Abstract: 

We investigate simple random walks on infinite (Bienaymé–) Galton–Watson trees. The main focus is on the  annealed return probability for these random walks. We prove that for all offspring distributions with finite first moment, the return probability decays subexponentially with power $t^{1/3}$ in the exponent, which is optimal whenever the offspring distribution does not forbid leaves or linear pieces in the tree. This complements the corresponding lower bound provided by Piau (1998). In the special case of a Poissonian offspring distribution, we apply this upper bound to deduce a Lifshits tail for the eigenvalue density of the graph Laplacian on supercritical sparse Erdös–Rényi random graphs.

Joint work with Peter Müller and Sara Terveer.

Abstract: 

Many models of interacting particle systems in a continuous medium, such as the eigenvalues of random matrices or Coulomb gases, exhibit a phenomenon called hyperuniformity: this means that the variance of the number of points in a "large ball" is negligible compared to the volume of the ball, as opposed to independent point systems, i.e., systems with no interaction. We will present an introduction to the general phenomenon of hyperuniformity, particularly recent results for infinite models, and show that the variance transfer principle also works for such finite-sized systems: if a weak version of hyperuniformity is shown only for the variance of smooth linear statistics, the rates obtained can be transferred via an argument in Fourier space to the variance of irregular linear statistics, such as the number variance on balls. This principle is applied to show non-asymptotic hyperuniformity results for Coulomb gases and Girko ensembles in random matrices. 

We consider the dilute Curie-Weiss model of size $N$, which is a generalization of the classical Curie-Weiss model where the dependency structure between the spins is not encoded by the complete graph but via the (directed) Erd\H{o}s-Rényi graph on $N$ vertices in which every edge appears independently with probability $p(N)$. In the high temperature with external magnetic field regime ($0<\beta<1,h\in\mathbb{R}$) we prove for $p^{3}N^{2}\to\infty$ sharp cumulant bounds for the magnetization under the annealed Gibbs measure implying a central limit theorem with rate, a moderate deviation principle, a concentration inequality, a normal approximation bound with Cramér correction and mod-Gaussian convergence.

Abstract: 

In the standard (hard) random geometric graph, the vertex set is given by a Poisson point process in a convex body, and the edges by pairs of points closer than a given threshold. The soft version of this model allows the existence of the edges to be decided not only by their lengths, but by i.i.d. random marks placed on each pair of vertices. In this talk, we choose the marks to have heavy tails, namely to be Pareto distributed, and consider the planarity of the arising graph. In particular, we focus on how the tail index of the mark distribution influences the general behaviour of the model, and specifically the regime in which the graph is planar.

Based on joint work with Hanna Döring.