## Seminários de probabilidade – Segundo Semestre de 2021

**Coordenação: **Professor Guilherme Ost e Professora Maria Eulalia Vares

As palestras ocorrem no Zoom (https://us02web.zoom.us/j/83698166702?pwd=ejFxYmh3dVk5b3J3SG0wMWxEVnVrUT09) **às segundas-feiras às 15h**, a menos de algumas exceções devidamente indicadas.

*Todas as palestras são em inglês.*

**Lista completa (palestras futuras podem sofrer alterações)**

Joint work with Peter Guttorp and Guilherme Ludwig.

*Access the slides here.*

*Access the slides here.*

#### 22/11

*Fast Consensus and Metastability in a Highly Polarized Social Network*

Antonio Galves (IME-USP)

The model we consider is a system with a large number of interacting marked point processes with memory of variable length. Each point process indicates the successive times in which a social actor expresses a “favorable” (+1) or “contrary” (-1) opinion on a certain subject. The social pressure on an actor determines the orientation and the rate at which he expresses opinions. When an actor expresses their opinion, social pressure on them is reset to 0, and simultaneously social pressure on the other actors is changed by one unit in the direction of the opinion that was just expressed.

The network has a polarization coefficient that indicates the tendency of social actors to express an opinion in the same direction of the social pressure exerted on them.

We show that when the polarization coefficient diverges consensus is reached almost instantaneously. Moreover, in a highly polarized network, consensus has a metastable behavior and changes its direction after a long and unpredictable random time.

This is a joint work and a joint talk with Kádmo de Souza Laxa.

*Access the slides here.*

*Access the slides here.*

*Access the slides here.*

*Access the slides here.*

*Access the slides here.*

We present an alternative proof of the so-called First Visit Time Lemma (FVTL), originally presented by Cooper and Frieze. We work in the original setting, considering a growing sequence of irreducible Markov chains on n states. We assume that the chain is rapidly mixing and with a stationary measure with no entry being either too small nor too large. Under these assumptions, the FVTL shows the exponential decay of the distribution of the hitting time of a given state x, for the chain started at stationarity, up to a small multiplicative correction. While the proof by Cooper and Frieze is based on tools from complex analysis, and it requires an additional assumption on a generating function, we present a completely probabilistic proof, relying on the theory of quasi-stationary distributions and on strong-stationary times arguments. In addition, under the same set of assumptions, we provide some quantitative control on the Doob’s transform of the chain on the complement of the state x.

I will also discuss the relation of this result with general results, previously obtained, providing an exact formula for the first hitting distribution via conditional strong quasi-stationary times.

*Access the slides here.*

Joint work with L. De Carlo and P. Goncalves.

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In this talk we will briefly present the model we are interested in, which is a fractional elliptic stochastic partial differential equation driven by Gaussian white noise. There is in the literature a standard way to approximate the covariance operator of the solution of such equations, the so-called rational approximation (Bolin and Kirchner, 2020), however this approach uses the solution to build such an approximation. By considering directly the covariance operator, we are able to provide a more computationally efficient approximation. We compute the rate of this approximation in terms of the Hilbert-Schmidt norm. Furthermore, we also obtain, rigorously, the rate of approximation of the so-called lumped mass method. This method is widely used by practitioners and is essential to make it computationally feasible to fit some models in spatial statistics. We obtain the rate of approximation of the lumped mass method in terms of the operator’s norm as well as, under some additional restrictions, the Hilbert-Schmidt norm. Finally, we present the usage of these approximations in maximum likelihood estimation. Joint work with David Bolin and Zhen Xiong.

*Access the slides here.*

The Dyson model is known to have a phase transition for decay parameters α between 1 and 2. We show that the metastate changes character at α =3/2. It is dispersed in both cases, but it changes between being supported on two pure Gibbs measures when α is less than 3/2 to being supported on mixtures thereof when α is larger than 3/2.

Joint work with Eric Endo and Arnaud Le Ny.

*Access the slides here.*

*Access the slides here.*

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

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).

*Access the slides here.*

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

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 ($gamm negamma_0$) are $(H_1,H_2)$-multi self-similar with one of $H_i$, $i=1,2$, equal to $0$ or $1$. The paper extends Pilipauskait.e 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 Vytautė Pilipauskaitė (University of Luxembourg).

*Access the slides here.*

Joint work with Daniel Figueiredo (COPPE/UFRJ) and Valmir Barbosa (COPPE/UFRJ).

Joint work with Ivailo Hartarsky.