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 maximal independent sets

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation

An algorithm for counting circuits: application to real-world and random graphs

We introduce an algorithm which estimates the number of circuits in a graph as a function of their length. This approach provides analytical results for the typical entropy of circuits in sparse random graphs. When applied to real-world networks, it allows to estimate exponentially large numbers of circuits in polynomial time. We illustrate the method by studying a graph of the Internet structure.Comment: 7 pages, 3 figures, minor corrections, accepted versio

A Theory of Cheap Control in Embodied Systems

We present a framework for designing cheap control architectures for embodied agents. Our derivation is guided by the classical problem of universal approximation, whereby we explore the possibility of exploiting the agent's embodiment for a new and more efficient universal approximation of behaviors generated by sensorimotor control. This embodied universal approximation is compared with the classical non-embodied universal approximation. To exemplify our approach, we present a detailed quantitative case study for policy models defined in terms of conditional restricted Boltzmann machines. In contrast to non-embodied universal approximation, which requires an exponential number of parameters, in the embodied setting we are able to generate all possible behaviors with a drastically smaller model, thus obtaining cheap universal approximation. We test and corroborate the theory experimentally with a six-legged walking machine. The experiments show that the sufficient controller complexity predicted by our theory is tight, which means that the theory has direct practical implications. Keywords: cheap design, embodiment, sensorimotor loop, universal approximation, conditional restricted Boltzmann machineComment: 27 pages, 10 figure

Deterministic generation of an on-demand Fock state

We theoretically study the deterministic generation of photon Fock states on-demand using a protocol based on a Jaynes Cummings quantum random walk which includes damping. We then show how each of the steps of this protocol can be implemented in a low temperature solid-state quantum system with a Nitrogen-Vacancy centre in a nano-diamond coupled to a nearby high-Q optical cavity. By controlling the coupling duration between the NV and the cavity via the application of a time dependent Stark shift, and by increasing the decay rate of the NV via stimulated emission depletion (STED) a Fock state with high photon number can be generated on-demand. Our setup can be integrated on a chip and can be accurately controlled.Comment: 13 pages, 9 figure

A List Referring Monte Carlo Method for Lattice Glass Models

We present an effcient Monte-Carlo method for lattice glass models which are characterized by hard constraint conditions. The basic idea of the method is similar to that of the $N$-fold way method. By using a list of sites into which we can insert a particle, we avoid trying a useless transition which is forbidden by the constraint conditions. We applied the present method to a lattice glass model proposed by Biroli and M{\'e}zard. We first evaluated the efficiency of the method through measurements of the autocorrelation function of particle configurations. As a result, we found that the efficiency is much higher than that of the standard Monte-Carlo method. We also compared the efficiency of the present method with that of the $N$-fold way method in detail. We next examined how the efficiency of extended ensemble methods such as the replica exchange method and the Wang-Landau method is inflnuenced by the choice of the local update method. The results show that the efficiency is considerably improved by the use of efficient local update methods. For example, when the number of sites $N_{\rm site}$ is 1024, the ergodic time $\tau_{\rm E}$ of the replica exchange method in the grand-canonical ensemble, which is the average round-trip time of a replica in chemical-potential space, with the present local update method is more than $10^2$ times shorter than that with the standard local update method. This result shows that the efficient local update method is quite important to make extended ensemble methods more effective.Comment: 16 pages, 21 figures; 1 subsection, 1 appendix, and 5 figures are added, abstract is changed, 1 figure is remove

HAMAP: a database of completely sequenced microbial proteome sets and manually curated microbial protein families in UniProtKB/Swiss-Prot

The growth in the number of completely sequenced microbial genomes (bacterial and archaeal) has generated a need for a procedure that provides UniProtKB/Swiss-Prot-quality annotation to as many protein sequences as possible. We have devised a semi-automated system, HAMAP (High-quality Automated and Manual Annotation of microbial Proteomes), that uses manually built annotation templates for protein families to propagate annotation to all members of manually defined protein families, using very strict criteria. The HAMAP system is composed of two databases, the proteome database and the family database, and of an automatic annotation pipeline. The proteome database comprises biological and sequence information for each completely sequenced microbial proteome, and it offers several tools for CDS searches, BLAST options and retrieval of specific sets of proteins. The family database currently comprises more than 1500 manually curated protein families and their annotation templates that are used to annotate proteins that belong to one of the HAMAP families. On the HAMAP website, individual sequences as well as whole genomes can be scanned against all HAMAP families. The system provides warnings for the absence of conserved amino acid residues, unusual sequence length, etc. Thanks to the implementation of HAMAP, more than 200 000 microbial proteins have been fully annotated in UniProtKB/Swiss-Prot (HAMAP website: http://www.expasy.org/sprot/hamap)

Age of Stratospheric Air: Progress on Processes, Observations, and Long‐Term Trends

Age of stratospheric air is a well established metric for the stratospheric transport circulation. Rooted in a robust theoretical framework, this approach offers the benefit of being deducible from observations of trace gases. Given potential climate-induced changes, observational constraints on stratospheric circulation are crucial. In the past two decades, scientific progress has been made in three main areas: (a) Enhanced process understanding and the development of process diagnostics led to better quantification of individual transport processes from observations and to a better understanding of model deficits. (b) The global age of air climatology is now well constrained by observations thanks to improved quality and quantity of data, including global satellite data, and through improved and consistent age calculation methods. (c) It is well established and understood that global models predict a decrease in age, that is, an accelerating stratospheric circulation, in response to forcing by greenhouse gases and ozone depleting substances. Observational records now confirm long-term forced trends in mean age in the lower stratosphere. However, in the mid-stratosphere, uncertainties in observational records are too large to confirm or disprove the model predictions. Continuous monitoring of stratospheric trace gases and further improved methods to derive age from those tracers will be crucial to better constrain variability and long-term trends from observations. Future work on mean age as a metric for stratospheric transport will be important due to its potential to enhance the understanding of stratospheric composition changes, address climate model biases, and assess the impacts of proposed climate geoengineering methods