## Courses

Calendar
Invited lecturers:

*F. Camia*“Brownian Loops and Conformal Fields”, [abstract]
The main topic of these lectures is the continuum scaling limit
of planar lattice models. One reason why this topic occupies an important
place in the theory of probability and mathematical statistical physics is
that scaling limits provide the link between statistical mechanics and
(Euclidean) field theory. In order to explain the main ideas behind the
concept of scaling limit, I will focus on a "toy" model that exhibits the
typical behavior of statistical mechanical models at and near the critical
point. This "toy" model, known as the random walk loop soup, turns out to
be interesting in its own right. It can be described as a Poissonian
ensemble of lattice loops, or a lattice gas of loops since it fits within
the ideal-gas framework of statistical mechanics. After introducing the
model and discussing some interesting connections with the discrete
Gaussian free field, I will present some results concerning its scaling
limit, which leads to a Poissonian ensemble of continuum loops known as the
Brownian loop soup. The latter was introduced by Lawler and Werner, and is
a very interesting object with connections to the Schramm-Loewner Evolution
and various models of statistical mechanics. In the second part of the
lectures, I will use the Brownian loop soup to construct a family of
functions that behave like correlation functions of a conformal field. I
will then use these functions and their derivation to introduce the concept
of conformal field and to explore the connection between scaling limits and
conformal fields.

*A. De Masi*“Stochastic particle systems with confining forces”, [abstract]
I will consider stochastic evolutions of particles that are confined in a
bounded region by means of different mechanisms at the boundary. The
motivation comes from one of the basic questions in statistical mechanics
concerning the structure of states which are stationary but not in thermal
equilibrium. These states are characterized by the presence of a steady
current flowing through the system and the Fouries (or Fick) law is expected
to hold. I will present different ways to impose a current and then I will
discuss and analyze the different effects so produced both at the microscopic
and macroscopic level. Rather surprisingly, these studies have generated
many interesting problems both mathematically and conceptually. There
are many papers on this subject and I will focus on some of them.

*H. Duminil-Copin*“Geometric representations of low-dimensional lattice spin systems”, [abstract]
In this course, we will review recent results in the theory of geometric
representations of low-dimensional spin systems such as Potts, spin O(n)
and Ising models. We will introduce random-cluster models, loop O(n)-models
and random current representations and will explain how these models can be
used to study the spin models at criticality and determine in particular
whether their phase transition is continuous or not.

*F. Guerra*“Equilibrium and off equilibrium properties of ferromagnetic and disordered statistical mechanics systems”, [abstract]
A self-contained review will be given about the equilibrium and off equilibrium properties of statistical mechanics systems, both in the case of ferromagnetic and disordered ones.
A systematic use of interpolation arguments and convexity will be the main tool.
The case of mean field will be treated throughly. The possible extensions to short range interaction will be pointed out.

*G. Jona Lasinio*“Macroscopic Fluctuation Theory”, [abstract]
Stationary non-equilibrium states describe steady flows through macroscopic systems. Although they represent the simplest generalization of equilibrium states, they exhibit a variety of new phenomena. Within a statistical mechanics approach, these states have been the subject of several theoretical investigations, both analytic and numerical. The macroscopic fluctuation theory, based on a formula for the probability of joint space-time fluctuations of thermodynamic variables and currents, provides a unified macroscopic treatment of such states for driven diffusive systems. The course will give a review of this theory including its main predictions and some relevant applications.

*A. Kupiainen*, “Quantum Field Theory for Probabilists”, [abstract]
The course consists of two parts. In the first one we give an
introduction to the Renormalization Group as a method to study quantum
field theory and
statistical mechanics models at critical temperature. In the second part
we apply these ideas to proving existence and uniqueness of solutions of
stochastic PDE's driven by space time white noise. Examples are the KPZ
and \phi^4_3 models.

*C. Newman*, “Riemann Hypothesis and Statistical Mechanics”, [abstract]
In this minicourse we discuss some old results concerning the location of zeros of partition functions (or moment generating functions) in certain statistical mechanics models and their possible connections to the Riemann Hypothesis (RH). A standard reformulation of the RH is as follows. The (two-sided) Laplace transform of a certain specific function \Psi on the real line is automatically an entire function on the complex plane; the RH is equivalent to this transform having only pure imaginary zeros. Also \Psi is a positive integrable function, so (modulo a multiplicative constant C) is a probability density function. A (finite) Ising model (with pair ferromagnetic interactions) is a specific type of probability measure P on the points S=(S_1,...,S_N) with each S_j = +1 or -1. The Lee-Yang theorem implies that that for non-negative a_1, ..., a_N, the Laplace transform of the induced probability distribution of a_1 S_1 + ... + a_N S_N has only pure imaginary zeros. The big question here is whether it's possible to find a sequence of Ising models so that the limit as N tends to \infty of such distributions has density exactly C \Psi. The course will focus on questions of this sort. Here are some background references: C. Newman, Z. f. Wahrschein. (Prob. Th. Re. Fields) 33 (1975) 25-93 (see, esp. p. 90) C. Newman, Proc. Amer. Math. Soc. 61 (1976) 245-251 C. Newman, Constr. Approx. 7 (1991) 389-399 A. Odlyzko, Num. Algorithms 25 (2000) 293-303

*E. Presutti*“Phase transitions in systems with spatially non homogeneous interactions”, [abstract]
I will describe recent results and works in progress on the
absence/presence of phase transitions in systems with spatially non
homogeneous interactions. In a first part I will consider the d ≥ 2 nearest
neighbor ferromagnetic Ising model under the action of a non negative,
space dependent magnetic field. It will be shown that at low temperatures
the occurrence of a phase transition depends critically on the rate at
which the magnetic field vanishes at infinity. In a second part I will
consider two dimensional systems with ”very small, short range vertical
interactions” while the horizontal interaction is described by a two body
Kac potential. I will first study the Ising case and then extend the
analysis to a continuous system where on each horizontal line there is a
system of hard rods with attractive Kac pair interaction.The goal is to
prove that such a system has the phase transition predicted by van
der Waals, which at this moment is still under study.

*C. Rovelli*“Statistical mechanics of gravity and thermodynamical origin of time”, [abstract]
Thermodynamics and statistical mechanics are universal, and we expect
them to apply also to gravitational phenomena. But our understanding
of gravity is given by general relativity, where temporal evolution is
given in a form incompatible with standard thermodynamics and
statistical mechanics. All principles of thermodynamics become wrong
or meaningless in gravity: temperature fails to be constant at
equilibrium (Tolman effect), energy is ill-defined, and it is not
clear which among the many notions of time, if any, may underpin the
second law.
I discuss partial successes and open problems in the effort to extend
the foundations of thermodynamics and statistical mechanics to a wider
context encompassing relativistic gravity.
Among the key ideas aiming to do so, is the idea that the physical
flow of time of the second law is determined by, rather than
determines, the statistical state of a covariant system. In the
quantum field theoretical context, it is given by the Tomita flow
generated by a state on the observables' algebra, and the origin of
time's flow is related, via a theorem by Alain Connes, to the
underlying quantum non-commutativity of the observable algebra.

*T. Sasamoto*“The one-dimensional KPZ equation: its height distribution and algebraic structures”, [abstract]
In 1986, Kardar, Parisi and Zhang introduced a non-linear stochastic PDE to describe dynamics of interface motion. This KPZ equation has been
studied intensively and extensively since then, but recently its one-dimensional version has been attracting particular attention because of its tractability and connections to various areas of mathematics and physics.
In this lecture I will explain part of these developments . I will mainly focus on the height distribution for the KPZ equation. I will also explain the underlying algebraic structure such as random matrix theory, Schur process and Macdonald process in connection to related discrete interacting particle systems like ASEP and q-boson zero range process.

*H. Spohn*“Integrable stochastic models in the Kardar-Parisi-Zhang universality class”, [abstract]
The course will cover:

-- determinantal processes and the statistical mechanics of line ensembles,

-- growth models and directed polymers in a random medium,

-- the Kardar-Parisi-Zhang stochastic PDE, sharp wedge initial conditions,

-- duality for interacting Brownian motions and other models in the KPZ class,

-- multi-component KPZ equations.

-- determinantal processes and the statistical mechanics of line ensembles,

-- growth models and directed polymers in a random medium,

-- the Kardar-Parisi-Zhang stochastic PDE, sharp wedge initial conditions,

-- duality for interacting Brownian motions and other models in the KPZ class,

-- multi-component KPZ equations.

*D. Stein*“Short-range spin glasses: results and applications”, [abstract]
The aim of this lecture series is to introduce the subject
of spin glasses, and more generally the statistical mechanics of
quenched disorder, as a problem of general interest to physicists and
mathematicians from multiple disciplines and backgrounds. Despite
years of study, the physics and mathematics of quenched disorder
remains poorly understood, and represents a major gap in our
understanding of the condensed state of matter. While there are many
active areas of investigation in this field, I will narrow the focus
of this series to some aspects of our current level of understanding
of the low-temperature equilibrium structure of realistic (i.e.,
finite-dimensional) spin glasses.
I will begin with a brief review of the basic features of spin glasses
and what is known experimentally. I will then turn to the problem of
understanding the nature of the spin glass phase --- if it exists.
The central question to be addressed is the nature of broken symmetry
in these systems. Parisi's replica symmetry breaking approach, now
mostly verified for mean field spin glasses, attracted great
excitement and interest as a novel and exotic form of symmetry
breaking. But does it hold also for real spin glasses in finite
dimensions? This has been a subject of intense controversy, and
although the issues surrounding it have become more sharply defined in
recent years, it remains an open question. I will explore this
problem, introducing new mathematical constructs such as the metastate
along the way, as well as related questions such as the number of pure
states and free energy fluctuations. If time permits, we will conclude
with an examination of some of the applications of spin glass
mathematics to problems in computer science, biology, and other
fields.

lectures 1 and 2
lectures 3 and 4
references
*R. Sun*“Brownian web, Brownian net, and their universality”, [abstract]
Abstract: The Brownian web is the collection of one-dimensional
coalescing Brownian motions starting from every point in space-time.
Originally conceived by Arratia in the context of the one-dimensional
voter model and its dual coalescing random walks, the Brownian web has
since been shown to arise in the scaling limit of many one-dimensional
interacting particle systems with coalescent interaction, including
zero-temperature dynamics of Ising and Potts models, true
self-avoiding random walks, drainage networks, Hastings-Levitov planar
aggregation models, and super-critical oriented percolation.
The Brownian net is an extension of the Brownian web, which also
allows for branching of the Brownian motions. It has been shown to
arise in the scaling limit of many one-dimensional interacting
particle systems with branching-coalescing interactions, including the
voter model with selection, dynamics of Ising and Potts models with
boundary nucleation, and one-dimensional random walks in i.i.d.
space-time random environments.
The goal of the lecture series is to introduce the Brownian web and
the Brownian net, discuss some of their properties, and study how they
arise in the scaling limits of various models of interest.

*O Zeitouni*“Extrema of log-correlated Gaussian fields: from branching processes to the Gaussian free field” [abstract]
This mini-course will review the use of the second
moment method in proving limit laws for extremes of the two dimensional
GFF; parallels and differences with branching random walks will be
discussed.