of a dynamical scaling law, which provides a implication for a universal relation between a size scale and a time scale near the nonergodic transition.Comment: 17 pages, 7 figures, v2; results have been refined, v3; A figure has been modified, v4; results have been more refine

Sparse random matrices: the eigenvalue spectrum revisited

We revisit the derivation of the density of states of sparse random matrices. We derive a recursion relation that allows one to compute the spectrum of the matrix of incidence for finite trees that determines completely the low concentration limit. Using the iterative scheme introduced by Biroli and Monasson [J. Phys. A 32, L255 (1999)] we find an approximate expression for the density of states expected to hold exactly in the opposite limit of large but finite concentration. The combination of the two methods yields a very simple simple geometric interpretation of the tails of the spectrum. We test the analytic results with numerical simulations and we suggest an indirect numerical method to explore the tails of the spectrum.Comment: 18 pages, 7 figures. Accepted version, minor corrections, references adde

Analytic determination of dynamical and mosaic length scales in a Kac glass model

We consider a disordered spin model with multi-spin interactions undergoing a glass transition. We introduce a dynamic and a static length scales and compute them in the Kac limit (long--but--finite range interactions). They diverge at the dynamic and static phase transition with exponents (respectively) -1/4 and -1. The two length scales are approximately equal well above the mode coupling transition. Their discrepancy increases rapidly as this transition is approached. We argue that this signals a crossover from mode coupling to activated dynamics.Comment: 4 pages, 4 eps figures. New version conform to the published on

Approximation schemes for the dynamics of diluted spin models: the Ising ferromagnet on a Bethe lattice

We discuss analytical approximation schemes for the dynamics of diluted spin models. The original dynamics of the complete set of degrees of freedom is replaced by a hierarchy of equations including an increasing number of global observables, which can be closed approximately at different levels of the hierarchy. We illustrate this method on the simple example of the Ising ferromagnet on a Bethe lattice, investigating the first three possible closures, which are all exact in the long time limit, and which yield more and more accurate predictions for the finite-time behavior. We also investigate the critical region around the phase transition, and the behavior of two-time correlation functions. We finally underline the close relationship between this approach and the dynamical replica theory under the assumption of replica symmetry.Comment: 21 pages, 5 figure

Computing a Knot Invariant as a Constraint Satisfaction Problem

We point out the connection between mathematical knot theory and spin glass/search problem. In particular, we present a statistical mechanical formulation of the problem of computing a knot invariant; p-colorability problem, which provides an algorithm to find the solution. The method also allows one to get some deeper insight into the structural complexity of knots, which is expected to be related with the landscape structure of constraint satisfaction problem.Comment: 6 pages, 3 figures, submitted to short note in Journal of Physical Society of Japa

Message passing for vertex covers

Constructing a minimal vertex cover of a graph can be seen as a prototype for a combinatorial optimization problem under hard constraints. In this paper, we develop and analyze message passing techniques, namely warning and survey propagation, which serve as efficient heuristic algorithms for solving these computational hard problems. We show also, how previously obtained results on the typical-case behavior of vertex covers of random graphs can be recovered starting from the message passing equations, and how they can be extended.Comment: 25 pages, 9 figures - version accepted for publication in PR

Exact solution of the Bose-Hubbard model on the Bethe lattice

The exact solution of a quantum Bethe lattice model in the thermodynamic limit amounts to solve a functional self-consistent equation. In this paper we obtain this equation for the Bose-Hubbard model on the Bethe lattice, under two equivalent forms. The first one, based on a coherent state path integral, leads in the large connectivity limit to the mean field treatment of Fisher et al. [Phys. Rev. B {\bf 40}, 546 (1989)] at the leading order, and to the bosonic Dynamical Mean Field Theory as a first correction, as recently derived by Byczuk and Vollhardt [Phys. Rev. B {\bf 77}, 235106 (2008)]. We obtain an alternative form of the equation using the occupation number representation, which can be easily solved with an arbitrary numerical precision, for any finite connectivity. We thus compute the transition line between the superfluid and Mott insulator phases of the model, along with thermodynamic observables and the space and imaginary time dependence of correlation functions. The finite connectivity of the Bethe lattice induces a richer physical content with respect to its infinitely connected counterpart: a notion of distance between sites of the lattice is preserved, and the bosons are still weakly mobile in the Mott insulator phase. The Bethe lattice construction can be viewed as an approximation to the finite dimensional version of the model. We show indeed a quantitatively reasonable agreement between our predictions and the results of Quantum Monte Carlo simulations in two and three dimensions.Comment: 27 pages, 16 figures, minor correction

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.Comment: Added references, updated notatio

A hard-sphere model on generalized Bethe lattices: Statics

We analyze the phase diagram of a model of hard spheres of chemical radius one, which is defined over a generalized Bethe lattice containing short loops. We find a liquid, two different crystalline, a glassy and an unusual crystalline glassy phase. Special attention is also paid to the close-packing limit in the glassy phase. All analytical results are cross-checked by numerical Monte-Carlo simulations.Comment: 24 pages, revised versio