Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

E. Löcherbach

We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Löcherbach (Stoch Process Appl 2017, http://arxiv.org/abs/1512.00265) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high-dimensional chain of differential equations driven by 2-dimensional Brownian motion. We study the large population, i.e., small noise limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.

Chromatic thresholds in dense random graphs

Peter Allen, Julia Böttcher, Simon Griffiths, Yoshiharu Kohayakawa and Robert Morris

The chromatic threshold δχ(H, p) of a graph H with respect to the random graphG(n, p) is the infimum over d > 0 such that the following holds with high probability: the familyof H-free graphs G ⊆ G(n, p) with minimum degree δ(G) ≥ dpn has bounded chromatic number.The study of the parameter δχ(H) := δχ(H,1) was initiated in 1973 by Erd˝os and Simonovits, andwas recently determined for all graphs H. In this paper we show that δχ(H, p) = δχ(H) for all fixedp ∈ (0, 1), but that typically δχ(H, p) ≠ δχ(H) if p = o(1). We also make significant progress towardsdetermining δχ(H, p) for all graphs H in the range p = n−o(1). In sparser random graphs the problem issomewhat more complicated, and is studied in a separate paper.

Bolsa para comunicação científica (BJC-2)

O Centro de Pesquisa, Inovação e Difusão em Neuromatemática (CEPID NeuroMat) oferece uma bolsa para profissionais de jornalismo científico interessados em fazer parte da equipe de difusão científica desse centro de excelência da FAPESP.

Ernst W. Hamburger's optimism: a NeuroMat video

The NeuroMat scientific dissemination has recently launched a short movie on physicist and science popularizer Ernst W. Hamburger, a NeuroMat PI. The video --which was produced by LAICA and supported by FAPESP-- is a development of an event on Hamburger's contributions that was organized at the University of São Paulo in 2016.

Vagas para pós-doutorado

O Centro de Pesquisa, Inovação e Difusão em Neuromatemática (NeuroMat), com sede na Universidade de São Paulo, Brasil, e financiado pela FAPESP, está selecionando candidatos a bolsas de pós-doutoramento para desenvolvimento de atividades de pesquisa na área de interface entre Matemática, Estatística, Computação e Neurociência.

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O Centro de Pesquisa, Inovação e Difusão em Neuromatemática está sediado na Universidade de São Paulo e é financiado pela FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo).

 

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