Oberseminar Stochastik

25.04.2023

Fu-Hsuan Ho, Université Toulouse III

Algorithmic perspectives of the continuous random energy model

Disordered systems have recently received much interest in the mathematical literature in terms of efficient algorithms for finding low-energy states, or sampling a typical state from the Gibbs measure. In this talk, I will discuss these algorithms in the context of the Continuous Random Energy Model (CREM), a toy model of disordered systems introduced by Derrida and Spohn in the 1980s. I will present a Gibbs Measure sampling algorithm and mention some properties of this algorithm. Then, if time permits, I will speak of a hardness result in the low-temperature regime.

09.05.2023 

Christian Mönch, JGU

Inhomogeneous long-range networks - an overview

We revisit inhomogenous long-range percolation models in Euclidean space and give an overview of results obtained in the recent past. Particular attention is given to 'kernel-based' variants, where edge probabilities are parametrised by spatial distance of the adjacent vertices and a bivariate kernel that takes as input a pair of independent 'fitnesses' intrinsic to each vertex. The talk is partly based on several joint works with Peter Gracar (U Leeds), Markus Heydenreich (U Augsburg), Lukas Lüchtrath (WIAS Berlin) and Peter Mörters (U Köln).

11.07.2023

Jan Lukas Igelbrink, JGU und Goethe-Universität Frankfurt/M.

Muller's ratchet with tournament selection:
near-criticality and links to the classical ratchet

Muller's ratchet is a prototype model in mathematical population genetics. In an asexual population of constant size N, individual lineages are assumed to slowly acquire slightly deleterious mutations over the generations. Due to randomness, every once and a while the individuals with the currently smallest number of mutations disappear from the population; this is a click of the ratchet. The classical variant of the model, which assumes so-called proportional selection, so far has resisted against a fully rigorous asymptotic analysis of the clicking rate. In [1] this hurdle has been overcome by considering tournament (instead of proportional) selection, where selective competition within pairs is won by the fitter individual.
In our talk we will explain the graphical construction which was used in [1] to obtain a hierarchy of dual processes for the tournament ratchet. We will apply this duality also in the "near-critical" regime. We will reveal the form of the type-frequency profile between clicks of the ratchet, as well as the asymptotic click rates in various subregimes. Finally, we will discuss the mapping which takes (m; s) into the corresponding parameter pair of the "classical" ratchet so that the click rates have similar asymptotics under appropriate approximations, and will illustrate this by simulations.
The talk is based on joint work in progress with A. Gonzalez
Casanova, Ch. Smadi and A. Wakolbinger.

18.07.2023

 Andreas Klippel, TU Darmstadt

Comparison of the random loop model to percolation and infinite
loops in the random link model

Peter Mühlbacher showed that the inverse temperature of the random loop
model can be compared to a percolation parameter which leads to a bound
where infinitely large loops do not occur. By a different approach, we
improve the bound for the inverse temperature. We also obtain a
numerical value such that the inverse temperature must be chosen larger
than it in order to obtain infinitely long loops. Furthermore, we
demonstrate how this comparison argument can be applied to compare the
random link model to percolation.

This is a joint work with Benjamin Lees and Mino Nicola Kraft.