Sophia-Marie Mellis (University of Bielefeld)
Coalescents with migration in the moderate regime

Multi-type models have recently experienced renewed interest in the stochastic modeling of evolution. This is partially due to their mathematical analysis often being more challenging than their single-type counterparts; an example of this is the site-frequency spectrum of a colony-based population with moderate migration.
In this talk, we model the genealogy of such a population via a multi-type coalescent starting with $N(K)$ colored singletons with $d \geq 2$ possible colors (colonies). The process is described by a continuous-time Markov chain with values on the colored partitions of the colored integers in $\{1, \ldots, N(K)\}$; blocks of the same color coalesce at rate $1$, while they are also allowed to change color at a rate proportional to $K$ (migration).
Given this setting, we study the asymptotic behavior, as $K\to\infty$ at small times, of the vector of empirical measures, whose $i$-th component keeps track of the blocks of color $i$ at time $t$ and of the initial coloring of the integers composing each of these blocks. We show that, in the proper time-space scaling, it converges to a multi-type branching process (case $N(K) \sim K$) or a multi-type Feller diffusion (case $N(K) \gg K$). Using this result, we derive an applicable representation of the site-frequency spectrum.
This is joint work with Fernando Cordero and Emmanuel Schertzer.
 

Rebecca Steiner (University of Mainz)
A Random Walk Approach to Broadcasting on Random Recursive Trees

In the broadcasting problem on trees, a $\{−1,1\}$-message originating in an unknown node is passed along the tree with a certain error probability $q$. The goal is to estimate the original message without knowing the order in which the nodes received the information. When using majority estimation, we establish connections to both Pólya urns and random walks with memory effects. We apply these approaches to study the error probability of the majority estimator and to identify an infeasibility regime on the entire group of very simple increasing trees as well as shape exchangeable trees, including majority estimation based solely on the leaf values. This extends the work of Addario-Berry et al. (2022), who investigated this estimator for uniform and linear preferential attachment random recursive trees.