Seminários de Probabilidade 2022 - 2º Semestre

• English
16/01 (remoto)

A very classical question in stochastic processes is to quantify the time it takes to first enter a given set, or the number of visits to this set in a fixed time window. The aim of the talk is to present some recent quantitative results concerning first hitting/return times and number of visits to rare sets (small/vanishing measure) in mixing processes. We will focus on asymptotic results (essentially convergence in distribution and large deviations). Most of the presentation will be about the classical case where the sets of interest are strings of symbols of increasing size. At the end, if time allows, we will give some glimpses of the case in which we target general sets having vanishing measure.

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09/01 (presencial)

Consideramos a questão de quantos cliques quase-maximais disjuntos podem ser encontrados num grafo aleatório G(n,p). Em 1992, Alon e Spencer conjecturaram que existe uma família de ordem n^2/(log n)^2 tais cliques disjuntos. Recentemente Acan e Kahn mostraram que a conjectura é falsa, provando uma cota superior da forma O(n^2/(log n)^3). Nós fornecemos uma cota inferior da mesma ordem.

19/12 (remoto)

We give a rigorous formulation of the Wick ordered stochastic heat equation, with planar white noise, whose solution is the free energy of an undirected random polymer, using a version of the Skorokhod integral. The solution is represented as an L^1 limit of a martingale given by the Feynman-Kac formula. We also show that the fundamental solution far from the center has fluctuations given by the 1+1 KPZ equation. This is a joint work with Jeremy Quastel and Balint Virag.

21/11 (presencial)

Nesta palestra recordamos alguns resultados sobre a transição de fase para modelos de percolação em ambientes aleatórios. O primeiro modelo (Hilário, Sá, Sanchis, Teixeira) consiste em eliminarmos da rede quadrada, os elos de colunas inteiras escolhidas de forma aleatória. Mostramos que a existência da transição de fase está relacionada com o momento da distância entre as colunas remanescentes. O segundo modelo (Hoffmann) consiste em eliminar também as linhas do mesmo modo, e mostramos que neste caso a transição ocorre quando a distância entre os remanescentes têm decaimento exponencial. Por fim, recordamos outros modelos como KSV e processos de contato.

07/11 (presencial)

A que distância está a lei conjunta de um vetor aleatório bidimensional da medida do produto induzida por suas marginais? Nesta palestra abordamos esta questão no contexto da equação de Kardar-Parisi-Zhang, onde a primeira coordenada do vetor é dada por um observável de uma condição inicial Browniana, e a segunda é um observável da solução em um tempo mais tarde. Para atacar esta tarefa usaremos ferramentas do cálculo de Malliavin e do Método de Stein, que nos permitirá obter um comportamento preciso de escala espaço-tempo para independência assintótica.

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24/10 (presencial)

A difusão de partículas em meios homogêneos é modelada como um processo que admite o deslocamento de partículas como um grupo uniforme, isto é, com mesma energia. No entanto se parte das partículas passa a um outro estado de energia elas passam a se deslocar sujeitas a um potencial diferente do potencial clássico que rege o movimento das partículas primitivas, isto é, o potencial de Fick. Mostra-se que esse processo de fluxo duplo com partículas em dois estados de energia é governado por uma equação de quarta ordem. Examinam-se alguns exemplos e sugestões de aplicação como em dinâmica populacional e fluxo de capitais.

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10/10 (presencial)

Many natural and artificial systems exhibit collective behaviors, which show in the form of spatio-temporal patterns. This has triggered the interests of scholars, who have proposed several theories to account for such diversity. One of the most popular mechanisms of pattern formation is due to Alan Turing, who showed that diffusion can disrupt a homogeneous stable state, triggering an instability [1]. The original theory has been conceived in the framework of reaction-diffusion PDEs, but it has been recently extended on networked systems [2]. Moreover, it has been shown that a Turing-like mechanism occurs in the framework of synchronized coupled oscillators, as diffusion can lead to a loss of synchronization [3].
In this seminar I will present an overview of Turing theory in networked systems, showing that the Turing framework is indeed not too far from that of coupled chaotic oscillators [4]. I will then focus the attention on two results we have recently obtained. The first one, about the study of reaction-diffusion systems on top of non-normal networks, i.e., networks whose adjacency matrix is non-normal [5]. Such topology makes the system more sensible to perturbation, leading to a loss of stability even when a linear stability analysis predicts otherwise [6, 7]. Finally, I will briefly introduce high-order structures, i.e., hypergraphs and simplicial complexes, and show a recent extension of Turing theory on such topologies [8].

References

[1] A M Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B, 237:37, 1952.
[2] Hiroya Nakao and Alexander S Mikhailov. Turing patterns in network-organized activator-inhibitor systems. Nature Physics, 6:544, 2010.
[3] J Challenger, D Fanelli, and R Burioni. Turing-like instabilities from a limit cycle. Phys. Rev. E, 92:022818, 2015.
[4] Louis M Pecora and Thomas L Carroll. Master stability functions for synchronized coupled systems. Physical Review Letters, 80(10):2109, 1998.
[5] Malbor Asllani, Renaud Lambiotte, and Timoteo Carletti. Structure and dynamics of non-normal networks. Sci. Adv., 4:Eaau9403, 2018.
[6] Riccardo Muolo, Malbor Asllani, Duccio Fanelli, Ph K Maini, and Timoteo Carletti. Patterns of non-normality in networked systems. Journal of Theoretical Biology, 480:81, 2019.
[7] Riccardo Muolo, Timoteo Carletti, James P Gleeson, and Malbor Asllani. Synchronization dynamics in non-normal networks: the trade-off for optimality. Entropy, 23:36, 2021.
[8] Riccardo Muolo, Luca Gallo, Vito Latora, Mattia Frasca, and Timoteo Carletti. Turing patterns in systems with high-order interaction. arXiv preprint arXiv:2207.03985, 2022.

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26/09 (remoto)

We develop a novel cluster expansion for finite-spin lattice systems subject to multi-body quantum —and, in particular, classical— interactions. Our approach is based on the use of “decoupling parameters”, advocated by Park, which relates partition functions with successive additional interaction terms. Our treatment, however, leads to an explicit expansion in a beta-dependent effective fugacity that permits an explicit evaluation of free energy and correlation functions at small beta. To determine its convergence region we adopt a relatively recent cluster summation scheme that replaces the traditional use of Kikwood-Salzburg-like integral equations by more precise sums in terms of particular tree-diagrams. Joint work with Nguyen Tong Xuan.

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12/09 (remoto)

This phenomenon can also be surprisingly observed in two-state Probabilistic Cellular Automata. In this talk I will review some of the results that can be proved in the framework of the nearest neighbor reversible Probabilistic Cellular Automaton and in that of the Blume-Capel model. The discussed results have been derived in collaboration with F.R. Nardi, E. Olivieri, and C. Spitoni.

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