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

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.

Read More## Contour methods for -dimensional Long-Range Ising Model.

Resumen: On the -dimensional lattice with , the phase transition of the nearest-neighbor ferromagnetic Ising model can be proved by using Peierls argument, that requires a notion of contours, geometric curves on the dual of the lattice to study the spontaneous symmetry breaking. It is known that the one-dimensional nearest-neighbor ferromagnetic Ising model does not undergo a phase transition at any temperature. On the other hand, if we add a polynomially decaying long-range interaction given by for , the works by Dyson and Fr\”{o}hlich-Spencer show the phase transition at low...

Read More## Effective Mass of the Fröhlich Polaron: Recent Progress and Open Question.

Resumen: Landau and Pekar (in 1948) and Spohn (in 1987) conjectured that the effective mass $m(\alpha)$ of the Fröhlich Polaron at coupling parameter $\alpha$ grows as $\alpha^4$ as $\alpha\to\infty$ with an explicit pre-factor. In a recent joint work with C. Mukherjee, M. Sellke, and S. R. S. Varadhan, we prove the lower bound $m(\alpha) \geq C \alpha^4$, which matches (up to a constant) the corresponding sharp upper bound shown recently by combining the results from Brooks and Seiringer (2022) and Polzer (2023).

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

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.

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

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...

Read More## On Bernoulli Trials with unequal harmonic success probability.

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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