Abstracts
(Joint with Ross Bowen, Dan Fretwell, Jessica Jay)
Asymmetric simple exclusion and nearest-neighbour asymmetric zero range processes both enjoy reversible product stationary distributions called blocking measures. But a careful look reveals that these two are really the same process. Exploitation of this fact gives rise to the Jacobi triple product formula - an identity previously known from algebra, combinatorics and number theory. I will show the main steps of deriving it from pure probability this time, and I hope to surprise my audience as much as we got surprised when this identity first popped up in our notebooks. I will then sketch more recent results on generalisations of the argument, where slightly trickier particle systems give rise to a zoo of identities that generalise Jacobi's triple product.
I will introduce stochastic and deterministic reaction network models, which describe the time evolution of a system of interacting particles in an homogeneous volume. I will show some results connecting graphical properties of the network with dynamical features of the underlying dinamical system. In this setting, I will underline similarities and differences in the long-term behaviour of stochastic and deterministic models. In particular, I will discuss the cases of complex balanced and absolute concentration robust networks.
We present a diffusive system with a slow barrier between the sets of negative numbers and non-negative numbers. and give some results regarding the hydrodynamical limit. Afterwards, we compare them with the expected results when we consider a slow barrier in a super-diffusive system
Leonardo de Carlo - Perspective in the hydrodynamics of diffusive non-gradient systems
The derivation of the hydrodynamics in diffusive systems is based on a gradient structure for the microscopic current. This property allows to apply entropy methods techniques to prove a law of large numbers. When this property is missing, under certain conditions it is still possible to prove the hydrodynamics without an explicit expression for the diffusion coefficient. Here we will propose a starting point for a theory allowing to derive explicit hydrodynamics in non-gradient systems. The theory is based on two decompositions of the microscopic current and gives an explicit minima in the Green-Kubo formula using a functional Hodge theory.
Claudio Landim - Static large deviations for a reaction-diffusion model
We examine the stationary state.of an interacting particle system whose macroscopic evolution is described by one-dimensional reaction-diffusion equations.
Alessandra Occelli - Discrete and continuous Muttalib–Borodin processes: the hard edge
In this note we study a natural measure on plane partitions giving rise to a certain discrete-time Muttalib--Borodin process (MBP): each time-slice is a discrete version of a Muttalib--Borodin ensemble (MBE). The process is determinantal with explicit time-dependent correlation kernel. Moreover, in the $q \to 1$ limit, it converges to a continuous Jacobi-like MBP with Muttalib--Borodin marginals supported on the unit interval. This continuous process is also determinantal with explicit correlation kernel. We study its hard-edge scaling limit (around 0) to obtain a discrete-time-dependent generalization of the classical continuous Bessel kernel of random matrix theory. We lastly their interpretations as models of directed last passage percolation (LPP). The aforementioned hard edge limits for our MBPs lead to interesting asymptotics for these LPP models. In particular, a special case of our LPP asymptotics gives rise to an extremal statistics distribution interpolating between the Tracy--Widom GUE and the Gumbel distributions.
Assaf Shapira - Hydrodynamic limit of the Kob-Andersen model - slides
The Kob-Andersen model is an interacting particle system on the lattice, in which sites can contain at most one particle. Each particle is allowed to jump to an empty neighboring site only if there are sufficiently many empty sites in its neighborhood. This way, when the density is very high, many particles are unable to move, and the system slows down. In particular, the time it takes particles to diffuse, moving from high density regions to lower density ones, is very long. We will discuss the diffusion coefficient, and see how it decays as the density approaches 1.