School on Disordered Systems, Random Spatial Processes and Some Applications 
January 5 -9, 2015

Elena Agliari (Roma La Sapienza) : “Random walks: theory, techniques and applications”

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-trivial phenomenologies (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
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
theCurie-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.

Louis-Pierre Arguin (Université de Montréal) : “Extrema of log-correlated random-variables: principles and examples”

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 ofextrema 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.

Andrea Cavagna (Roma La Sapienza) : « Collective behaviour in biological systems »

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?

Iwan Corwin (MIT) : Integrable probability”

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.

Sydney Redner (Boston University) : “Applications of Statistical Physics to Coarsening and the Dynamics of Social Systems”

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 groundstate is never
reached. Instead domains are topologically complex and contain a small
fraction of « blinker » spins that can flip perpetuallywith 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.