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Abstract We give a sufficient criterion for the weak disorder regime of directed polymers in random environment, which extends a well-known second moment criterion. We use a stochastic representation of the size-biased law of the partition function. We consider the so-called directed polymer in random environment, being defined as follows: Let p(x, y) =… (More)

- Matthias Birkner, Jochen Blath, Bjarki Eldon
- Genetics
- 2013

A large offspring-number diploid biparental multilocus population model of Moran type is our object of study. At each time step, a pair of diploid individuals drawn uniformly at random contributes offspring to the population. The number of offspring can be large relative to the total population size. Similar "heavily skewed" reproduction mechanisms have… (More)

Let Λ be a finite measure on the unit interval. A Λ-Fleming-Viot process is a probability measure valued Markov process which is dual to a coalescent with multiple collisions (Λ-coalescent) in analogy to the duality known for the classical Fleming-Viot process and Kingman’s coalescent, where Λ is the Dirac measure in 0. We explicitly construct a dual… (More)

- Matthias Birkner, Jochen Blath
- Journal of mathematical biology
- 2008

One of the central problems in mathematical genetics is the inference of evolutionary parameters of a population (such as the mutation rate) based on the observed genetic types in a finite DNA sample. If the population model under consideration is in the domain of attraction of the classical Fleming-Viot process, such as the Wright-Fisher- or the Moran… (More)

- M. Birkner, J. Blath, +4 authors A. Wakolbinger
- 2004

We determine that the continuous-state branching processes for which the genealogy, suitably time-changed, can be described by an autonomous Markov process are precisely those arising from α-stable branching mechanisms. The random ancestral partition is then a time-changed Λ-coalescent, where Λ is the Beta-distribution with parameters 2− α and α, and the… (More)

- Bjarki Eldon, Matthias Birkner, Jochen Blath, Fabian Freund
- Genetics
- 2015

The ability of the site-frequency spectrum (SFS) to reflect the particularities of gene genealogies exhibiting multiple mergers of ancestral lines as opposed to those obtained in the presence of population growth is our focus. An excess of singletons is a well-known characteristic of both population growth and multiple mergers. Other aspects of the SFS, in… (More)

- Matthias Birkner, Jochen Blath, Matthias Steinrücken
- Theoretical population biology
- 2011

We present and discuss new importance sampling schemes for the approximate computation of the sample probability of observed genetic types in the infinitely many sites model from population genetics. More specifically, we extend the 'classical framework', where genealogies are assumed to be governed by Kingman's coalescent, to the more general class of… (More)

- Matthias Steinrücken, Matthias Birkner, Jochen Blath
- Theoretical population biology
- 2013

We apply recently developed inference methods based on general coalescent processes to DNA sequence data obtained from various marine species. Several of these species are believed to exhibit so-called shallow gene genealogies, potentially due to extreme reproductive behaviour, e.g. via Hedgecock's "reproduction sweepstakes". Besides the data analysis, in… (More)

- Matthias Birkner, Jochen Blath, Bjarki Eldon
- Genetics
- 2013

Statistical properties of the site-frequency spectrum associated with Λ-coalescents are our objects of study. In particular, we derive recursions for the expected value, variance, and covariance of the spectrum, extending earlier results of Fu (1995) for the classical Kingman coalescent. Estimating coalescent parameters introduced by certain Λ-coalescents… (More)

- Matthias Birkner, José Alfredo López-Mimbela, Anton Wakolbinger

We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power… (More)