Strongly Correlated Random Interacting Processes

The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including selfinteracting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations

Large Scale Stochastic Dynamics

The goal of this workshop was to explore the recent advances in the mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. There were 55 participants, including postdocs and graduate students, working in diverse intertwining areas of probability and statistical mechanics. During the meeting, 29 talks of 45 minutes were scheduled and an evening session was organised with 10 more short talks of 10 minutes, mostly by younger participants. These talks addressed the following topics : randomness emerging from deterministic dynamics, hydrodynamic limits, interface growth models and slow convergence to equilibrium in kinetically constrained dynamics

Critical phenomena in complex networks

The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, researchers have made important steps toward understanding the qualitatively new critical phenomena in complex networks. We review the results, concepts, and methods of this rapidly developing field. Here we mostly consider two closely related classes of these critical phenomena, namely structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. We also discuss systems where a network and interacting agents on it influence each other. We overview a wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, k-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks. We also discuss strong finite size effects in these systems and highlight open problems and perspectives.Comment: Review article, 79 pages, 43 figures, 1 table, 508 references, extende

Glassy dynamics of kinetically constrained models

We review the use of kinetically constrained models (KCMs) for the study of dynamics in glassy systems. The characteristic feature of KCMs is that they have trivial, often non-interacting, equilibrium behaviour but interesting slow dynamics due to restrictions on the allowed transitions between configurations. The basic question which KCMs ask is therefore how much glassy physics can be understood without an underlying ``equilibrium glass transition''. After a brief review of glassy phenomenology, we describe the main model classes, which include spin-facilitated (Ising) models, constrained lattice gases, models inspired by cellular structures such as soap froths, models obtained via mappings from interacting systems without constraints, and finally related models such as urn, oscillator, tiling and needle models. We then describe the broad range of techniques that have been applied to KCMs, including exact solutions, adiabatic approximations, projection and mode-coupling techniques, diagrammatic approaches and mappings to quantum systems or effective models. Finally, we give a survey of the known results for the dynamics of KCMs both in and out of equilibrium, including topics such as relaxation time divergences and dynamical transitions, nonlinear relaxation, aging and effective temperatures, cooperativity and dynamical heterogeneities, and finally non-equilibrium stationary states generated by external driving. We conclude with a discussion of open questions and possibilities for future work.Comment: 137 pages. Additions to section on dynamical heterogeneities (5.5, new pages 110 and 112), otherwise minor corrections, additions and reference updates. Version to be published in Advances in Physic

Logarithmic Conformal Field Theory: Beyond an Introduction

This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is on continuum methods and their generalisation from the familiar rational case. In particular, the modular properties of the characters of the spectrum play a central role and Verlinde formulae are evaluated with the results compared to the known fusion rules. Moreover, bulk modular invariants are constructed, the structures of the corresponding bulk state spaces are elucidated, and a formalism for computing correlation functions is discussed.Comment: Invited review by J Phys A for a special issue on LCFT; v2 updated references; v3 fixed a few minor typo

The roles of random boundary conditions in spin systems

Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these result

Theoretical and experimental approaches for the initiation and propagation of activity in spatially embedded neuronal cultures

[eng] Spatial embedding and inherited metric constraints are a fundamental trait of biological neuronal circuits. However their role in shaping connectivity and dynamics has been often disregarded, with models of neuronal networks paying much more attention to the distribution of connections in the quest for understanding network's behavior. In this thesis we aim at filling this gap by studying the importance of metric features in the complex connectivity- dynamics-noise interplay that shapes spontaneous neuronal activity. This thesis combines experiments in rat dissociated neuronal cultures with theoretical analyses to better comprehend the relevance of spatial embedding. We developed a new theoretical model grounded on Ising Models to assess metric effects in neuronal cultures' behavior, and in the context of percolation approaches. Once metric effects were settled, we illustrated their relevance in shaping spontaneous activity by perturbing the structural connectivity blueprint of neuronal cultures. This was achieved by patterning the substrate where neurons grow, and by using topographical molds that dictated the connectivity of the network. Next, and since the initiation of bursting activity is governed in great manner by a complex amplification mechanism that involves metric correlations and noise, we focused on the metric-driven amplification of spontaneous single-neuron noise to derive an analytical model that predicts the frequency of bursting events in neuronal cultures. We then further investigated in an experimental context the contribution of noise to the observed activity patterns, and by implementing a moderate electrical stimulation protocol that increases the level of activity noise in cultures. Finally, the latter study was completed with experiments regarding the specific role of inhibition in neuronal networks, to provide a wider understanding of the mechanisms that govern the initiation and propagation of activity fronts in cortical cultures.[cat] L'objectiu d'aquesta tesis és investigar els mecanismes que generen l'activitat espontània i estimulada en xarxes neuronals, més concretament en cultius corticals dissociats, i fent un especial èmfasi en l’efecte de les correlacions mètriques. En aquest marc, l’activitat col·lectiva consisteix en episodis esporàdics de dispars quasi sincronitzats entre totes les neurones del cultiu, anomenats “esclats de xarxa”. Tres elements principals en determinen les característiques: connectivitat entre neurones, dinàmica intrínseca neuronal, i soroll (activacions neuronals aleatòries). La investigació s’ha centrat en cinc línies de recerca: l’estudi de correlacions mètriques en cultius neuronals; el desenvolupament d’un model teòric per descriure i predir l’esclat de xarxa; l’anàlisi de la propagació dels fronts d’activitat experimentals sota pertorbacions estructurals de la connectivitat del cultiu; l’estudi de l’efecte de la inhibició en la iniciació i propagació dels esclats ‘in vitro’; i l’estudi de la resposta experimental dels cultius sota una estimulació elèctrica moderada de baixa freqüència. En la primera línia de recerca hem comprovat que les correlacions mètriques dominen el comportament dinàmic del cultiu, fins al punt d’emmascarar la contribució de la distribució del nombre de connexions. En la segona línia hem desenvolupat un model analític que prediu semi- quantitativament la freqüència dels esclats observada experimentalment. La tercera línia s’ha centrat en l’efecte de pertorbacions estructurals en la connectivitat; la dinàmica resultant ha mostrat una gran riquesa en patrons d’activitat, esclats de xarxa a diferents escales, i propagació altament específica de cada cultiu. La quarta línia de recerca ha demostrat que les xarxes sense inhibició disminueixen la seva freqüència d’esclat respecte a les xarxes control, que la velocitat de propagació de l’activitat incrementa lleugerament quan s’ha bloquejat la inhibició, i que els punts on s’inicien ens esclats varien respecte als controls. I, finalment, la cinquena línia de recerca ha constatat que l’aplicació d’un camp elèctric feble augmenta el soroll d’activitat de la xarxa, generant un increment en la freqüència dels esclats de xarxa