Conference Agenda

We consider a Dirichlet process $\zeta$ on a general measurable space equipped with a finite measure $\rho$. This is a random probability measure whose finite dimensional distributions are Dirichlet with parameters determined by $\rho$.

It was proven by Peccati (2008) that every square-integrable function of $\zeta$ can be written as an orthogonal series of multiple integrals w.r.t. $\zeta$, where the kernels (i.e. the integrands) are degenerate in a certain sense.

In this talk, we revisit this fundamental chaos expansion by providing an alternative proof via a Mecke-type equation for a Dirichlet process along with explicit formulas for the kernels. As an application we give a direct proof of the Poincare inequality, which was derived by Stannat (2000) from the corresponding inequality for the Dirichlet distribution by a suitable approximation.

The talk is based on joint work with Günter Last.

In this talk, we explore transports of stationary random measures on $\mathbb{R}^d$. Under a suitable mixing assumption on two different transports of a single random measure we prove that the resulting random measures have the same asymptotic variance. An important consequence is a mixing criterion that ensures the persistence of the asymptotic variance under a transport. We pay special attention to the case of a vanishing asymptotic variance, known as hyperuniformity, which implies a suppression of long-range density fluctuations. Our approach enables us to rigorously establish hyperuniformity for many point processes and random measures that are relevant, among others, to random self-organization and material design. In particular, we construct a perturbation that turns any ergodic point process of finite intensity into a hyperuniform one.

We study a hierarchical model of non-overlapping cubes of sidelengths $2^j$, $j \in \mathbb{Z}$. The model allows for cubes of arbitrarily small size and the activities need not be translationally invariant. It can also be recast as a spin system on a tree with a long-range hard-core interaction. We prove necessary and sufficient conditions for the existence and uniqueness of Gibbs measures, discuss fragmentation and condensation, and prove bounds on the decay of two-point correlation functions. (Preprint: arXiv:2406.06249)

We work with a class of random Hamiltonians on $\mathbb R^d,$ $H^\omega,$ whose kinetic part is a L\'{e}vy operator, and the independent potential part comes from an alloy-type potential. We study the asymptotic behavior of the integrated density of states at the bottom of the spectrum of $H^\omega$.

When the profile function of the potential had compact support, in [K.Kaleta, K.Pietruska-Paluba, Lifshitz tail for continuous Anderson models driven by Lévy operators, Comm. Contemp. Math. 2020, 2050065] we have thoroughly examined the asymptotics behaviour of the IDS at the bottom of the spectrum: we gave precise rate of its decay, being of Lifschitz-tail type. However, the approach relied on the compactness of the support of W, the profile function.

In present work we show how to proceed withouh the compact support assumption. In this case (the long range alloy-type Hamiltonians) we also give precise rates of decay for the IDS at the bottom of the spectrum.

The class of L\'{e}vy operators condsidered here contains fractional laplacians, relativistic stable laplacians, and the lattice random variables (those that `amplify' the profile function) are quite general.