DISORDERED SYSTEMS, RANDOM SPATIAL PROCESSES AND SOME APPLICATIONS

The first part of these lectures is devoted to diffusion processes and related models. In particular, we focus on random walks on graphs and we review the main analytical techniques for their study. Non-trivial phenomenologies (e.g., splitting between local and average properties, two-particles type problem) emerging when random walks are set in highly inhomogeneous structures (e.g., quasi-self-similar graphs, combs) are also discussed. In the second part of these lectures we highlight a close connection between the random walk problem and a series of fundamental statistical-mechanics models (e.g., the oscillating network, the free scalar field, the spherical model). In fact, the latter are described by a Hamiltonian which is linear in the adjacency matrix related to the embedding structure, in such a way that 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 analogies between diffusion theory and (mean-field) statistical mechanics is further deepened from a methodological perspective: using the Curie-Weiss model and the Sherrington-Kirckpatrick model, as prototypes for simple and complex behaviors, respectively, we will show how to solve for their free energy by mapping this problem into a random-walk framework, so to use techniques originally meant for the latter. Finally, we present two examples of statistical-mechanics models where the topics described above come into play. Both examples are inspired by quantitive sociology applications.

The study of the distributions of extrema of a large collection of random variables dates back to the early 20th century and is well established in the case of independent or weakly correlated variables. Until recently, few sharp results were known in the case where the random variables are strongly correlated. In the last few years, there have been conceptual progress in describing the distribution of extrema of the log-correlated Gaussian fields. This class of fields includes important examples such as branching Brownian motion and 2D the Gaussian free field. In this series of lectures, we will study the statistics of extrema of the log-correlated Gaussian fields. The focus will be on explaining the guiding principles behind the results. We will also discuss why these techniques are expected to be applicable to a variety of problems such as the maxima of characteristic polynomials of random matrices and more, boldly, the maxima of the Riemann Zeta function.

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: relevant observables, polarization and global order, radial correlation function, spatial distribution of the neighbours, topological vs metric interaction, cognitive vs sensory bottlenecks, the problem of the border. Correlation: interaction vs correlation, relevance of behavioural fluctuations, velocity correlation function, what is the correlation length, scaling relations, when the group is more than the sum of its parts, scale-free correlations, orientation vs speed correlations, spontaneous symmetry breaking, statistical inference, basic relations in probability, general Bayesian framework, what does it mean to fit a model, the problem of the prior, model selection and the Occam razor, why you should keep your model simple, maximum entropy method for living groups, the minimal model compatible with the data, how to cope with motion - spins vs birds, spin wave approximation, maximum entropy for orientation, maximum entropy for speed, near a critical point?

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

When the Ising model, initially at infinite temperature, is suddenly cooled to zero temperature, a rich coarsening dynamics occurs that exhibits surprising features. In two dimensions, the ground state is reached only about 2/3 of the time, and the evolution is characterized by two distinct time scales, the longer of which arises from topological defects. There is also a deep connection between domain topologies and continuum percolation. In three dimensions, the ground state is never reached. Instead domains are topologically complex and contain a small fraction of "blinker" spins that can flip perpetually with no energy cost. Moreover, the relaxation time grows exponentially with system size. Insights gained from the coarsening kinetics of spin systems will then be applied to social dynamics. I will first discuss the voter model, a paradigmatic description of consensus formation in a population of interacting agents. Each voter can be in one of two opinion states and continuously updates its opinion at a rate proportional to the fraction of neighbors of the opposite opinion. Exact results for the voter model on regular lattices will be reviewed. I'll then discuss extensions of the voter model that attempt to incorporate elements of reality, while remaining within the domain of analytically tractable. These will include: (i) the voter model on complex graphs, where consensus is generally achieved quickly and via an interesting route, (ii) the voter model with more than two states, where stasis can arise, (iii) the bounded compromise model, in which two agents average their real-valued opinions if the difference is less than a threshold and do not evolve otherwise, and (iv) the Axelrod model, in which agents possess a multi-dimensional opinion variable and two agents interact only if they share at least one voting trait.