This workshop gathers mathematicians around
two mini-courses (i) Julien Poisat on "charged polymer",
and (ii) Serguei Popov on "Coupling and decoupling with
soft local times".
PARTICIPANTS (*=to be confirmed)
Bruno SCHAPIRA, and Julien SOHIER.
Charge Polymer, Julien Poisat, Paris-Dauphine
I will present recent results on the charged polymer model, which is a discrete statistical mechanics model of a polymer chain (a random walk) interacting with itself via random charges distributed along the chain. The focus will be mainly on the one-dimensional annealed model, for which a fairly complete description of the phase diagram and phase transitions is now available, but I will also mention extensions/open questions for the quenched model as well as for higher dimensions. The mini-course will be based on joint work with Caravenna, den Hollander, Pétrélis (http://arxiv.org/pdf/1509.02204.pdf, Math. Phys. Anal. Geom. 2016) and work in progress with Berger and den Hollander. A temptative plan is:
(I) The charged polymer model, results and related models.
(II) Local times and spectral representation of the free energy
(III) Weak interaction limits
Coupling and decoupling with soft local times, by Serguei Popov, University of Campinas, Brazil.
This mini-course is an introduction to the method of constructing couplings of stochastic processes using an auxiliary Poisson process, also called "soft local time method" [Popov-Teixeira, JEMS-2015]. We plan also to discuss its applications to random walks and random interlacements.
Monday and Tuesday 9:30-12:30 2 courses, and 14:00 to 17:00 three talks.
Wednesday 9:30-12:30 2 courses, and we end after lunch.
Monday 6th of June 2016:
9:15-9:30 Welcome, by A.Asselah
9:30-10:50 S.Popov, I
11:10-12:30 J.Poisat, I
14:00-17:00, Three 45mn Talks: B.Schapira, A.Sapozhnikov,Q.Berger
Tuesday 7th of June 2016:
9:30-10:50 J.Poisat, II
11:10-12:30 S.Popov, II
14:00-17:00, Three 45mn Talks: F.Comets, J.Cerny, J.Sohier.
Wednesday 8th of June 2016:
9:30-10:50 S.Popov, III
11:10-12:30 J.Poisat, III
Quentin Berger, Paris
High-temperature behavior of the directed polymer model
We will introduce the directed polymer model, and discuss recent progress on the high-temperature behavior of the model.
We will in particular focus on the free energy, which is related to localization properties of the polymer.
Bruno Schapira, Marseille
Boundary of the range of a transient random walk
We will review recent results on the typical (and atypical) behaviour of the boundary (and capacity) of the range of a transient simple random walk on the Euclidean lattice. Based on joint works with Amine Asselah and Perla Sousi.
Francis Comets, Paris
Localization for directed polymers in random medium
Directed polymers in random environment are known to localize when the disorder is strong. After a review
of polymer models, we will analyse this phenomenon in a precise manner for an exactly solvable model:
the nearest neighbor polymer in one space dimension in log-gamma distributed environment with boundary conditions, introduced by Timo Seppäläinen. In the equilibrium case, as the length increases, the end point of the polymer converges to a distribution proportional to the exponent of a zero-mean random walk.
Joint work with Vu-Lan Nguyen.
Artem Sapozhnikov, Leipzig
Large-scale invariance in percolation models (with strong correlations)
I will discuss recent progress in understanding supercritical percolation models on lattices, particularly in the presence of strong spatial correlations. This includes quenched Gaussian heat kernel bounds, Harnack inequalities, and local CLT for the random walk on infinite percolation clusters. The results apply to the random interlacements at all levels, the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of local uniqueness.
Jiri Cerny, Vienna
The maximum particle of branching random
walks in spatially random branching environment
Branching random walks and Brownian motion have
been the subject of intensive research during the last decades.
We consider branching random walks and investigate the effect
of introducing a spatially random branching environment. We are primarily
interested in the position of the maximum particle, for which we prove a CLT.
Our result correspond, on an analytic level, to a CLT for the front of
the solutions to a randomized Fisher-KPP equation, and also to a CLT for the
parabolic Anderson model.
Julien Sohier, Paris
Convergence to equilibrium for a directed (1+d)-dimensional polymer
We consider a flip dynamics for directed (1+d)-dimensional lattice paths
with length L. The model can be interpreted as a higher dimensional version of the
simple exclusion process, the latter corresponding to the case d=1.
We prove that the
mixing time of the associated Markov chain scales like LxLxlog(L) up to a d-dependent
multiplicative constant. The key step in the proof of the upper bound is to show that
the system satisfies a logarithmic Sobolev inequality on the diffusive scale LxL for every
fixed d, which we achieve by a suitable induction over the dimension together with an
estimate for adjacent transpositions.
The lower bound is obtained with a version of
Wilson's argument for the one-dimensional case.
Joint work with P. Caputo.