Thermodynamics of the L\'evy spin glass

We investigate the L\'evy glass, a mean-field spin glass model with power-law distributed couplings characterized by a divergent second moment. By combining extensively many small couplings with a spare random backbone of strong bonds the model is intermediate between the Sherrington-Kirkpatrick and the Viana-Bray model. A truncated version where couplings smaller than some threshold \eps are neglected can be studied within the cavity method developed for spin glasses on locally tree-like random graphs. By performing the limit \eps\to 0 in a well-defined way we calculate the thermodynamic functions within replica symmetry and determine the de Almeida-Thouless line in the presence of an external magnetic field. Contrary to previous findings we show that there is no replica-symmetric spin glass phase. Moreover we determine the leading corrections to the ground-state energy within one-step replica symmetry breaking. The effects due to the breaking of replica symmetry appear to be small in accordance with the intuitive picture that a few strong bonds per spin reduce the degree of frustration in the system

Phase space diffusion and low temperature aging

We study the dynamical evolution of a system with a phase space consisting of configurations with random energies. The dynamics we use is of Glauber type. It allows for some dynamical evolution ang aging even at very low temperatures, through the search of configurations with lower energies.Comment: 11 pages latex, 1 ps figure adde

Partition Function Expansion on Region-Graphs and Message-Passing Equations

Disordered and frustrated graphical systems are ubiquitous in physics, biology, and information science. For models on complete graphs or random graphs, deep understanding has been achieved through the mean-field replica and cavity methods. But finite-dimensional `real' systems persist to be very challenging because of the abundance of short loops and strong local correlations. A statistical mechanics theory is constructed in this paper for finite-dimensional models based on the mathematical framework of partition function expansion and the concept of region-graphs. Rigorous expressions for the free energy and grand free energy are derived. Message-passing equations on the region-graph, such as belief-propagation and survey-propagation, are also derived rigorously.Comment: 10 pages including two figures. New theoretical and numerical results added. Will be published by JSTAT as a lette

Phase Diagram of a Classical Fluid in a Quenched Random Potential

We consider the phase diagram of a classical fluid in the presence of a random pinning potential of arbitrary strength. Introducing replicas for averaging over the quenched disorder, we use the hypernetted chain approximation to calculate the correlations in the replicated liquid. The freezing transition of the liquid into a nearly crystalline state is studied using a density functional approach, and the liquid-to-glass transition is studied using a phenomenological replica symmetry breaking approach introduced by Mezard and Parisi. The first-order liquid-to-crystal transition is found to change to a continuous liquid-to-glass transition as the strength of the disorder is increased above a threshold value.Comment: 7 pages, 4 figures, to appear in EuroPhysics Letter

Zero temperature solutions of the Edwards-Anderson model in random Husimi Lattices

We solve the Edwards-Anderson model (EA) in different Husimi lattices. We show that, at T=0, the structure of the solution space depends on the parity of the loop sizes. Husimi lattices with odd loop sizes have always a trivial paramagnetic solution stable under 1RSB perturbations while, in Husimi lattices with even loop sizes, this solution is absent. The range of stability under 1RSB perturbations of this and other RS solutions is computed analytically (when possible) or numerically. We compute the free-energy, the complexity and the ground state energy of different Husimi lattices at the level of the 1RSB approximation. We also show, when the fraction of ferromagnetic couplings increases, the existence, first, of a discontinuous transition from a paramagnetic to a spin glass phase and latter of a continuous transition from a spin glass to a ferromagnetic phase.Comment: 20 pages, 10 figures (v3: Corrected analysis of transitions. Appendix proof fixed

Chaos in Glassy Systems from a TAP Perspective

We discuss level crossing of the free-energy of TAP solutions under variations of external parameters such as magnetic field or temperature in mean-field spin-glass models that exhibit one-step Replica-Symmetry-Breaking (1RSB). We study the problem through a generalized complexity that describes the density of TAP solutions at a given value of the free-energy and a given value of the extensive quantity conjugate to the external parameter. We show that variations of the external parameter by any finite amount can induce level crossing between groups of TAP states whose free-energies are extensively different. In models with 1RSB, this means strong chaos with respect to the perturbation. The linear-response induced by extensive level crossing is self-averaging and its value matches precisely with the disorder-average of the non self-averaging anomaly computed from the 2nd moment of thermal fluctuations between low-lying, almost degenerate TAP states. We present an analytical recipe to compute the generalized complexity and test the scenario on the spherical multi-$p$ spin models under variation of temperature.Comment: 12 pages, 2 figure

Statistics of low energy excitations for the directed polymer in a 1+d1+d random medium (d=1,2,3d=1,2,3)

We consider a directed polymer of length $L$ in a random medium of space dimension $d=1,2,3$. The statistics of low energy excitations as a function of their size $l$ is numerically evaluated. These excitations can be divided into bulk and boundary excitations, with respective densities $\rho^{bulk}_L(E=0,l)$ and $\rho^{boundary}_L(E=0,l)$. We find that both densities follow the scaling behavior $\rho^{bulk,boundary}_L(E=0,l) = L^{-1-\theta_d} R^{bulk,boundary}(x=l/L)$, where $\theta_d$ is the exponent governing the energy fluctuations at zero temperature (with the well-known exact value $\theta_1=1/3$ in one dimension). In the limit $x=l/L \to 0$, both scaling functions $R^{bulk}(x)$ and $R^{boundary}(x)$ behave as $R^{bulk,boundary}(x) \sim x^{-1-\theta_d}$, leading to the droplet power law $\rho^{bulk,boundary}_L(E=0,l)\sim l^{-1-\theta_d}$ in the regime $1 \ll l \ll L$. Beyond their common singularity near $x \to 0$, the two scaling functions $R^{bulk,boundary}(x)$ are very different : whereas $R^{bulk}(x)$ decays monotonically for $0<x<1$, the function $R^{boundary}(x)$ first decays for $0<x<x_{min}$, then grows for $x_{min}<x<1$, and finally presents a power law singularity $R^{boundary}(x)\sim (1-x)^{-\sigma_d}$ near $x \to 1$. The density of excitations of length $l=L$ accordingly decays as $\rho^{boundary}_L(E=0,l=L)\sim L^{- \lambda_d}$ where $\lambda_d=1+\theta_d-\sigma_d$. We obtain $\lambda_1 \simeq 0.67$, $\lambda_2 \simeq 0.53$ and $\lambda_3 \simeq 0.39$, suggesting the possible relation $\lambda_d= 2 \theta_d$.Comment: 15 pages, 25 figure

The Wandering Exponent of a One-Dimensional Directed Polymer in a Random Potential with Finite Correlation Radius

We consider a one-dimensional directed polymer in a random potential which is characterized by the Gaussian statistics with the finite size local correlations. It is shown that the well-known Kardar's solution obtained originally for a directed polymer with delta-correlated random potential can be applied for the description of the present system only in the high-temperature limit. For the low temperature limit we have obtained the new solution which is described by the one-step replica symmetry breaking. For the mean square deviation of the directed polymer of the linear size L it provides the usual scaling $L^{2z}$ with the wandering exponent z = 2/3 and the temperature-independent prefactor.Comment: 14 pages, Late

Near optimal configurations in mean field disordered systems

We present a general technique to compute how the energy of a configuration varies as a function of its overlap with the ground state in the case of optimization problems. Our approach is based on a generalization of the cavity method to a system interacting with its ground state. With this technique we study the random matching problem as well as the mean field diluted spin glass. As a byproduct of this approach we calculate the de Almeida-Thouless transition line of the spin glass on a fixed connectivity random graph.Comment: 13 pages, 7 figure

Spectra of Euclidean Random Matrices

We study the spectrum of a random matrix, whose elements depend on the Euclidean distance between points randomly distributed in space. This problem is widely studied in the context of the Instantaneous Normal Modes of fluids and is particularly relevant at the glass transition. We introduce a systematic study of this problem through its representation by a field theory. In this way we can easily construct a high density expansion, which can be resummed producing an approximation to the spectrum similar to the Coherent Potential Approximation for disordered systems.Comment: 10 pages, 4 figure