## Seminários de Probabilidade 2023 - 1º Semestre

• ##### English
###### 03/07 (presencial)

The Euler method for approximating an ODE is known to be of order 1. For Stochastic ODEs with multiplicative noise, however, it drops to 1/2. What about for Random ODEs? Current works tell us it is also order 1/2, or even less, depending on the Holder exponent of the noise sample paths. Here we show that in many typical situations, it is actually of order 1. This applies to a variety of noises, such as additive or multiplicative Itô processes, transport processes with sample paths of bounded variation, and even processes with discontinuous sample paths, as in point-process noises. For fractional Brownian motion processes, we may not reach order 1, depending on the Hurst parameter, but we still improve the order compared with the current belief. The proofs rely on writing a global error formula instead of estimating the local error; using Fubini to move the critical regularity term to the larger scales; and using the Itô isometry or some other form of global estimate to control that critical term. In this talk, we discuss these improvements, sketch the proofs, and illustrate the results numerically with a number of interesting models. This is a joint work with Peter Kloeden (University of Tübingen, Germany).

###### 19/06 (presencial)

The set-colouring Ramsey number Rr,s(k) is defined to be the minimum n such that if each edge of the complete graph Kn is assigned a set of s colours from {1, . . . , r}, then one of the colours contains a monochromatic clique of size k. The case s = 1 is the usual r-colour Ramsey number, and the case s = r − 1 was studied by Erdős, Hajnal and Rado in 1965, and by Erdős and Szemerédi in 1972.

The first significant results for general s were obtained only recently, by Conlon, Fox, He, Mubayi, Suk and Verstraëte, who showed that Rr,s(k) = 2^{Θ(kr)} if s/r is bounded away from 0 and 1. In the range s = r − o(r), however, their upper and lower bounds diverge significantly. In this talk we introduce a new (random) colouring, and use it to determine Rr,s(k) up to polylogarithmic factors in the exponent for essentially all r, s and k.

This is a joint work with Lucas Aragão, Maurício Collares, João Pedro Marciano and Rob Morris.

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###### 05/06 (remoto)

In this talk we study the scaling limit of a random field which is a non-linear transformation of the gradient Gaussian free field. More precisely, our object of interest is the recentered square of the norm of the gradient Gaussian free field at every point of the square lattice. Surprisingly, in dimension 2 this field bears a very close connection to the height-one field of the Abelian sandpile model studied in Dürre (2009). In fact, with different methods we are able to obtain the same scaling limits of the height-one field: on the one hand, we show that the limiting cumulants are identical (up to a sign change) with the same conformally covariant property, and on the other that the same central limit theorem holds when we view the interface as a random distribution. We generalize these results to higher dimensions as well.

Joint work with Rajat Subhra Hazra (Leiden), Alan Rapoport (Utrecht) and Wioletta Ruszel (Utrecht).

###### 22/05 (presencial)

Neste seminário estudaremos um passeio aleatório em ambientes aleatórios em uma dimensão. Supondo que o ambiente satisfaça uma desigualdade de desacoplamento e que o passeio se mova balisticamente, mostraremos uma lei forte dos grandes números para este passeio. Exemplos de ambientes que satisfazem a primeira hipótese são o processo zero-range e o processo de exclusão assimétrico. Esse seminário será baseado em um trabalho conjunto com Rangel Baldasso, Marcelo Hilário e Renato Santos.

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###### 08/05 (remoto)

In this talk, we analyse the genealogy of a sample of k particles without replacement from a population alive at large time in a critical Galton Watson process in varying environment. Our approach uses k distinguished spine particles and a suitable change of measure similar to a k-size biasing with a discounting rate proportional to the total population size. Under this measure, we provide the law of the splitting times of the spines and the offspring distribution of particles on and off the spines.

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###### 24/04 (remoto)

We present an estimator of the covariance matrix of a random d-dimensional vector from an i.i.d. finite sample. Our only assumption is that this vector satisfies a bounded L^p-L^2 marginal moment for p greater or equal than 4, and we allow an adversary to modify arbitrary a fraction of the sample. Given this, we show that the covariance can be estimated with the same high-probability error rates that the sample covariance matrix achieves in the case of Gaussian data. This talk is based on a joint work with Roberto I. Oliveira (IMPA).

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###### 10/04 (presencial)

Let V be the vertex set of the regular tree of degree d + 1. We consider an oriented, dependent, and long-range bond percolation model. In this model, the underlying graph is the complete and oriented graph in which its vertex set is V. We derive a new upper bound for the critical probability of this model, and as a consequence, we retrieve and improve previous bounds for the critical probability of some particular models in the literature.

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###### 27/03 (presencial)

We study noise sensitivity of the consensus opinion of the voter model on finite graphs with respect to noise affecting the initial opinions and noise affecting the dynamics. We prove that the final opinion is stable with respect to small perturbations of the initial configuration and is sensitive to perturbations of the dynamics governing the evolution of the process. This talk is based on a joint work with G. Amir, O. Angel, and R. Peretz.

###### 13/03 (presencial)

In the literature on directed polymers in random environments, two notions of strong disorder emerged: the first one is based on whether or not the partition function converges to zero and is known as strong disorder. The second depends on whether the Lyapunov exponent associated with the partition function is negative or not and is known as very strong disorder. Each of these notions corresponds to a critical temperature. Very strong disorder implies strong disorder and a conjecture for directed polymers is that these two critical points coincide. However, we will present special cases for which there is not a very strong disorder phase, while strong disorder is maintained at sufficiently low temperature. This shows that the conjecture cannot be held to complete generality.

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