Archiv OS 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.

Götz Kersting, Uni Frankfurt:

Kugelwellen

Kugelwellen sind uns als Schallwellen wohlbekannt und scheinen uns in ihrem Verhalten völlig einleuchtend. Dabei sind etwa die Eigenschaften von Lautsprechern überraschend und intuitiv weniger einsichtig. Erst eine mathematische Behandlung führt zu einem tieferen Verständnis, und es zeigt sich, dass Kugelwellen, so wie wir sie erleben, ein Phänomen speziell des 3-dimensionalen Raums sind.

Die Behandlung von Kugelwellen gelingt mit elementaren Hilfsmitteln aus den Anfängervorlesungen der Analysis. Das Thema ist geeignet, den Sinn gerade auch von Inhalten der Analysis II-Vorlesung zu illustrieren — etwa für Lehramtsstudierende, die hier in der Lehre oft auf sich allein gestellt bleiben.

Frederic Alberti, Uni Bielefeld:

The selection-recombination equation and its solution

The deterministic selection-recombination equation describes the evolution of the genetic type composition of a population under selection and recombination in a law of large numbers regime. This equation is notoriously difficult to solve; only in the special case of three sites with selection acting on one of them has an approximate solution been found, but without an obvious path to generalisation. The matter finally became transparent by considering a dual ancestral process which combines the ancestral recombination graph with the ancestral selection graph. This approach led to an explicit solution of the selection-recombination equation, in the special case of one selected site linked to arbitrarily many neutral sites. In this talk, I will discuss a related approach based on a discretisation scheme, along with an application to genetic hitchhiking.

Timo Schlüter, JGU Mainz:

Grenzwertsätze für gerichtete Irrfahrten in dynamischer zufälliger Umgebung

Frederik Klement (JGU Mainz):

Poisson Representations for Population Models with Competition

https://us02web.zoom.us/j/87543076825?pwd=ZFY5eE41SXBTblZIRDdNOWZDSEhoZz09
Meeting ID: 875 4307 6825
Passwort: 381139

Abstract:

The Dawson-Watanabe superprocess is an important population model in genetics and it can be obtained as the high-density limit of a sequence of finite branching particle systems, similar how the Feller diffusion is obtained as the limit of time-continuous Galton-Watson processes. Unfortunately all information of the individual particles is lost, when we take the limit. But Thomas G. Kurtz and Eliane R. Rodrigues presented 2011 a so called Poisson representation for the Dawson-Watanabe superprocess. This representation gives us not only the superprocess but maintains the particles. This result is not only very fascinating, but allows one to derive many facts about the Dawson-Watanabe process in a much easier fashion as before.

Unfortunately the Dawson-Watanabe superprocess and the Kurtz-Rodrigues representation do not allow interactions between the particles like competition. We want to discuss how to modify the Kurtz-Rodrigues representation, so that it becomes a Poisson representation for populations models with competition between the particles. For this purpose we will study how one can combine ideas from Perkins historical stochastic calculus with the Kurtz-Rodrigues representation. This will not only allows us to derive a Poisson representation for competitive models, but also gives a natural coupling of the competitive model as a subpopulation of the Dawson-Watanabe superprocess.

We will also discuss a small application.

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Matthias Birkner (JGU Mainz):

Onsagers Lösung des zweidimensionalen Ising-Modells

https://us02web.zoom.us/j/82723290131?pwd=anBrVkxYRkYyRlU5OEVxSlZzVkdrQT09
Meeting ID: 827 2329 0131
Passwort: 099205

19.06.2019

10 ct