Títulos: The Parabolic Anderson Model on a Galton-Watson Tree e Local Scaling Limits of Lévy Driven Fractional Random Fields

Palestrantes: Frank den Hollander (Leiden University) e Donatas Surgailis (Vilnius University)
Data: 23/08/2021
Horário: 14:00h
Local: Transmissão online.

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Resumos:

Frank den Hollander (Leiden University) - The Parabolic Anderson Model on a Galton-Watson Tree

We consider the parabolic Anderson model on a supercritical Galton-Watson tree with an i.i.d.\ random potential whose marginal distribution is close to the double-exponential. Under the assumption that the degree distribution has a sufficiently thin tail, we derive an asymptotic expansion for large times of the total mass of the solution given that initially a unit mass sits at the root. We derive the expansion both under the quenched law (i.e., conditional on the realisation of the random tree and the random potential) and under the half-annealed law (i.e., conditional on the realisation of the random tree but averaged over the random potential). The two expansions turn out to be different, but both contain a coefficient that is given by a variational formula indicating that the solution concentrates on a subtree with minimal degree according to a computable profile. A key tool in the analysis is the large deviation principle for the empirical distribution of a Markov renewal process.

Joint work with Wolfgang König (Berlin), Renato dos Santos (Belo Horizonte), Daoyi Wang (Leiden).

Donatas Surgailis (Vilnius University) - Local Scaling Limits of Lévy Driven Fractional Random Fields

We obtain a complete description of local anisotropic scaling limits for a class of fractional random fields $X$ on ${\mathbb{R}}^2$ written as stochastic integral with respect to an infinitely divisible random measure. The scaling procedure involves increments of $X$ over points the distance between which in the horizontal and vertical directions shrinks as $O(\lambda) $ and $O(\lambda^\gamma)$ respectively as $\lambda \downarrow 0$, for some $\gamma>0$. We consider two types of increments of $X$: usual increment and rectangular increment, leading to the respective concepts of $\gamma$-tangent and $\gamma$-rectangent random fields. We prove that for above $X$ both types of local scaling limits exist for any $\gamma>0$ and undergo a transition, being independent of $\gamma>\gamma_0$ and $\gamma<\gamma_0$, for some $\gamma_0>0$; moreover, the `unbalanced' scaling limits ($\gamma\ne\gamma_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskaite and Surgailis (2017) and Surgailis (2020) on large-scale anisotropic scaling of random fields on ${\mathbb{Z}}^2$ and Benassi et al. (2004) on $1$-tangent limits of isotropic fractional Lévy random fields.

This is joint work with Vytaute Pilipauskaite (University of Luxembourg)

Título: Local Symmetry in Erdős–Rényi Graphs

Palestrante: Jefferson Elbert Simões (DIA/UNIRIO)
Data: 16/08/2021
Horário: 15:00h
Local: Transmissão online.

Resumo: Real-world networks are often understood as being symmetrical, meaning that vertices can be found which perform similar or equivalent structural roles (such as hubs from different communities in social networks, or functional regions in neuronal networks). These roles are usually associated with their topological placement relative to its surroundings; however, traditional mathematical formulations of graph symmetry are based on automorphism groups, which depend fundamentally on global structure and do not account for similarities in local structures. In this work, we introduce the concept of local symmetry, which reflects the structural equivalence of vertices' egonets while generalizing classical conceptualizations of symmetry such as automorphism and isomorphism. We also study the emergence of local asymmetry in Erdős–Rényi graphs, identifying regimes of both asymptotic local symmetry and asymptotic local asymmetry. We find that local symmetry persists at least to an average degree of n^{1/3} and local asymmetry emerges at an average degree not greater than n^{1/2}, which are regimes of much larger average degree than for traditional, global asymmetry.
Joint work with Daniel Figueiredo (COPPE/UFRJ) and Valmir Barbosa (COPPE/UFRJ).
All the talks are held in English.

The videos of the online seminars are available:
2020 
2021-1

For the second semester, a few days after each meeting the video should be available at HERE.

Título: Fluctuations of homology of diffusion processes on Riemannian manifolds

Palestrante: Mauro Mariani, HSE University, Moscow.
Data: 18/08/2021
Horário: 13:00h
Local: Transmissão online.

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ID da reunião: 958 0581 3232

Resumo: I will discuss the long time behavior and symmetry properties of the DeRham homology of Brownian curves on compact manifolds. Some basic results will be shown and open problems discussed.

Título: Generalised oriented site percolation

Palestrante: Réka Szabó (Université Paris Dauphine)
Data: 09/08/2021
Horário: 15:00h
Local: Transmissão online.

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Resumo: We consider a generalised oriented site percolation model on Z^d with arbitrary neighbourhood. The key additional difficulties as compared to standard oriented percolation are the lack of symmetry and, in two dimensions, of planarity. We establish that, despite these deficiencies, in the supercritical regime GOSP behaves qualitatively like OP. Joint work with Ivailo Hartarsky. All the talks are held in English.

The videos of the online seminars are available:
2020
2021-1

For the second semester, a few days after each meeting the video should be available at HERE.