Workshop "Self-interacting random walks, Polymers and Folding"

September 9-13, 2019. CIRM, Marseille.

Organizers: Quentin Berger (quentin.berger@sorbonne-universite.fr), Francesco Caravenna (francesco.caravenna@unimib.it), Julien Poisat (poisat@ceremade.dauphine.fr).

Scientific Committee: Amine Asselah (Université de Créteil), Hubert Lacoin (IMPA, Rio de Janeiro).

The focus of this workshop will be on self-interacting random walks/polymer models, with an emphasis on those models which exhibit a folding or unfolding transition. The goal of the workshop is to bring together researchers working on this topic to share ideas and techniques, around three mini-courses and several talks.

For the schedule, we refer to this page.

Mini-courses:

Roland Bauershmidt (Cambridge): Tricritical polymer collapse
Description: I will give a pedagogical introduction to the use of supersymmetry to study interacting random walks. Concretely, I will derive the phase diagram of a mean-field model with tricritical behaviour using this approach. I will then indicate in which way the general approach has been used, in much more sophisticated form, for the study of lattice models in 3 and 4 dimensions.

Ryoki Fukushima (Kyoto): Random Walks among Bernoulli obstacles
Description: Random walk in random potential has been studied as a stochastic process related to the Anderson Hamiltonian since 1970's. The aim of this course is to explain some recent progress in the case of hard core potentials. In this case, the annealed law of the random walk can be regarded as a model of a polymer with self-attractive interaction. In 1990's, it was proved that the polymer is localized (or folded) in a ball at all temperature. But the structure inside that ball remained unclear in dimensions three and higher. Recently, it has been proved that the ball is filled by the polymer, which implies that the polymer macroscopically looks like a solid ball. In addition, a partial result on the boundary fluctuation has been obtained. I will explain the basic ideas of the proofs of the localization, ball covering, and boundary fluctuation. Main part of this course will be based on a joint work with Jian Ding, Rongfeng Sun, and Changji Xu.

Nicolas Pétrélis (Nantes): Interacting Partially Directed Self-Avoiding Walk
Description: We will describe and study the Interacting Partially-Directed Self-Avoiding Walk (IPDSAW), a model initially introduced in 1968 by Zwanzig and Lauritzen to investigate the collapse transition of an homopolymer dipped in a poor solvent. In size L, the allowed configurations of the model are given by the L-step trajectories of a self-avoiding walk on Z2 that takes unitary steps up, down and to the right. The coupling parameter is β≥0 and the Hamiltonian associated with every configuration is β times the number of "self-touching" of that configuration. This model was known to undergo a collapse transition at some critical parameter βc>0 (see e.g. [Brak, Guttmann & Whittington 92]) between an extended phase {β<βc} and a collapsed phase {β≥βc}.
We will begin by displaying a new probabilistic representation of the partition function, based on an auxiliary random walk, which relates some geometric features of IPDSAW trajectories to that of a particular random walk conditioned on enclosing a prescribed geometric area. This method allowed us to push the mathematical understanding of IPDSAW some steps further in the last years. We will illustrate this new family of results by providing a sharp geometric description of a typical configuration of IPDSAW both inside the collapse phase {β>βc}, the extended phase {β<βc} and also at criticality {β=βc}.
Finally, we will consider the 2-dimensional Interacting Prudent Walk (IPRW), an extension of the IPDSAW that is not directed. Using some particular decomposition of prudent paths into partially directed paths, we will prove that the IPRW undergoes a collapse transition as well.

Talks:

Erik Bates (UC Berkeley)
Path localization of polymers in Gaussian disorder

Nathanaël Berestycki (Vienna)
Localization for random walk in dimension d≥3

Alessandra Cipriani (Delft)
Scaling limit of semiflexible polymers: a phase transition

Denis Denisov (Manchester)
Random walks in cones and Lipschitz domains

Pablo Ferrari (Buenos Aires)
Gaussian random permutation and the boson point process

Alan Hammond (UC Berkeley)
Polymer models, area constraints, and the collapse transition

Dima Ioffe (Technion Haifa)
Critical prewetting and 1-2-3 diffusive scaling of interfaces in the 2D Ising model

Rongfeng Sun (Singapore)
The 2-dimensional KPZ equation in the entire subcritical regime

Niccolò Torri (Paris Créteil)
Directed polymer in a heavy-tail random environment

Other participants:

Amine Asselah (Paris Créteil)
Quentin Berger (Paris Sorbonne)
Matthias Birkner (Mainz)
Francesco Caravenna (Milan Bicocca)
Fabienne Castell (Marseille)
Francis Comets (Paris Diderot)
Dirk Erhard (Salvador)
Giambattista Giacomin (Paris Diderot)
Tyler Helmut (Bristol)
Yueyun Hu (Paris Villetaneuse)
Chien-Hao Huang (Taiwan)
Hubert Lacoin (IMPA, Rio)
Alexandre Legrand (Nantes)
Makoto Nakashima (Nagoya)
Sébastien Ott (Genève)
Julien Poisat (Paris Dauphine)
Matteo Quattropani (Roma Tre)
Bruno Schapira (Marseille)
Nadia Sidorova (London)
François Simenhaus (Paris Dauphine)
Julien Sohier (Paris Créteil)
Lorenzo Taggi (Weierstraß Institut, Berlin)
Nikos Zygouras (Warwick)

The workshop is part of a series of conferences organized by the ANR project SWiWS.

Sponsors: