21.07.2023

Stefan Adams: Scaling limits for non-convex interaction

Abstract: We introduce a currently hot topic in probability theory, the theory of scaling limits for random fields of gradients in all dimensions. The random fields are a class of model systems arising in the studies of random interfaces, random geometry, Euclidean field theory, the theory of regularity structures, and elasticity theory. After explaining how non-convex energy terms can influence the scaling limit, we outline our result on the scaling limit to the continuum Gaussian Free Field in dimension d=2,3 for a class of non-convex interaction energies. We show that the Hessian of the free energy governs the continuum Gaussian Free Field. The second result concern the Gaussian decay of correlations. All our results hold in the low-temperature regime and moderate boundary tilts. We outline how multi-scale/renormalisation group methods provide means of proving our statements. if time permits we discuss isomorphism theorems for the model.

Elia Bisi: Non-intersecting path constructions for inhomogeneous TASEP and the KPZ fixed point

Abstract: The KPZ fixed point is conjectured to be the universal space-time scaling limit of the models belonging to the KPZ universality class and it was rigorously constructed by Matetski, Quastel and Remenik (Acta Math., 2021) as a scaling limit of TASEP (Totally Asymmetric Simple Exclusion Process) with arbitrary initial configuration. We set up a new, alternative approach to the KPZ fixed point, based on combinatorial structures and non-intersecting path constructions, which also allows studying inhomogeneous interacting particle systems. More specifically, we consider a discrete-time TASEP, where each particle jumps according to Bernoulli random variables with particle-dependent and time-inhomogeneous parameters, starting from an arbitrary initial configuration. We provide an explicit, step-by-step route from the very definition of the model to a Fredholm determinant representation of the joint distribution of the particle positions in terms of certain random walk hitting probabilities. Our tools include the combinatorics of the Robinson-Schensted-Knuth correspondence, intertwining relations, non-intersecting lattice paths, and determinantal point processes.