Systèmes désordonnés, processus spatiaux aléatoires et certaines applications
5 au 9 janvier 2015
Elena Agliari (Roma La Sapienza) : “Random walks: theory, techniques and applications”
The first part of these lectures is devoted to diffusion processes andrelated models. In particular, we focus on random walks on graphs andwe review the main analytical techniques for their study. Non-trivialphenomenologies (e.g., splitting between local and average properties,two-particles type problem) emerging when random walks are set inhighly inhomogeneous structures (e.g., quasi-self-similar graphs,combs) are also discussed. In the second part of these lectures wehighlight a close connection between the random walk problem and aseries of fundamental statistical-mechanics models (e.g., theoscillating network, the free scalar field, the spherical model). Infact, the latter are described by a Hamiltonian which is linear in theadjacency matrix related to the embedding structure, in such a waythat the main concepts and parameters characterizing random walks(e.g., recurrence and transience, as well as the spectral dimension)also affect the properties of these models. The strong analogiesbetween diffusion theory and (mean-field) statistical mechanics isfurther deepened from a methodological perspective: using theCurie-Weiss model and the Sherrington-Kirckpatrick model, asprototypes for simple and complex behaviors, respectively, we willshow how to solve for their free energy by mapping this problem into arandom-walk framework, so to use techniques originally meant for thelatter. Finally, we present two examples of statistical-mechanicsmodels where the topics described above come into play. Both examplesare inspired by quantitive sociology applications.
Louis-Pierre Arguin (Université de Montréal) : “Extrema of log-correlated random-variables: principles and examples”
The study of the distributions of extrema of a large collection of random variables dates back to the early 20th century and is wellestablished in the case of independent or weakly correlated variables.Until recently, few sharp results were known in the case where therandom variables are strongly correlated. In the last few years, therehave been conceptual progress in describing the distribution ofextrema of the log-correlated Gaussian fields. This class of fieldsincludes important examples such as branching Brownian motion and 2Dthe Gaussian free field. In this series of lectures, we will study thestatistics of extrema of the log-correlated Gaussian fields. The focuswill be on explaining the guiding principles behind the results. Wewill also discuss why these techniques are expected to be applicableto a variety of problems such as the maxima of characteristicpolynomials of random matrices and more, boldly, the maxima of theRiemann Zeta function.
Andrea Cavagna (Roma La Sapienza) : « Collective behaviour in biological systems »
Introduction: a phenomenon on many scales, fundamental questions,physics vs biology, the problem of scalability, more is different -small vs large groups, empirical observations. Structure: relevantobservables, polarization and global order, radial correlationfunction, spatial distribution of the neighbours, topological vsmetric interaction, cognitive vs sensory bottlenecks, the problem ofthe border. Correlation: interaction vs correlation, relevance ofbehavioural fluctuations, velocity correlation function, what is thecorrelation length, scaling relations, when the group is more than thesum of its parts, scale-free correlations, orientation vs speedcorrelations, spontaneous symmetry breaking, statistical inference,basic relations in probability, general Bayesian framework, what doesit mean to fit a model, the problem of the prior, model selection andthe Occam razor, why you should keep your model simple, maximumentropy method for living groups, the minimal model compatible withthe data, how to cope with motion – spins vs birds, spin waveapproximation, maximum entropy for orientation, maximum entropy forspeed, near a critical point?
Iwan Corwin (MIT) : Integrable probability”
A number of probabilistic systems which can be analyzed in greatdetail due to certain algebraic structures behind them. These systemsinclude certain directed polymer models, random growth process,interacting particle systems and stochastic PDEs; their analysisyields information on certain universality classes, such as theKardar-Parisi-Zhang; and these structures include Macdonald processesand quantum integrable systems. We will provide background on thisgrowing area of research and delve into a few of the recentdevelopments.
Sydney Redner (Boston University) : “Applications of Statistical Physics to Coarsening and the Dynamics of Social Systems”
When the Ising model, initially at infinite temperature, is suddenlycooled to zero temperature, a rich coarsening dynamics occurs thatexhibits surprising features. In two dimensions, the ground state isreached only about 2/3 of the time, and the evolution is characterizedby two distinct time scales, the longer of which arises fromtopological defects. There is also a deep connection between domaintopologies and continuum percolation. In three dimensions, the groundstate is never reached. Instead domains are topologically complex andcontain a small fraction of « blinker » spins that can flip perpetuallywith no energy cost. Moreover, the relaxation time growsexponentially with system size. Insights gained from the coarseningkinetics of spin systems will then be applied to social dynamics. Iwill first discuss the voter model, a paradigmatic description ofconsensus formation in a population of interacting agents. Each votercan be in one of two opinion states and continuously updates itsopinion at a rate proportional to the fraction of neighbors of theopposite opinion. Exact results for the voter model on regularlattices will be reviewed. I’ll then discuss extensions of the votermodel that attempt to incorporate elements of reality, while remainingwithin the domain of analytically tractable. These will include: (i)the voter model on complex graphs, where consensus is generallyachieved quickly and via an interesting route, (ii) the voter modelwith more than two states, where stasis can arise, (iii) the boundedcompromise model, in which two agents average their real-valuedopinions if the difference is less than athreshold and do not evolve otherwise, and (iv) the Axelrod model, in which agents possess a multi-dimensional opinion variable and twoagents interact only if they share at least one voting trait.