Mini-Courses

Oren Louidor, Technion

Extreme value theory for logarithmically correlated fields

Random fields whose correlations decay (approximately) logarithmically in the distance between two test points, are shown increasingly prevalent in numerous physical and mathematical contexts. One active line of research in this area, is the study of statistical features of the extreme (near maximum/minimum) values of such fields. Aside a from pure mathematical significance (including an emerging universality picture), extreme values govern much of phenomena observed in applications where such fields are encountered.

In this mini-course, I intend to (time-permitting): 1. Define such fields, discuss their fundamental properties, and survey some of the canonical examples, 2. Review recent progress in the emerging extreme value theory, 3. Highlight key and common arguments in the proofs of the latter, and 4. Discuss various applications of the theory.

Key objects which will appear frequently in the presentation include the branching random walk, branching Brownian motion and the Gaussian free field.

No preliminary background is needed except for a graduate level knowledge of probability theory.

Leonardo Rolla, University of São Paulo

An introduction to the contact process

Dieter Mitsche, Universidad Católica de Chile

• uniqueness of the infinite cluster in supercritical bond percolation
• box-crossing property and FKG-inequality in bond percolation
• Russo’s formula and its application to prove the criticality threshold of bond-percolation
• renormalization technique to study the existence of an infinite-cluster in long-range percolation in Z

Literature:

• 1. Vincent Tassion. “Planarity and locality in percolation theory”, PhD thesis, ENS Lyon, 2014.
• 2. Hugo Duminil-Copin. “Introduction to Bernoulli percolation”, Lecture notes available at https://www.ihes.fr/~duminil/publi/2017percolation.pdf
• 3. H. Duminil-Copin, C. Garban, V. Tassion. “Long-range percolation in 1D revisited”. Available at https://arxiv.org/abs/2011.04642.

Talks

Tamara Fernández, Universidad Adolfo Ibáñez

A general framework for the analysis of kernel-based tests

Kernel-based tests provide a simple yet effective framework that utilizes the theory of reproducing kernel Hilbert spaces to design non-parametric testing procedures. In this presentation, we introduce new theoretical tools for studying the asymptotic behavior of kernel-based tests across various data scenarios and testing problems. Unlike current approaches, our methods circumvent the use of U and V-statistics expansions, which often result in lengthy and tedious computations and asymptotic approximations, especially in complex scenarios. Instead, we work directly with random functionals on the Hilbert space to analyse kernel-based tests. By harnessing the use of random functionals, our framework leads to a much cleaner analysis, involving less tedious computations. Additionally, it offers the advantage of accommodating pre-existing knowledge regarding test-statistics as many of the random functionals considered in applications are known statistics that have been studied thoroughly.

Pablo López-Rivera, Université Paris Cité

Preservation of functional inequalities under log-Lipschitz perturbations

Given a probability measure satisfying some functional inequalities (Poincaré, log-Sobolev, etc.), it is natural to wonder if these remain valid for a perturbation of the measure. In particular, if there exists a globally Lipschitz map pushing forward the source measure towards its perturbation, then it is easy to transport certain functional inequalities. For example, Caffarelli’s contraction theorem states that the optimal transport map between the Gaussian measure and a log-concave perturbation is 1-Lipschitz.

In this talk I will show how such a map exists if we consider log-Lipschitz perturbations of a measure on a Riemannian manifold, via the interpolation given by the Langevin diffusion associated to the source measure (aka Kim-Milman’s heat flow transport map), assuming as well control on the curvature of the manifold at first and second order in the sense of Bakry-Émery-Ricci. No prerequisites in geometry are needed.

Gonzalo Panizo, IMCA

Random Interlacements of directed paths

We present the model of Random Interlacements and a variation of it where the simple symmetric random walk is replaced by a directed random walk. We talk about the existence and percolation problems of this model and its relationship with the so-called escape problems.

Jaime San Martín, Universidad de Chile

Some New and Classical results for Schrödinger operators in dimension 1

We will see classical and new results for Schrödinger operators in dimension 1, in the context of Branching Diffusions. We will give asymptotic for the mean number of  particles surviving to time t and its limiting conditional Laplace transform.

Leonardo Videla, Universidad de Santiago

Simplified Wright-Fisher processes and some non-linear extensions

We propose a family of Markov kernels on the unit interval which, after rescaling, are proved to converge in law to a large family of diffusion processes of the Wright-Fisher type. Discrete-time line-counting processes (the ancestral lineage processes) are obtained, and we prove moment duality formulae for the discrete-time processes. This unified, parsimonious derivation of some well-known population genetic processes allows one to easily extend the dynamics to the case where mutations and selection rates depend on the instantaneous distribution of the process. In this connection we prove both, propagation of chaos for the associated particle system in mean-field regime, and the convergence of the rescaled discrete-time Wright-Fisher conglomerate to diffusions of the McKean-Vlasov type.

Joint work with Héctor Olivero (Universidad de Valparaíso), Fernando Cordero (Universität Bielefeld) and Christian Jorquera (Universidad de Santiago de Chile) .

Tianqi Wu, Technion

Diameter of a long-range percolation graph for the critical exponent s=d

Many real-world networks exhibit the small-world phenomenon: their typical distances are much smaller than their sizes. A natural way to model this phenomenon is a long-range percolation graph on the lattice $Z^d$, in which edges are added between far-away vertices with probability falling off to the $s$-th power of the Euclidean distance. How does the resulting graph distance scale with the Euclidean distance? The question has been intensely studied in the past and the answer depends on the exponent $s$ in the connection probabilities, for which five regimes of behavior have been identified. In this talk I will give an overview on past results and then discuss our current understanding of the critical regime $s=d$.