Speaker: Fabio Toninelli (Technical University of Vienna)

Title: Driven diffusive systems and stochastic PDEs

Abstract: In the introductory lecture, I will discuss a bit of statistical physics background/motivations about "driven diffusive systems", and explain how they can be mathematically modeled, either via discrete Markov chains (interacting particle systems, like the "Asymmetric Simple Exclusion Process") or via stochastic PDEs (like the Stochastic Burgers equation). I will also give a panorama of expected and proven results about "large-scale Gaussian fluctuations in dimension d\ge2". This part will include no proofs or technical details.

In the more technical lecture, I will focus on the stochastic Burgers equation (and similar stochastic PDEs) in the critical dimension d=2. I will discuss what is the "weak coupling limit" and I will formulate a  theorem (based on 2 joint recent works ( arXiv:2304.05730, arXiv:2108.09046) with G. Cannizzaro, D. Erhard, M. Gubinelli in different combinations) that says that these equations, in the weak coupling regime, have a Gaussian scaling limit at large scales. I plan to explain a bit the ideas behind the proof. 

Speaker: Julia Komjathy (Delft University of Technology)

Title: Cluster-size decay in spatial random graphs

Abstract: We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation/geometric inhomogeneous random graphs and the age-dependent random connection model. Assume that the parameters are such that there is an infinite component. We identify the stretch-exponent zeta in (0,1) of the subexponential decay of the cluster-size distribution. That is, with C(0) denoting the number of vertices in the component of the vertex at the origin,

  P ( C is larger than k but finite ) = exp ( -k^zeta) as k tends to infinity

 The value of zeta undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent tau of the degree distribution and a long-range parameter alpha governing the presence of long edges in Euclidean space.

In this seminar I will describe the connection to the second largest component, and present the key ideas of the proof. Based on joint work with Joost Jorritsma (Tu/e) and Dieter Mitsche (Lyon).