Vinita Mulay (University Duisburg-Essen)
Elephant Random Walk with Tampered Memory

Abstract: The Elephant Random Walk (ERW), introduced by Schütz and Trimper (2004), has gained a lot of attention in recent years. We consider a memory-based variation of the classical ERW. The aim is to understand how much memory is 'enough' to observe an ERW-like behaviour.

In our model, we partition the memory into two subsets, $D_n$ and $D_n^c$, such that the elephant behaves like the classical ERW model when a step is chosen from $D_n^c$ and behaves differently when it is chosen from $D_n$. We illustrate that in order to obtain a phase transition, the size of $D_n^c$ needs to be more than half the entire memory. This is joint work with Neeraja Sahasrabudhe (IISER Mohali) and Debleena Thacker (Durham University).

Mikhail Urusov (University Duisburg-Essen)
Representation property for 1d general diffusion semimartingales.

Abstract: A general diffusion semimartingale is a 1d continuous semimartingale that is also a regular strong Markov process. The class of general diffusion semimartingales is a natural generalization of the class of (weak) solutions to SDEs. A continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingale part. We show that the representation property holds for a general diffusion semimartingale if and only if its scale function is (locally) absolutely continuous in the interior of the state space.
Surprisingly, this means that not all general diffusion semimartingales possess the representation property, which is in contrast to the SDE case. Furthermore, it follows that the laws of general diffusion semimartingales with absolutely continuous scale functions are extreme points of their semimartingale problems. We construct a general diffusion semimartingale whose law is not an extreme point of its semimartingale problem. This contributes to the solution of the problems posed by Jacod and Yor and by Stroock and Yor on the extremality of strong Markov solutions (to martingale problems). This is a joint work with David Criens.

Anton Klimovsky (University Würzburg)
Opinion dynamics on evolving networks: a measure-valued Markov process and its scaling limit

Abstract: We study a voter-type opinion dynamics on a network whose community structure evolves endogenously through two simple mechanisms: individuals may migrate between communities (“poaching”) or split off to form singleton communities (“self-employment”), while opinions evolve by within-community resampling.
Using a two-level measure-valued representation -- communities as integer-valued measures on the opinion set and the network as a point measure on community measures -- we obtain an explicit generator and natural polynomial sampling observables.
In a diffusion scaling limit for two opinions, we derive a coupled SDE: the global opinion frequency follows a Wright–Fisher-type diffusion with stochastic volatility driven by an evolving homozygosity statistic of the community partition.

15:15 – 16:00 Gernot Akemann (Bielefeld): Universality classes in Hermitian and non-Hermitian random matrix theory
16:15 – 17:00 Hanna Döring (Osnabrück): Fine asymptotics of the magnetization of the annealed dilute Curie-Weiss model
17:15 – 18:00 Alexander Drewitz (Köln): (Near-)critical behavior of a strongly correlated percolation model
 

Abstract: In this talk we discuss a variant of the voter model on a co-evolving network in which interactions of two individuals with differing opinions only lead to an agreement on one of these opinions with a fixed probability $q$.  Otherwise, with probability $1-q$, both individuals become offended in the sense that they never interact again, i.e.\ the corresponding edge is removed from the underlying network.  Eventually, these dynamics reach an absorbing state at which there is only one opinion present in each connected component of the network. If globally both opinions are present at absorption we speak of ``segregation'', otherwise of ``consensus''.
We show that segregation and a weaker form of consensus both occur with positive probability for every $q \in (0,1)$ and that the segregation probability tends to $1$ as $q \to 0$. Furthermore, we establish that, if $q \to 1$ fast enough, with high probability the population reaches consensus while the underlying network is still densely connected.
If time allows we briefly discuss results from simulations to assess the obtained results and to discuss further research directions.
This talk is based on joint work with Raphael Eichhorn and Felix Hermann.

Kinga Nagy (University Osnabrück)
Number of crossings, and the planarity of random geometric graphs with heavy-tailed mark distributions.

Consider a random graph that is geometrically defined in $d$-dimensional space, such that the vertices are given by a random point configuration, and the existence of each edge (represented by the line segment between the points) is decided by independent marks on the edge and its endpoints. We can generate a drawing of the graph by projecting the construction onto a plane. In this talk, we consider the number of crossings in such a projection, focusing on the case when the markings have heavy tails. In particular, the distribution of the i.i.d. marks exhibits polynomial decay.
We study the overall behaviour of the model depending on the tail index of the markings. As a related problem, we also consider the graph-theoretical planarity of the graph, and how this relates to the number of crossings in the projection.
Joint work with Hanna Döring.

A central limit theorem for a random walk on Galton-Watson trees with random conductances

We consider limit theorems for a random walk on a weighted Galton-Watson tree, where the edges of the tree are assigned randomly uniformly elliptic conductances and the random walker crosses an edge with probability proportional to its conductance. For this process, the effective speed or variance depend in an intricate way on the law of the conductances. In order to study this dependence, we assign to a positive fraction of edges a small conductance $\varepsilon$. We show a functional central limit theorem for the distance of the walker to the root and, provided that the tree formed by larger conductances is supercritical, we show that the variance is nonvanishing as $\varepsilon\to 0$, which implies that the slowdown induced by few edges with small conductance is not too strong. The proof utilizes a specific regeneration structure, which leads to escape estimates uniform in the edge weights.

Large Deviation Principle for Friendship Biases in Galton-Watson Trees

The friendship paradox is a structural bias in networks stating that, on average, the neighbours of a vertex tend to have a higher degree than the vertex itself. Although this is counterintuitive, the phenomenon is deeply rooted in the geometry of networks and holds for every finite (simple or multiple) graph. However, at the level of individual vertices, it is not a priori guaranteed that most vertices have a smaller degree than their neighbours.
The friendship bias of a vertex is defined as the difference between the average degree of its neighbours and its own degree. According to the sign of this quantity, we can naturally classify the vertices as "negative", "neutral", or "positive". In this talk, we first briefly review some previous probabilistic results on the friendship paradox. We then analyse the fractions of vertex types along a random downward path in an infinite Galton-Watson tree and derive a large deviation principle for these fractions as the branching depth grows. The rate function is characterised through a variational problem involving relative entropy under a linear constraint. We discuss its properties in the case of binary branching.