arXiv subjects

Goetz Kersting

Publications and source records attributed to Goetz Kersting.

2 recordsLinked to original sources

arXiv2024

Branching processes in varying environment with generation-dependent immigration

A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and immigration laws may vary over the generations. This flexibility makes the process more appropriate to model real populations due to the fact that the stability in the reproductive capacity and in the immigration laws are not usually fulfilled. In this setting, we study the extinction problem, providing a necessary and sufficient condition for the certain extinction of the population. The asymptotic behaviour of the model is analysed for those processes with critical offspring distributions, according with the classification established in Kersting (2017), and when the immigration means stabilize to a positive value. More specifically, we establish that the asymptotic distribution of the process -- under a suitable normalization -- belongs to the gamma distribution family.

math.PR↗

arXiv2019

From Parisi to Boltzmann

We sketch a new framework for the analysis of disordered systems, in particular mean field spin glasses, which is variational in nature and within the formalism of classical thermodynamics. For concreteness, only the Sherrington-Kirkpatrick model is considered here. For this we show how the Parisi solution (replica symmetric, or when replica symmetry is broken) emerges, in large but finite volumes, from a high temperature expansion to second order of the Gibbs potential with respect to order parameters encoding the law of the effective fields. In contrast with classical systems where convexity in the order parameters is the default situation, the functionals employed here are, at infinite temperature, concave: this feature is eventually due to the Gaussian nature of the interaction and implies, in particular, that the canonical Boltzmann-Gibbs variational principles must be reversed. The considerations suggest that thermodynamical phase transitions are intimately related to the divergence of the infinite expansions.

math.PR↗