Daniel Perales (Texas A&M University)

Title: Finite free probability

Abstract: Finite free additive and multiplicative convolutions, which are binary operations of polynomials that behave well with respect to the roots. These operations have gained interest in recent years due to its interpretation as expected characteristic polynomials of random matrix operations and their connection to free probability, geometry of polynomials, representation theory and combinatorics.

In the first talk, we will introduce the additive and multiplicative convolutions and we will discuss in detail their basic properties:

interpretation in terms of differential operators, interpretation as characteristic polynomials, bilinearity, commutativity, associativity, identities, inverses, location of the roots, interlacing and root separation. 

In the second talk, we will discuss some important families of real-rooted polynomials, such as Hermite, Laguerre, Jacobi, and hypergeometric polynomials. We will define finite free cumulants and discuss some limit theorems in the finite free world, such as LLN, CLT and Poisson limit theorem. We will briefly mention some related convolutions of polynomials. 

For the last talk, we will let the degree of the polynomials tend to infinity and explore the connection to free probability in the limit. We will explain why the finite free convolutions tend to the free convolutions, why differentiation of polynomials tends to free fractional convolution, and some other limit theorems.

Víctor Perez Abreu (UVEG)

Title: Sobre la historia del SIMA (En Memoria de Mario Alberto Diaz Torres)

Abstract: En esta charla recordaré varios aspectos, razones, hechos, dificultades y personas que han

sido fundamentales para la existencia y el impacto del Seminario Interinstitucional de Matrices

Aleatorias, esta vez en su décima edición.

Natalia Cardona (Universidad Nacional de Colombia)

Title: Yaglom’s limit for Galton-Watson processes in varying environment 

Abstract: A Galton–Watson process in a varying environment is a discrete-time branching process in which the offspring distributions vary across generations. In this talk, we will discuss the Yaglom-type limit for such a family of processes. The result states that, in the critical regime, a suitable normalization of the process, conditioned on non-extinction, converges in distribution to a standard exponential random variable. Furthermore, we will discuss the rate of convergence of the Yaglom limit with respect to the Wasserstein metric. The proof is based on a one-spine decomposition technique and Stein's method for exponential approximations. This talk is based on joint work with Arturo Jaramillo (CIMAT) and Sandra Palau (UNAM).

Osvaldo Angtuncio Hernandez (CIMAT)

Title: Convergence of the Aldous-Broder Markov chain on high-dimensional graphs.

Abstract: The continuum random tree is the scaling limit of the uniform spanning tree on the complete graph with $N$ vertices. The Aldous-Broder Markov chain on a graph $G=(V,E)$ is a Markov chain with values in the space of rooted trees whose vertex set is a subset of $V$ with the uniform distribution on the space of rooted trees spanning $G$. In Evans, Pitman and Winter (2006) the so-called root growth with regrafting process (RGRG) was constructed. Further it was shown that the suitable rescaled Aldous-Broder Markov chain converges to the RGRG weakly with respect to the Gromov-Hausdorff topology. It was shown in Peres and Revelle (2005) that (up to a dimension depending constant factor) the continuum random tree is with respect to the Gromov-weak topology the scaling limit of the uniform spanning tree on the torus $\mathbb{Z}^d_N$, for $d\geq 5$. In the present talk we discuss that the rescaled Aldous-Broder Markov chain on high-dimensional graphs (in particular, $\mathbb{Z}^d_N$, $d\geq 5$), converges to the RGRG weakly with respect to the Gromov-Hausdorff topology when initially started in the trivial rooted tree. 

Juan Carlos Pardo Millán (CIMAT)

Title: On the speed of coming down from infinity for subcritical branching processes with pairwise interactions

Abstract: Abstract In this talk, we investigate the phenomenon of coming-down from infinity for (sub)critical cooperative branching processes with pairwise interactions under suit- able conditions. A process in this class behaves as a pure branching process with the difference that competition and cooperation events between pairs of individuals are also allowed. In particular, we are interested in the speed of BPI-processes when their initial population is very large, as well as in their second order fluctuations. This is a joint work with Gabriel Berzunza. 

Jorge Garza Vargas (CalTech)

Title: Una demostración simple del teorema de Friedman con corolarios fuertes 

Abstract:El célebre resultado de Friedman (2004) establece que, a medida que el número de vértices tiende a infinito, las gráficas aleatorias d-regulares  son (con probabilidad  alta) expansores casi óptimos, lo que significa que el mayor eigenvalor no trivial de su matriz de adyacencia converge en probabilidad a 2 sqrt(d-1). Dado que los expansores son de gran interés en matemáticas y ciencias de la computación, la prueba de Friedman (que tomó 100 páginas) ha atraído mucha atención en las últimas dos décadas y ha sido mejorada y generalizada. Sin embargo, todos los enfoques al teorema de Friedman y sus extensiones se basaron en consideraciones combinatorias muy delicadas y sofisticadas, lo que dificulta aplicar dichas ideas a otros contextos de interés.

En esta charla discutiré un enfoque fundamentalmente nuevo (analítico) al teorema de Friedman que proporciona una demostración elemental que se puede escribir en un par de páginas. Nuestra técnica nos permite también establecer convergencia fuerte (es decir, estimaciones precisas de norma) para modelos mucho más generales de tuplas de matrices aleatorias (gráficas regulares aleatorias corresponden al caso particular en donde se suman permutaciones aleatorias independientes). Estos resultados pueden utilizarse para demostrar que ciertos objetos infinitos admiten aproximaciones finitas precisas, lo que tiene importantes implicaciones en álgebras de operadores, geometría espectral y geometría diferencial.

Este es trabajo conjunto con Chi-Fang Chen, Joel Tropp y Ramon van Handel.  

James Melbourne (CIMAT)

Title:Sobre los mecanismos óptimos en la privacidad diferencial:

Abstract: La privacidad diferencial es un marco matemático para divulgar información estadística sobre conjuntos de datos y, al mismo tiempo, proteger la privacidad de individuos que tienen su informacion dentro de esos conjuntos de datos.Una técnica estándar en la aplicación es devolver una versión ruidosa de una respuesta a una consulta en lugar de la respuesta verdadera. Sin embargo, existe una "tradeoff": un mayor ruido conduce a una mayor privacidad, pero a respuestas menos informativas. En 2015, Geng et al. demostraron que en las dimensiones 1 y 2, en un sentido que se precisará durante la charla, un  "mecanismo de escalera" es óptimo y conjeturaron que este resultado se mantenía en una dimensión arbitraria. En esta charla confirmamos esta conjetura. Sin embargo, también analizaremos los problemas prácticos con esta "solución óptima", que motivan una noción de "privacidad robusta a la sensibilidad" bajo la cual demostramos que la distribución de Laplace se comporta de manera óptima.

Este es un trabajo conjunto con Mario Diaz y Shahab Asoodeh.

Julian Zazueta Obeso (CIMAT)

Title: Multivariate moments of the Markov-Krein Transform 

Abstract: The Markov-Krein transform is a bijective transformation that links probability measures with certain signed measures on the real line. Its origins trace back to fundamental work in classical probability, and it was first introduced in 1997 by Sergei Kerov.

 Recently, this transform has emerged in areas related to noncommutative probability and random matrices, and these developments have led to an extension of the transform to a multivariate version. In this talk, we will provide an overview of what is known about the MarkovKrein transform in its univariate version and explain how the multivariate version has been connected to non-commutative probability, particularly highlighting the relationships between multivariate moments and the different types of cumulants: classical, boolean, and free. 

Saylé Sigarreta Ricardo (BUAP)

Title: Third-Order Cumulants of Products and Their Implications

Abstract: Free independence, introduced by Voiculescu, extends the classical notion of independence to a broader algebraic framework, with free cumulants serving as multilinear objects that describe this concept. Krawczyk and Speicher addressed the problem of computing cumulants of products in terms of individual cumulants.

The theory of higher-order freeness, an extension of Voiculescu’s Free Probability Theory, emerged from studying large random matrices, generalizing properties of first-order cumulants. Under this line, Mingo, Speicher, and Tan later computed second-order cumulants of products. In this work, we extend the previous result to third-order cumulants, including applications to aa∗ when a is a third-order R-diagonal operator.

Samuel Gurrola (CIMAT)

Title: Bound for the energy of graphs in terms of degrees and leaves.

Abstract: We establisha new bound for the energy of graphs in terms of its degrees and number of leaves joined to each vertex. In addition, we review some application in random graph models. Particularly, we will talk about the Barabasi-Albert model, the Erdos-Renyi model and their behavior in the limit when the number of nodes tend to infinity.

Sheng Yin (Baylor University)

Title: Absolute continuity of operator-valued random variables

Recently, the regularity of noncommutative random variables has been studied in many aspects. In this talk, we will discuss one of these regularity problems that addresses the absolute continuity of the density. We will consider operator-valued random variables that have dual operators relative to a completely positive map. A particular interesting example is given by the matrix-valued case in which we can draw more conclusions. This is based on an ongoing project with Tobias Mai.

Katsunori Fujie (Kyoto University)

Title: Principal minor of random matrix, Markov--Krein correspondence, and type B' free probability. 

Abstract: The connection between random matrix theory and free probability increases more and more.

For example, it is well known that an independent family of random matrices becomes freely independent as the dimension goes to infinity.

We consider the principal minors of such convergent random matrices with invariant distribution under unitary conjugation.

Their asymptotic picture can be obtained by using the Markov--Krein correspondence.

In the latter part of the talk, we introduce type B' free probability, which captures the difference between the identity matrix and projection with codimension 1, and hence, it is useful to describe the principal minors.

As an application, in that setting, MK correspondence describes their infinitesimal additive and multiplicative convolutions.

This talk is based on joint works with Takahiro Hasebe (Hokkaido University). 

Chiara Amorino (Universitat Pompeu Fabra)

Title: Polynomial rates via deconvolution for nonparametric estimation in McKean-Vlasov SDEs

Abstract: This paper investigates the estimation of the interaction function for a class of McKean-Vlasov stochastic differential equations. The estimation is based on observations of the associated particle system at time $T$, considering the scenario where both the time horizon $T$ and the number of particles $N$ tend to infinity. Our proposed method recovers polynomial rates of convergence for the resulting estimator. This is achieved under the assumption of exponentially decaying tails for the interaction function. Additionally, we conduct a thorough analysis of the transform of the associated invariant density as a complex function, providing essential insights for our main results.

Arturo Jaramillo (CIMAT)

Título: Método de Stein libre y el teorema de Berry-Esseen.

Resumen: Se presentará una formulación del método Stein para la distribución semicircular diseñado para el estudio de variables no conmutativas. Como aplicación fundamental, damos una cuantificación de la semicircularidad asintótica para sumas de variables débilmente dependientes, medidos en términos de la distancia en variación total, con mejora de decaimiento en presencia de coincidencia de momentos de orden arbitrario.