From Large Scale Rearrangements to Mode Coupling Phenomenology

We consider the equilibrium dynamics of Ising spin models with multi-spin interactions on sparse random graphs (Bethe lattices). Such models undergo a mean field glass transition upon increasing the graph connectivity or lowering the temperature. Focusing on the low temperature limit, we identify the large scale rearrangements responsible for the dynamical slowing-down near the transition. We are able to characterize exactly the dynamics near criticality by analyzing the statistical properties of such rearrangements. Our approach can be generalized to a large variety of glassy models on sparse random graphs, ranging from satisfiability to kinetically constrained models.Comment: 4 pages, 4 figures, minor corrections, accepted versio

Possible world-wide middle miocene iridium anomaly and its relationship to periodicity of impacts and extinctions

In a study of one million years of Middle Miocene sediment deposition in ODP Hole 689B in the Weddell Sea near Antarctica, a single iridium (Ir) anomaly of 44 (+ or - 10) x 10 to the 12th gram Ir per gram rock (ppt) was observed in core 6H, section 3, 50 to 60 cm, after background contributions associated with manganese precipitates and clay are subtracted. The ODP Hole 689B is 10,000 km away from another site, DSDP Hole 588B in the Tasman Sea north of New Zealand, where a single Ir anomaly of 144 + or - 7 ppt over a background of 11 ppt was found in an earlier study of 3 million years of deposition. From chemical measurements the latter deposition was thought to be impact-related. Ir measurements were made, following neutron activation, with the Iridium Coincidence Spectrometer. The age vs depth calibration curves given in the DSDP and ODP preliminary reports indicate the ages of the Iranomalies are identical, 11.7 million years, but the absolute and relative uncertainties in the curves are not known. Based on the newest age data the age estimate is 10 million years. As the Ir was deposited at the two sites at about the same time and they are one quarter of the way around the world from each other it seems likely that the deposition was world-wide. The impact of a large asteroid or comet could produce the wide distribution, and this data is supportive of the impact relationship deduced for Deep Sea Drilling Project (DSDP) 588B from the chemical evidence. If the surface densities of Ir at the two sites are representative of the world-wide average, the diameter of a Cl type asteroid containing the necessary Ir would be 3 + or - 1 km, which is large enough to cause world-wide darkness and hence extinctions although the latter point is disputed

Cavity method for quantum spin glasses on the Bethe lattice

We propose a generalization of the cavity method to quantum spin glasses on fixed connectivity lattices. Our work is motivated by the recent refinements of the classical technique and its potential application to quantum computational problems. We numerically solve for the phase structure of a connectivity $q=3$ transverse field Ising model on a Bethe lattice with $\pm J$ couplings, and investigate the distribution of various classical and quantum observables.Comment: 27 pages, 9 figure

Random subcubes as a toy model for constraint satisfaction problems

We present an exactly solvable random-subcube model inspired by the structure of hard constraint satisfaction and optimization problems. Our model reproduces the structure of the solution space of the random k-satisfiability and k-coloring problems, and undergoes the same phase transitions as these problems. The comparison becomes quantitative in the large-k limit. Distance properties, as well the x-satisfiability threshold, are studied. The model is also generalized to define a continuous energy landscape useful for studying several aspects of glassy dynamics.Comment: 21 pages, 4 figure

Loop expansion around the Bethe-Peierls approximation for lattice models

We develop an effective field theory for lattice models, in which the only non-vanishing diagrams exactly reproduce the topology of the lattice. The Bethe-Peierls approximation appears naturally as the saddle point approximation. The corrections to the saddle-point result can be obtained systematically. We calculate the lowest loop corrections for magnetisation and correlation function.Comment: 8 page

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

Lagrangian planetary equations in Schwarzschild space--time

We have developed a method to study the effects of a perturbation to the motion of a test point--like object in a Schwarzschild spacetime. Such a method is the extension of the Lagrangian planetary equations of classical celestial mechanics into the framework of the full theory of general relativity. The method provides a natural approach to account for relativistic effects in the unperturbed problem in an exact way.Comment: 7 pages; revtex; accepted for publication in Class. Quantum Gra

BKT-like transition in the Potts model on an inhomogeneous annealed network

We solve the ferromagnetic q-state Potts model on an inhomogeneous annealed network which mimics a random recursive graph. We find that this system has the inverted Berezinskii--Kosterlitz--Thouless (BKT) phase transition for any $q \geq 1$, including the values $q \geq 3$, where the Potts model normally shows a first order phase transition. We obtain the temperature dependences of the order parameter, specific heat, and susceptibility demonstrating features typical for the BKT transition. We show that in the entire normal phase, both the distribution of a linear response to an applied local field and the distribution of spin-spin correlations have a critical, i.e. power-law, form.Comment: 7 pages, 3 figure

Probabilistic Reconstruction in Compressed Sensing: Algorithms, Phase Diagrams, and Threshold Achieving Matrices

Compressed sensing is a signal processing method that acquires data directly in a compressed form. This allows one to make less measurements than what was considered necessary to record a signal, enabling faster or more precise measurement protocols in a wide range of applications. Using an interdisciplinary approach, we have recently proposed in [arXiv:1109.4424] a strategy that allows compressed sensing to be performed at acquisition rates approaching to the theoretical optimal limits. In this paper, we give a more thorough presentation of our approach, and introduce many new results. We present the probabilistic approach to reconstruction and discuss its optimality and robustness. We detail the derivation of the message passing algorithm for reconstruction and expectation max- imization learning of signal-model parameters. We further develop the asymptotic analysis of the corresponding phase diagrams with and without measurement noise, for different distribution of signals, and discuss the best possible reconstruction performances regardless of the algorithm. We also present new efficient seeding matrices, test them on synthetic data and analyze their performance asymptotically.Comment: 42 pages, 37 figures, 3 appendixe