*Moderator: *Dmitry Ioffe (Haifa)

Pietro Caputo (Roma),
Frank den Hollander (Leiden),
Dmitry Ioffe (Haifa),
François Simenhaus (Paris),
Pierre Tarrès (Paris),
Bálint Tóth (Bristol & Budapest),
Atilla Yılmaz (İstanbul).

We consider the (2+1)-dimensional solid-on-solid (SOS) interface pinned at the boundary of a lattice box in the low temperature phase. In the presence of a positivity constraint (hard wall), the interface is pushed away from the wall at a height H=H(L) proportional to log(L), where L is the side of the box. We give sharp estimates which quantify this entropic repulsion phenomenon, and show that the ensemble of level lines of heights H, H-1, ... has a macroscopic limit as L diverges. The scaling limit is given by nested loops formed via translates of Wulff shapes. Moreover we show that for the unconstrained model, the probability of satisfying the positivity constraint in the box scales exponentially with asymptotic rate c L log(L), where the constant c coincides with the so-called step free energy of the model. This is based on a series of joint works with E.Lubetzky, F.Martinelli, A.Sly and F.L.Toninelli.

**Frank den Hollander** (Leiden):
*Annealed Scaling for a Charged Polymer.*
pdf

We study an undirected polymer chain living on the d-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 analyze the annealed free energy per monomer in the
limit as the length of the polymer chain tends to infinity.
We show that there is a critical curve in the plane of charge bias versus
inverse temperature, separating an extended phase from a collapsed
phase. We 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
(joint work with Q. Berger and J. Poisat).

Our proof is built on an analysis of the downward large deviations of
the self-intersection local time of weakly self-avoiding simple random
walk. A comparison will be made with earlier more detailed results in
one dimension (joint work with F. Caravenna, N. Petrelis and J. Poisat),
and conjectures will be formulated for the behaviour of the polymer chain
in the two phases (work in progress with D. Ioffe).

**Dmitry Ioffe** (Haifa):
*Scaling limits for ordered walks under area tilts.*

I shall discuss scaling limits for a class of ordered random walks subject to positivity constraint and self-potentials, which look like generalized area tilts. Such polymers arise, for instance, as effective models for: (a) Phase segregation lines in 2D Ising model with negative b.c. and positive magnetic fields. (b) Level sets of 2+1 discrete SOS models coupled with Bernoulli bulk fields. The limiting objects happen to be ergodic Ferrari-Spohn diffusions, conditioned on non-intersection. Invariant measures for n such diffusions are given in terms of Slater determinants constructed from n first eigenfunctions of appropriate Sturm-Liouville operators. Based on joint works with S. Shlosman, Y.Velenik and V.Wachtel.

**François Simenhaus** (Paris):
*Random walk driven by the simple exclusion process.*

We consider the one-dimensional simple exclusion process (SEP) with jump parameter gamma>0. We then consider a random walk on Z with transitions given by this SEP : when the walker sits on a particle it jumps to the right with probability alpha and to the left with probability 1-alpha ; if it sits on an empty site then it uses transitions beta and 1-beta instead. First, we establish that if the asymptotic velocity of the walker is non-zero in the limiting case gamma = infty, where the environment gets fully refreshed between each step of the walker, then, for gamma large enough, the walker still has a non-zero asymptotic velocity in the same direction. Second, we establish that if the walker is transient in the limiting case gamma = 0, then, for gamma small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience. There exists some choice of parameters so that these two limiting velocities are of opposite sign. In both cases, we prove an annealed invariance principle to show that the fluctuations are normal. This is a joint work with F. Huveneers.

**Pierre Tarrès** (Paris):
*Edge reinforced random walk and statistical physics.*

**Bálint Tóth**
(Bristol &
Budapest):
*Super-diffusivity of the periodic Lorentz-gas in the Boltzmann-Grad limit.
*

We prove central limit theorem and invariance principle under superdiffusive scaling $\sqrt{t \log t}$ for the displacement of particle in the $Z^d$-based periodic Lorentz gas, in the Boltzmann-Grad limit. The result holds in any dimension. This is joint work with Jens Marklof (Bristol).

**Atilla Yılmaz** (İstanbul):
*Variational formulas and disorder regimes of random walks in random
potentials.*
pdf

I will start by providing three variational formulas for the quenched free energy of a random walk in random potential (RWRP) when the underlying walk is directed or undirected, the environment is stationary & ergodic, and the potential is allowed to depend on the next step of the walk which covers RWRE. Next, in the directed i.i.d. case, I will give two variational formulas for the annealed free energy of RWRP. These five formulas are the same except that they involve infima over different sets, and I will say a few words about how they are derived. Then, I will present connections between the existence & uniqueness of the minimizers of these variational formulas and the weak & strong disorder regimes of RWRP. I will end with a conjecture regarding the fine asymptotics of the quenched free energy under very strong disorder. Based on joint work with Firas Rassoul-Agha and Timo Seppalainen.