# P.C. Mahalanobis Lectures 2015-16

## Professor Frank den Hollander

Mathematical Institute
Universiteit Leiden
The Netherlands
http://www.math.leidenuniv.nl/~denholla

will deliver the P.C. Mahalanobis Lectures 2015-16 at

## P.C. Mahalanobis Lecture

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

## At Indian Statistical Institute, Delhi Centre    [Bangalore][Kolkata][Top]

January 6th
(PCM Lecture)
3:30pm - 4:30pm
Title: Metastability for interacting particle systems.
Venue: Conference Hall

Slides of the talk.

January 7th
3:30pm - 4:30pm
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 very different.

Based on joint work with Michiel van den Berg (Bristol), Erwin Bolthausen (Zurich) and Jesse Goodman (Auckland).

Slides of the talk.

## At Indian Statistical Institute, Bangalore Centre    [Delhi][Kolkata][Top]

January 11th
(PCM Lecture)
2:00pm - 3:00pm
Title: Metastability for interacting particle systems.
Venue: Second Floor Auditorium, Main Building

Slides of the talk.

January 11th
3:15pm - 4:15pm
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).

Slides of the talk.

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.

This mini-course consists of two parts

January 14th
14:00-15:00,
15:15-16:15

PART I: A quick overview of the basic theory:

(1) Large Deviation Principle (LDP), Varadhan's lemma, Bryc's lemma, Contraction Principle, Dawson-Gartner projective limit LDP, Gartner-Ellis theorem.

(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.

January 15th
14:30-15:30,
15:45-16:45

PART II: An application to drawing random words from random letter sequences:

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 sequence.

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.

Slides of the talk.

## At Indian Statistical Institute, Kolkata    [Bangalore][Delhi][Top]

January 18th
(PCM Lecture)
4:15pm-5:30pm
Title: Metastability for interacting particle systems.
Venue: Platinum Jubilee Auditorium.

Slides of the talk.

January 19th 4:15pm-5:15pm 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).

Slides of the talk.