Title: Metastability for interacting particle systems.
Abstract: Metastability is the phenomenon where a system, under the
influence of a stochastic dynamics, moves between different subregions
of its state space on different time scales. Metastability is
encountered in a wide variety of stochastic systems. The challenge is
to devise realistic models and to explain the experimentally observed
universality that is displayed by metastable systems, both
qualitatively and quantitatively.
In statistical physics, metastability is the dynamical manifestation
of a first-order phase transition. In this talk I give a brief
historical account of metastability in this context. After that I
describe the metastable behaviour of one particular model, namely, the
Widom-Rowlinson model on a two-dimensional torus subject to a
Metropolis stochastic dynamics. In this model, particles are randomly
created and annihilated inside the torus as if the outside of the
torus were an infinite reservoir with a given chemical potential. The
particles are viewed as points carrying disks, and the energy of a
particle configuration is equal to the volume of the union of the
disks, called the "halo" of the configuration. Consequently, the
interaction between the particles is attractive.
We are interested in the metastable behaviour at low temperature when
the chemical potential is supercritical. In particular, we start with
the empty torus and are interested in the first time when we reach the
full torus, i.e., the torus is fully covered by disks. In order to
achieve the transition from empty to full, the system needs to create
a sufficiently large droplet of overlapping disks, which plays the
role of a "critical droplet'' that triggers the crossover. In the
limit as the temperature tends to zero, we compute the asymptotic
scaling of the average crossover time, show that the crossover time
divided by its average is exponentially distributed, and identify the
size and the shape of the critical droplet. It turns out that the
critical droplet exhibits "surface fluctuations", which need to be
understood in order to obtain a fine estimate of the crossover time
Title: How Porous is a Brownian motion ? Venue: Seminar Room 2
Abstract: The path of a Brownian motion on a $d$-dimensional torus run up to time $t$ is a random compact subset of the torus. In this
talk we look at the geometric and spectral properties of the
complement $C(t)$ of this set when $t$ tends to infinity.
Questions we address are the following:
1. What is the linear size of the largest region in $C(t)$?
2. What does $C(t)$ look like around this region?
3. Does $C(t)$ have some sort of "component-structure"?
4. What are the largest capacity, largest volume and smallest
principal Dirichlet eigenvalue of the "components" of $C(t)$
We discuss both $d \geq 3$ and $d=2$, which turn out to be
Based on joint work with Michiel van den Berg (Bristol), Erwin
Bolthausen (Zurich) and Jesse Goodman (Auckland).
Title: How Does a Charged Polymer Collapse ? Venue: Second Floor Auditorium, Main Building
Abstract: In this talk we consider an undirected polymer chain living
on the one-dimensional integer lattice and carrying i.i.d. random
charges. Each self-intersection of the polymer chain contributes an
energy to the "interaction Hamiltonian" that is equal to the product
of the charges of the two monomers that meet. The joint probability
distribution for the polymer chain and the charges is given by the
"Gibbs distribution" associated with the interaction Hamiltonian. We
analyse the "free energy" per monomer in the limit as the length of
the polymer chain tends to infinity.
We derive a spectral representation for the free energy and use
this to show that there is a "critical curve" in the (charge bias, inverse
temperature)-plane separating a ballistic phase from a subballistic
phase. We show that the phase transition is first order, identify the
scaling behaviour of the critical curve for small and for large charge
bias, and also identify the scaling behaviour of the free energy for
small charge bias and small inverse temperature. In addition, we
prove a "large deviation principle" for the joint law of the empirical
speed and the empirical charge, and derive a spectral representation
for the associated "rate function". This in turn leads to a law of large
numbers and a central limit theorem.
Based on joint work with F. Caravenna (Milano), N. Petrelis (Nantes)
and J. Poisat (Paris).
Title : Large Deviations (mini-course) Venue: Second Floor Auditorium, Main Building
Large deviation theory describes how stochastic processes may deviate
substantially from their typical behaviour. Such large deviations are
always done in "the least unlikely of all the unlikely ways". This fact
serves as a guiding principle that allows for a precise characterisation
of large deviations in many concrete examples.
(2) Cramer's theorem and Sanov's theorem for i.i.d. random variables,
their extension to an LDP for the empirical process involving relative
entropy. Extension to Markov processes.
PART II: An application to drawing random words from random letter
When we cut an i.i.d. sequence of letters into words according to
an independent renewal process, we obtain an i.i.d. sequence of
words. In the annealed LDP for the empirical process of words,
the rate function is the specific relative entropy of the observed
law of words w.r.t. the reference law of words. In this lecture, we
consider the quenched LDP, i.e., we condition on a typical letter
The rate function of the quenched LDP turns out to be a sum of
two terms, one being the annealed rate function, the other being
proportional to the specific relative entropy of the observed law
of letters, obtained after a "randomised concatenation'' of words,
w.r.t. the reference law of letters. The proportionality constant turns
out to be equal to the tail exponent of the renewal process.
Title: Breaking of Ensemble Equivalence in Complex Networks Venue: $L_\infty$ lecture hall, 5th Floor, Kolmogorov Bhavan.
Abstract: It is generally believed that for physical systems in the thermodynamic limit,
the microcanonical description as a function of energy coincides with the
canonical description as a function of temperature. However, various examples
of systems for which the microcanonical and canonical ensembles are not
equivalent have been identified. A complete theory of this intriguing phenomenon is still missing.
In this talk we show that ensemble nonequivalence can manifest itself also
in random graphs with topological constraints. We find that, while graphs
with a given number of links are ensemble-equivalent, graphs with a given
degree sequence are not. This result holds irrespective of whether the
"energy" associated with the constraint is nonadditive (as in unipartite
graphs) or additive (as in bipartite graphs). In contrast with previous
expectations, our results show that: (1) physically, nonequivalence can
be induced by an extensive number of local constraints, and not necessarily
by long-range interactions or nonadditivity; (2) mathematically, nonquivalence
is determined by a different large-deviation behaviour of microcanonical and
canonical probabilities for a single microstate, and not necessarily for almost
all microstates. The latter criterion, which is entirely local, is not restricted to networks and holds in general.
Based on joint work with Diego Garlaschelli (Leiden), Joey de Mol (Leiden),
Tiziano Squartini (Rome) and Andrea Roccaverde (Leiden).