Summary of the talks

Nina Holden: Natural measures on random fractals
Several fractals associated with the scaling limit of (interacting) random walks and other statistical physics models come equipped with a natural measure. This measure can often be defined equivalently in multiple ways: axiomatically, via Minkowski content, or as the limit of counting measure for the discrete model. We study such measures for the Schramm-Loewner evolution, percolation pivotal points, and various fractals in the geometry of Liouville quantum gravity.
Based on joint works with Bernardi, Lawler, Li, and Sun.

Alex Drewitz: Geometry of Gaussian free field sign clusters and random interlacements
We consider two fundamental percolation models with long-range correlations: Level set percolation for the Gaussian free field (GFF) and percolation of the vacant set of random interlacements. Both models have been the subject of intensive research during the last decades. In this talk we focus on structural properties of the level set percolation of the GFF. In particular, we establish the non-triviality of a phase in which two infinite sign clusters dominate, and their complement only has small connected components. While the respective results are new for Z^d as the underlying graph, we also cover more intricate geometries such as transient graphs with subdiffusive random walk behavior. As a consequence, we answer an open problem on the non-triviality of the phase transition of the vacant set of Random Interlacements on such geometries.
Based on joint works with A. Prévost (Universität zu Köln) and P.-F. Rodriguez (IHES).

Hubert Lacoin: The renormalization of the log-gas via its Sine-Gordon representation (in dimension one)
Consider a model gas of electrically charged particles in D a bounded subset of R^d. The interaction given by λ_iλ_jV(|x_i-x_j|) where λ_i,λ_j∈{-1,1} stands for the particle charge and x_i,x_j∈D for the particle position and V represents the electrical potential (which is a function of the distance). Given α and β two positive parameters, our model is described by the following partition function which induces a probability distribution on the set of charged particles
We are interested specifically in the case when V diverges logarithmically in zero V(r)=-log(r)+O(1). In that case, we observe that the above partition function is finite if and only if β≥d. The aim of this talk is to introduce renormalization techniques using connection with Gaussian fields which allows for a definition of the Log-gas when β∈[d,2d). (joint work with V.Vargas and R.Rhodes).

Yvan Vélénik: Ising/Potts interface above a wall
I'll review two recent results about entropic repulsion of an interface in 2d Ising/Potts models. In the first one (joint work with D. Ioffe, S. Ott and V. Wachtel), we consider a 2d Potts model below its critical temperature, with a boundary condition inducing the presence of an interface along the bottom wall. We prove that, under diffusive scaling, this interface converges to A Brownian excursion. In the second one (joint work with D. Ioffe, S. Ott and S. Shlosman), we consider the same setting, but restrict our attention to the Ising model and investigate the effect of a magnetic field penalizing the "phase" located below the interface. We prove that, in a suitable scaling limit in which the strength of the magnetic field tends to zero as the system's size diverges, the interface converges to a Ferrari-Spohn diffusion.

Aran Raoufi: Exponential decay of truncated correlations for the Ising model for any non-critical temperature
We prove the truncated two-point function of the ferromagnetic Ising model on Z^d (d≥3) decay exponentially fast throughout the ordered regime (T<T_c and h=0). Together with the previously known results, this implies that the exponential clustering property holds throughout the model's phase diagram except for the critical point. The proof utilizes the two different geometric representations of the Ising model, namely the FK-Ising and the random-current representation and the relationships between them.

Clément Cosco: Gaussian Fluctuations and Rate of Convergence of the Kardar-Parisi-Zhang equation in Weak Disorder in d ≥ 3
Trying to give a proper definition of the KPZ (Kardar-Parisi-Zhang) equation in dimension d ≥ 3 is a challenging question. A plan to do so, is to first consider the well-defined KPZ equation when the white noise is smoothed in space. For d ≥ 3 and small noise intensity, the solution is known to converge to some random variable as the smoothing is removed. It is also known that the limiting random variable can be related to the free energy of a Brownian polymer, in a smoothed white noise environment. In this talk, we will state some recent results about the fluctuations of the convergence of the solution. In particular, we will show that the fluctuation of the solution, around the rescaled free energy of the polymer, converges pointwise towards a Gaussian limit.

Vincent Tassion: Renormalization in supercritical percolation