Affine processes are common generalizations of continuous state and continuous time branching processes with immigration and Ornstein-Uhlenbeck type processes. Roughly speaking, the affine property means that the logarithm of the characteristic function of the process at any time is affine with respect to the initial state. We will consider a special two-factor affine process given by a jump-type stochastic differential equation. Based on the asymptotic behavior of the expectation vector, we introduce a classification of the process by distinguishing subcritical, critical and supercritical cases. In the subcritical case we prove that the affine model in question has a unique strictly stationary solution, further, in the subcritical diffusion case, ergodicity is also shown. As an application, in the latter case, we study asymptotic behaviour of the maximum likelihood estimator of the drift parameters of the process based on continuous time observations proving strong consistency and asymptotic normality.
Joint work with Leif Doring, Zenghu Li and Gyula Pap.

One model of real-life spreading processes is First Passage Percolation (also called SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with non-Markovian heavy-tailed edge weights. On the other hand, random graphs are often locally tree-like, and spreading on trees is very slow, because of bottleneck edges with huge weights. We show the surprising phenomenon that adding a single random edge to a tree typically accelerates the process severely. We examine this acceleration effect on some natural models of random trees: critical Galton-Watson trees conditioned to be large in some way, uniform random trees using Loop Erased Random Walks and Pólya urn ideas, and will also discuss what should happen on near-critical Erdős-Rényi graphs.
This is joint work with Gábor Pete.

Legyen B olyan halmaz R^3-ben (vagy általánosabban R^n-ben), amely a tér minden pontja körül (mint középpont körül) tartalmaz kocka élvázat (vagy általánosabban n-dimenziós kocka k-dimenziós vázát). Három esetet vizsgálunk:

(1) csak tengelypárhuzamos kockát engedünk meg,
(2) bármilyen kockát megengedünk, azaz elforgatottat is,
(3) csak egységkockákat engedünk meg, vagyis csak forgatni tudunk.

Az fog kiderülni, hogy B mindhárom esetben lehet nullmértékű, sőt a legtöbb esetben a Hausdorff dimenzió lehet n-nél kevesebb. A konstrukciók a Baire kategória tételt használják: azt mutatjuk meg, hogy tipikus jó elrendezés esetén B kicsi.
Alan Chang-gel, Csörnyei Mariannával és Héra Kornéliával közös eredmények.

I will discuss recent progress in understanding supercritical percolation models on lattices, particularly in the presence of strong spatial correlations. This includes quenched Gaussian heat kernel bounds, Harnack inequalities, and local CLT for the random walk on infinite percolation clusters. The results apply to the random interlacements at all levels, the vacant set of random interlacements and the level sets of the Gaussian free field in the regime of local uniqueness.

Random Apollonian networks are graphs which can be constructed as a sort of dual to the Apollonian circle packing problem. They have a tractable hierarchical structure. We show how to exploit this to prove a central limit theorem for the shortest path between two randomly chosen vertices of the graph.
Joint work with Júlia Komjáthy and Lajos Vágó.

We study factor of i.i.d. processes on the d-regular tree. We show that if such a process is restricted to two distant connected subgraphs of the tree, then the two parts are basically uncorrelated. More precisely, we give an explicit bound for the correlation of any measurable functions of the two parts, the bound only depending on d and the distance of the subgraphs. This result can be considered as a quantitative version of the fact that factor of i.i.d. processes have trivial 1-ended tails.
  Joint work with Ágnes Backhausz, Balázs Gerencsér, and Máté Vizer.

The Furstenberg measure is the stationary measure of a Markov chain on the projective space. The geometric measure theoretical properties of the Furstenberg measure allows us to give sufficient conditions for calculating the dimension of self-affine sets and measures, in a very natural way. We will see how.

In this talk I will give investigate the question of explosion of branching processes, i.e., when is it possible that a BP produces infinitely many offspring in finite time. Two important cases in terms of application are age-dependent BPs and BPs arising from epidemic models where individuals are only contagious in a possibly random interval after being infected. This imposes dependencies between the birth-time of the children of an individual.
The motivation for studying the explosiveness question is to understand weighted distances in locally tree-like random graphs, such as the configuration model, in the regime where the degree distribution is a power-law with exponent between (2,3). Here, the local neighborhood of a vertex and thus the initial stages of the spreading can be the approximated by an infinite mean offspring BP. I will explain the recent results on this area. This part is joint work with Enrico Baroni and Remco van der Hofstad.

We consider a system of non-intersecting squared Bessel processes which all start from one point at time 0 and they all return to the same point at time 1. Under the scaling of the starting and ending points when the macroscopic boundary of the paths touches the hard edge, we describe the limiting critical process in the neighbourhood of the touching point. In the talk, I will introduce the notion of non-intersecting paths and determinantal processes showing the connection to random matrices.
Joint work with Steven Delvaux.

Motivated by Diffusion Limited Aggregation being too hard to understand, Itai Benjamini suggested a way to construct a fully conformally invariant growth process in the plane: an aggregation process of chordal SLE(κ) excursions in the unit disk, starting from the boundary, growing towards all inner points simultaneously, invariant under all conformal self-maps of the disk. We prove that this conformal growth process of excursions, abbreviated as CGE(κ), exists iff κ∈ [0,4), and that, maybe sadly, it does not create additional fractalness: the Hausdorff dimension of the closure of all the SLE(κ) arcs attached is 1+κ/8 almost surely.
I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).
Joint work with Hao Wu.

Motivated by Diffusion Limited Aggregation being too hard to understand, Itai Benjamini suggested a way to construct a fully conformally invariant growth process in the plane: an aggregation process of chordal SLE(κ) excursions in the unit disk, starting from the boundary, growing towards all inner points simultaneously, invariant under all conformal self-maps of the disk. We prove that this conformal growth process of excursions, abbreviated as CGE(κ), exists iff κ∈ [0,4), and that, maybe sadly, it does not create additional fractalness: the Hausdorff dimension of the closure of all the SLE(κ) arcs attached is 1+κ/8 almost surely.
I will start with a brief introduction to conformally invariant processes and the Schramm-Loewner Evolution SLE(κ).
Joint work with Hao Wu.