**School on Disordered Systems, Random Spatial Processes and Some Applications**

**January 5 -9, 2015**

**ABSTRACTS**

**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-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”**

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 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 »**

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?

**Iwan Corwin (MIT) : I****ntegrable probability”**

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.

**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 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.