Seminario Probabilidades CMM

Limiting distributions of Spherical and Spin O(N) models: Appearance of GFF.

Event Date: Jun 12, 2024 in Seminario de Probabilidades de Chile, Seminars

Resumen:   Spherical model is a mathematical model of a ferromagnet introduced by Berlin and Kac in 1952 as a rough but analytically convenient modification of the Ising model. Since its inception it has enjoyed considerable popularity among the mathematicians and physicists as an exactly soluble model exhibiting a phase transition. In this talk we will explain its relation to the Gaussian free field in the infinite volume limit and to the spin O(N) model in the infinite spin-dimensionality limit of the latter.

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On Wu’s Inequality and the Poisson-Föllmer Process.

Event Date: May 15, 2024 in Seminario de Probabilidades de Chile, Seminars

Resumen:  In the discrete setting the Poisson distribution is a ubiquitous object, as the Gaussian distribution is in the Euclidean setting. In spite of that, it does not satisfy Gross’ log-Sobolev inequality. Nevertheless, Bobkov and Ledoux were able to prove that it satisfies a “modified” version of it, which was subsequently reinforced by Wu. In the first part of this talk we will exhibit a new stochastic proof of Wu’s modified log-Sobolev, via an entropy-minimizing process constructed by Klartag and Lehec, which we call the Poisson-Föllmer process. We will also see how this stochastic...

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On Bernoulli Trials with unequal harmonic success probability.

Event Date: Apr 24, 2024 in Seminario de Probabilidades de Chile, Seminars

Resumen:  A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some if its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, parameter estimation, and relations to Sibuya distributions and Yule–Simon distributions are briefly discussed. Stirling numbers play a key role in the analysis. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and...

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Sharp large deviation principles for descents of random permutations.

Event Date: Apr 17, 2024 in Seminario de Probabilidades de Chile, Seminars

Resumen:   A permutation $\pi_n$ of size $n$ is said to have a descent at position $k$ if $\pi_n(k) > \pi_n(k + 1)$. We define $D_n$ as the number of descents of $\pi_n$, and $D_n’$ as the number of descents of $\pi_n^{-1}$, the inverse permutation of $\pi_n$. In this talk, I will present recent developments regarding the sharp large deviations for $D_n$ and for the pair $(D_n, D_n’)$, along with other probabilistic results when $\pi_n$ is taken uniformly at random from $S_n$, the set of permutations of size $n$. Work in collaboration with B. Bercu, M. Bonnefont, and A....

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Optimal methods for combining discrete p-values.

Event Date: Apr 03, 2024 in Seminario de Probabilidades de Chile, Seminars

Abstract: Combining the significance of multiple experiments regarding the same scientific hypothesis is a crucial method for global hypothesis testing, with applications in meta-analysis, signal detection, and other data-integrative studies.  Such procedures consider a group of p-values, $P_1, \cdots, P_n$, to form a summary statistic to determine the overall evidence against a global null hypothesis. Mathematical studies often assume that the underlying statistics are continuous and independent, due to their homogeneous and straightforward mathematical structure. In reality, however, data...

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Wandering Architects: The Dual Perspectives of Self-Creating Random Walks.

Event Date: Mar 20, 2024 in Seminario de Probabilidades de Chile, Seminars

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