Glass models on Bethe lattices

We consider ``lattice glass models'' in which each site can be occupied by at most one particle, and any particle may have at most l occupied nearest neighbors. Using the cavity method for locally tree-like lattices, we derive the phase diagram, with a particular focus on the vitreous phase and the highest packing limit. We also study the energy landscape via the configurational entropy, and discuss different equilibrium glassy phases. Finally, we show that a kinetic freezing, depending on the particular dynamical rules chosen for the model, can prevent the equilibrium glass transitions.Comment: 24 pages, 11 figures; minor corrections + enlarged introduction and conclusio

Quiet Planting in the Locked Constraint Satisfaction Problems

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio

Amorphous-amorphous transition and the two-step replica symmetry breaking phase

The nature of polyamorphism and amorphous-to-amorphous transition is investigated by means of an exactly solvable model with quenched disorder, the spherical s+p multi-spin interaction model. The analysis is carried out in the framework of Replica Symmetry Breaking theory and leads to the identification of low temperature glass phases of different kinds. Besides the usual `one-step' solution, known to reproduce all basic properties of structural glasses, also a physically consistent `two-step' solution arises. More complicated phases are found as well, as temperature is further decreased, expressing a complex variety of metastable states structures for amorphous systems.Comment: 8 pages, 7 figures, longer version, new references adde

Assessment of subseasonal-to-seasonal (S2S) ensemble extreme precipitation forecast skill over Europe

Heavy precipitation can lead to floods and landslides, resulting in widespread damage and significant casualties. Some of its impacts can be mitigated if reliable forecasts and warnings are available. Of particular interest is the subseasonal-to-seasonal (S2S) prediction timescale. The S2S prediction timescale has received increasing attention in the research community because of its importance for many sectors. However, very few forecast skill assessments of precipitation extremes in S2S forecast data have been conducted. The goal of this article is to assess the forecast skill of rare events, here extreme precipitation, in S2S forecasts, using a metric specifically designed for extremes. We verify extreme precipitation events over Europe in the S2S forecast model from the European Centre for Medium-Range Weather Forecasts. The verification is conducted against ERA5 reanalysis precipitation. Extreme precipitation is defined as daily precipitation accumulations exceeding the seasonal 95th percentile. In addition to the classical Brier score, we use a binary loss index to assess skill. The binary loss index is tailored to assess the skill of rare events. We analyze daily events that are locally and spatially aggregated, as well as 7 d extreme-event counts. Results consistently show a higher skill in winter compared to summer. The regions showing the highest skill are Norway, Portugal and the south of the Alps. Skill increases when aggregating the extremes spatially or temporally. The verification methodology can be adapted and applied to other variables, e.g., temperature extremes or river discharge.</p

Statistical Mechanics of the Hyper Vertex Cover Problem

We introduce and study a new optimization problem called Hyper Vertex Cover. This problem is a generalization of the standard vertex cover to hypergraphs: one seeks a configuration of particles with minimal density such that every hyperedge of the hypergraph contains at least one particle. It can also be used in important practical tasks, such as the Group Testing procedures where one wants to detect defective items in a large group by pool testing. Using a Statistical Mechanics approach based on the cavity method, we study the phase diagram of the HVC problem, in the case of random regualr hypergraphs. Depending on the values of the variables and tests degrees different situations can occur: The HVC problem can be either in a replica symmetric phase, or in a one-step replica symmetry breaking one. In these two cases, we give explicit results on the minimal density of particles, and the structure of the phase space. These problems are thus in some sense simpler than the original vertex cover problem, where the need for a full replica symmetry breaking has prevented the derivation of exact results so far. Finally, we show that decimation procedures based on the belief propagation and the survey propagation algorithms provide very efficient strategies to solve large individual instances of the hyper vertex cover problem.Comment: Submitted to PR

Cusps and shocks in the renormalized potential of glassy random manifolds: How Functional Renormalization Group and Replica Symmetry Breaking fit together

We compute the Functional Renormalization Group (FRG) disorder- correlator function R(v) for d-dimensional elastic manifolds pinned by a random potential in the limit of infinite embedding space dimension N. It measures the equilibrium response of the manifold in a quadratic potential well as the center of the well is varied from 0 to v. We find two distinct scaling regimes: (i) a "single shock" regime, v^2 ~ 1/L^d where L^d is the system volume and (ii) a "thermodynamic" regime, v^2 ~ N. In regime (i) all the equivalent replica symmetry breaking (RSB) saddle points within the Gaussian variational approximation contribute, while in regime (ii) the effect of RSB enters only through a single anomaly. When the RSB is continuous (e.g., for short-range disorder, in dimension 2 <= d <= 4), we prove that regime (ii) yields the large-N FRG function obtained previously. In that case, the disorder correlator exhibits a cusp in both regimes, though with different amplitudes and of different physical origin. When the RSB solution is 1-step and non- marginal (e.g., d < 2 for SR disorder), the correlator R(v) in regime (ii) is considerably reduced, and exhibits no cusp. Solutions of the FRG flow corresponding to non-equilibrium states are discussed as well. In all cases the regime (i) exhibits a cusp non-analyticity at T=0, whose form and thermal rounding at finite T is obtained exactly and interpreted in terms of shocks. The results are compared with previous work, and consequences for manifolds at finite N, as well as extensions to spin glasses and related models are discussed.Comment: v2: Note added in proo

The cavity method for large deviations

A method is introduced for studying large deviations in the context of statistical physics of disordered systems. The approach, based on an extension of the cavity method to atypical realizations of the quenched disorder, allows us to compute exponentially small probabilities (rate functions) over different classes of random graphs. It is illustrated with two combinatorial optimization problems, the vertex-cover and coloring problems, for which the presence of replica symmetry breaking phases is taken into account. Applications include the analysis of models on adaptive graph structures.Comment: 18 pages, 7 figure

Statistical mechanics of error exponents for error-correcting codes

Error exponents characterize the exponential decay, when increasing message length, of the probability of error of many error-correcting codes. To tackle the long standing problem of computing them exactly, we introduce a general, thermodynamic, formalism that we illustrate with maximum-likelihood decoding of low-density parity-check (LDPC) codes on the binary erasure channel (BEC) and the binary symmetric channel (BSC). In this formalism, we apply the cavity method for large deviations to derive expressions for both the average and typical error exponents, which differ by the procedure used to select the codes from specified ensembles. When decreasing the noise intensity, we find that two phase transitions take place, at two different levels: a glass to ferromagnetic transition in the space of codewords, and a paramagnetic to glass transition in the space of codes.Comment: 32 pages, 13 figure

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