Discrepancy and eigenvalues of Cayley graphs

Yoshiharu Kohayakawa, Vojtěch Röd and Mathias Schacht

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

Ten research directions for the NeuroMat team in the coming years

As the NeuroMat research team worked on the project renewal proposal, ten research teams were listed. These teams have specific research topics, that will in the coming years provide the direction of the RIDC agenda. These teams were normally already active prior to 2018.

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.

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NeuroMat

The Research, Innovation and Dissemination Center for Neuromathematics is hosted by the University of São Paulo and funded by FAPESP (São Paulo Research Foundation).

 

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