## Introductory School

Invited lecturers:

*E. Agliari**“Random walks: theory, techniques and applications”*[abstract]
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.

*L.P. Arguin**“Extrema of log-correlated random-variables: principles and examples”*[abstract]
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.

*A. Cavagna**“Collective behaviour in biological systems”*[abstract]
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?

*I. Corwin**“Integrable probability”*[abstract]
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.

*S. Redner**“Applications of Statistical Physics to Coarsening and the Dynamics of Social Systems”*[abstract]
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.