Monday  Tuesday  Wednesday  Thursday  Friday  
10:0011:15  Mitsche  Rolla  Mitsche  Louidor  Mitsche 
11:1511:45  Coffee  Coffee  Coffee  Coffee  Coffee 
11:4513:00  Rolla  Louidor  Louidor  Rolla  Louidor 
13:0015:00  Lunch  Lunch  Lunch  Lunch  Lunch 
15:0016:00  San Martín  Panizo  Wu  Fernández  
16:0016:30  Coffee  Coffee  Coffee  
16:3017:30  LópezRivera  Rolla  Videla  Mitsche  
Reception  Dinner 
MiniCourses
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 minicourse, I intend to (timepermitting): 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
 boxcrossing property and FKGinequality in bond percolation
 Russo’s formula and its application to prove the criticality threshold of bondpercolation

renormalization technique to study the existence of an infinitecluster in longrange percolation in Z
Literature:
 1. Vincent Tassion. “Planarity and locality in percolation theory”, PhD thesis, ENS Lyon, 2014.
 2. Hugo DuminilCopin. “Introduction to Bernoulli percolation”, Lecture notes available at https://www.ihes.fr/~duminil/
publi/2017percolation.pdf  3. H. DuminilCopin, C. Garban, V. Tassion. “Longrange 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 kernelbased tests
Kernelbased tests provide a simple yet effective framework that utilizes the theory of reproducing kernel Hilbert spaces to design nonparametric testing procedures. In this presentation, we introduce new theoretical tools for studying the asymptotic behavior of kernelbased tests across various data scenarios and testing problems. Unlike current approaches, our methods circumvent the use of U and Vstatistics 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 kernelbased 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 preexisting knowledge regarding teststatistics as many of the random functionals considered in applications are known statistics that have been studied thoroughly.
Pablo LópezRivera, Université Paris Cité
Preservation of functional inequalities under logLipschitz perturbations
Given a probability measure satisfying some functional inequalities (Poincaré, logSobolev, 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 logconcave perturbation is 1Lipschitz.
In this talk I will show how such a map exists if we consider logLipschitz perturbations of a measure on a Riemannian manifold, via the interpolation given by the Langevin diffusion associated to the source measure (aka KimMilman’s heat flow transport map), assuming as well control on the curvature of the manifold at first and second order in the sense of BakryÉmeryRicci. 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 socalled 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 WrightFisher processes and some nonlinear 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 WrightFisher type. Discretetime linecounting processes (the ancestral lineage processes) are obtained, and we prove moment duality formulae for the discretetime processes. This unified, parsimonious derivation of some wellknown 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 meanfield regime, and the convergence of the rescaled discretetime WrightFisher conglomerate to diffusions of the McKeanVlasov 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 longrange percolation graph for the critical exponent s=d
Many realworld networks exhibit the smallworld phenomenon: their typical distances are much smaller than their sizes. A natural way to model this phenomenon is a longrange percolation graph on the lattice $Z^d$, in which edges are added between faraway 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 $s=d$.