Universality and scaling limits of interacting random systems.

Tuesday 23 January 2018, IHP

Moderator: Milton Jara (Rio de Janeiro)

Francesco Caravenna (Milano), Dmitry Chelkak (Paris & St. Petersburg), Bertrand Duplantier (Saclay-Paris), Massimiliano Gubinelli (Bonn), Benoît Laslier (Paris), Otávio Menezes (Lisboa), Wioletta Ruszel (Delft).

Milton Jara (Rio de Janeiro): Non-equilibrium fluctuations of interacting particle systems.

Applying the formalism of hydrodynamic limit of interacting particle systems (see the book of Kipnis-Landim for a comprehensive review), the fluctuations of conserved quantities around their mean is well understood in the case of equilibrium systems. For systems out of equilibrium, up to recently the only general approach available was the analysis of correlation functions (as covered in the LNM of De Masi-Presutti), which is extremely demanding from a technical point of view, and only works for solvable models (like exclusion processes or systems of independent particles), or perturbations of these models. Starting from Gonçalves-J.'14, we have developed a general strategy to deal with this problem. I will discuss the application of this method to the derivation of a functional central limit theorem for three examples: one-dimensional models in the KPZ universality class, driven diffusive systems in dimensions up to three, and reaction-diffusion models. Depending on time, I will also discuss the relation of the corresponding limiting objects with the Gaussian Free Field.
Joint works with Patricia Gonçalves and Otávio Menezes.

Francesco Caravenna (Milano): Pinning model, universality and rough paths.

We consider a stochastic renewal equation, which describes the partition function of the pinning model in the weak-disorder scaling limit. Using ideas from rough paths, we present a robust analysis of this equation. This sheds light on the effect of disorder and leads naturally to universality results.

Dmitry Chelkak (Paris & St. Petersburg): S-embeddings of planar graphs carrying the Ising model.

During the last decade, a number of results on conformal invariance of the critical nearest-neighbor Ising model in 2D appeared, both for the scaling limits of correlation functions (fermions, energy densities, spins, etc) and interfaces (aka domain walls) or loop ensembles arising in the model. Embarrassingly enough, these results are not truly universal with respect to the choice of an underlying planar graph: the highest generality in which the machinery developed so far can be applied is the self-dual Ising model on isoradial graphs. The primary goal of this talk is to discuss a new (much broader) class of embeddings of generic weighted planar graphs into the complex plane, which naturally arises in the Ising model context and, in particular, could pave a way to the better understanding of the universality in the 2D Ising model.

Bertrand Duplantier (Saclay-Paris): Integral Means Spectrum of Whole-Plane SLE.

We will briefly review the notion of integral means spectrum for the Schramm-Loewner Evolution (SLE), and present a generalized definition relevant for whole-plane SLE, depending on the moments of both the SLE conformal map and its derivative. Multiple phase transition lines exist in the moment plane between SLE standard integral means spectra (bulk, tip, linear) and the average generalized spectrum. A universal generalized spectrum can also be defined, and its precise form partially proved or conjectured.
Joint works with Dmitry Belyaev (Oxford), Xuan Hieu Ho (Orléans), Le Tanh Binh (Quy Nhon) and Michel Zinsmeister (Orléans).

Massimiliano Gubinelli (Bonn): Weak universality of singular SPDE's.

I will discuss the problem of controlling the large scale limit of weakly non-linear SPDEs (of parabolic or hyperbolic type) in the regime where the non-linearity survives at the macroscopic scale giving rise to singular non-linear SPDEs.

Benoît Laslier (Paris): Universal and non-universal properties in the dimer model.

The dimer model is one of the few lattice models where any kind of universality with respect to change in the underlying lattice is proved. However, it can also display a very large sensibility to some microscopic details. I will review some old and new results to try to shed some light on the interplay of universal and non universal behaviours, with respect to change in the lattice and change in boundary conditions.

Otávio Menezes (Lisboa): Invariance principle for a slowed random walk driven by symmetric exclusion.

We establish an invariance principle for a random walk driven by the symmetric exclusion process in one dimension. The walk has a drift to the left (resp. right) when it sits on a particle (resp. hole). The environment starts from equilibrium and is speeded up with respect to the walker. After a suitable space-time rescaling, the random walk converges to a sum of a Brownian motion and a Gaussian process with stationary increments, independent of the Brownian motion. Our main tool in the proof is an estimate on the relative entropy between the law of the environment process and the equilibrium measure of the exclusion process. Joint work with Milton Jara.

Wioletta Ruszel (Delft): Scaling limits of odometers in sandpile models.

The divisible sandpile model is a special case of the class of continuous fixed energy sandpile models on some lattice or graph where the initial configuration is random and the evolution deterministic. One question which arises is under which conditions the model will stabilize or not. The amount of mass u(x) emitted from a certain vertex x during stabilization is called the odometer function. The classical Abelian sandpile model is related to uniform spanning trees. Recently Lawler, Sun and Wu proved that the scaling limit of components of a uniform spanning tree where boundary connections are removed is again a bi-Laplacian model. This suggests that the sandpile model belongs to the same universality class as log-correlated models.
In this talk we will explain the connections and construct the scaling limit of the odometer function of a divisible sandpile model on a torus. This is joint work with A. Cipriani (U Bath/TU Delft) and R. Hazra (ISI Kolkatta).