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We consider a class of general SDEs with a jump integral term  driven by a time-inhomogeneous random Poisson measure. We propose a Euler-type scheme for this SDE class and prove an optimal rate of convergence for the L^p. One of the primary issues to address in this context is the approximation of the noise structure when it can no longer be expressed as the increment of random variables. We extend the Asmussen-Rosinski approach to the case of a three-parameters jump coefficient and time-dependent Lévy measure, handling contribution of jumps smaller than epsilon with an appropriate Gaussian increment and exact simulation for the large  jumps contribution. For any p <= 2, with hypothesis to control the L^p-moments of the process, we obtain a convergence rate of order 1/p confirmed by numerical experiments. We apply this model class for some anomalous diffusion model related to the dynamics of rigid fibres in turbulence.

Video 

We consider a class of general SDEs with a jump integral term  driven by a time-inhomogeneous random Poisson measure. We propose a Euler-type scheme for this SDE class and prove an optimal rate of convergence for the L^p. One of the primary issues to address in this context is the approximation of the noise structure when it can no longer be expressed as the increment of random variables. We extend the Asmussen-Rosinski approach to the case of a three-parameters jump coefficient and time-dependent Lévy measure, handling contribution of jumps smaller than epsilon with an appropriate Gaussian increment and exact simulation for the large  jumps contribution. For any p <= 2, with hypothesis to control the L^p-moments of the process, we obtain a convergence rate of order 1/p confirmed by numerical experiments. We apply this model class for some anomalous diffusion model related to the dynamics of rigid fibres in turbulence.

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We consider a one-dimensional discrete Dirac operator in a potential given by a family of i.i.d. random variables modulated by a decreasing envelope. In [1], we showed that these models exhibit a rich phase diagram in terms of their spectrum as a function of the rate of decay of the random potential, where the spectrum of the operator

• is absolutely continuous for fast decay,

• is pure point for slow decay,

• presents a spectral transition for critical decay.

We show dynamical localization in the sub-critical region by means of the fractional moments method and provide control on the eigenfunctions. We studied spectral statistics in [2], we show that, in the fast decay case, the rescaled spectrum of the operator converges to the clock process while for critical decay, it converges to the Schrödinger point process. In this way, we recovered the results of Kritchevski, Valkó and Virag established for the Anderson model. Our proof is based on the scaling limit of the Prüfer transform associated with the system and uses the monotonicity to deduce the convergence. In the slow decay, the spectral statistics are expected

to be given by a Poisson process. 

[1] O. Bourget, G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics, Mathematical Physics, Analysis

and Geometry, Volume 23, Article number 20, 2020.

[2] G. R. Moreno Flores, A. Taarabt, One-dimensional Discrete Dirac Operators in a Decaying Random Potential II: Clock, Schrödinger and Sine statistics, submitted.

Calculamos la función de correlación de 2-puntos del crecimiento de la ecuación KPZ que empieza con un Browniano en la línea recta como condición inicial. Usando herramientas básicas de cálculo de Malliavin podemos calcular esta función en términos de la distribución annealed del punto final de un polímero aleatorio asociada a la ecuación de calor estocástica. También mostramos cotas superiores para tener independencia asintótica entre la condición inicial y la evolución al tiempo t. Esta charla está basada en un trabajo conjunto con  Leandro Pimentel (Universidade Federal do Rio de Janeiro 

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The bisexual Galton-Watson process [Daley, '68] is an extension of the classical Galton-Watson process, but taking into account the mating of females and males, which form couples that can accomplish reproduction. Properties such as extinction conditions and asymptotic behaviour have been studied in the past years, but multi-type versions have only been treated in some particular cases. In this work we deal with a general multi-dimensional version of Daley’s model, where we consider different types of females and males, which mate according to a "mating function". We consider that this function is superadditive, which in simple words implies that two groups of females and males will form a larger number of couples together rather than separate. One of the main difficulties in the study of this process is the absence of a linear operator that is the key to understand its behavior in the asexual case, but in our case it turns out to be only concave. To overcome this issue, we use a concave Perron-Frobenius theory [Krause, '94] which ensures the existence of eigen-elements for some concave operators. Using this tool, we find a necessary and sufficient condition for almost sure extinction as well as a law of large numbers. Finally, we study the convergence of the process in the long-time through the identification of a supermartingale.

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Consideramos soluciones de la ecuación planar estocástica del calor interpretadas por medio de la integral de Skorokhod en la representación integral de Duhamel, ya estudiadas hasta el tiempo crítico dado por la mejor constante de la desigualdad de Gagliardo-Nirenberg-Sobolev por Nualart-Zakai (1989) y Hu (2002). Extendemos esta solución más allá del tiempo crítico por aproximaciones de Fourier convergentes en . Además, probamos que las fluctuaciones lejos del centro están dadas por la ecuación del calor estocástica en dimensión d=1. 

Esta charla se basada en un trabajo conjunto con Jeremy Quastel y Balint Virag.

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El modelo de marchas aleatorias activadas (Activated Random Walks, ARW) es un sistema de partículas introducido con el fin de estudiar la criticalidad auto-organizada. Este consiste en una familia de partículas que se mueven aleatoriamente y que pueden dormirse espontáneamente, lo que detiene el movimiento. La interacción ocurre cuando una partícula activa visita a una durmiendo, lo que gatilla que la partícula dormida se active y continúe su movimiento. Este modelo presenta una transición de fase en términos de la densidad de partículas. Si hay muchas partículas la actividad en el sistema continúa de manera indefinida, mientras que si la densidad es baja la proporción de partículas activas decae a 0. En esta charla estudiaremos ARW's en dos casos límite que, si bien simplifican el modelo, a la vez permiten establecer resultados rigurosos en el punto crítico.

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Consideramos un modelo de polímeros dirigidos en ambiente aleatorio a valores complejos sobre el árbol, introducido por Cook y Derrida y posteriormente estudiado por Derrida, Evans y Speer. El diagrama de fases del modelo presenta tres regiones, una de las cuales es causada por el efecto de las fases aleatorias y no se encuentra en el modelo con ambientes a valores positivos.

En este trabajo en colaboración con L. Medina Espinosa (UC), extendemos los resultados de Derrida, Evans y Speer a ambientes más generales y obtenemos una convergencia más fuerte hacia la energía libre. 

 Esta charla concierne la aproximación de juegos de campo medio de primer orden, o deterministas, introducidos por J.-M. Lasry y P.-L. Lions en el año 2007. Luego de introducir este tipo de juegos, nos concentraremos en la aproximación de la función valor de un jugador típico, elemento clave de la discretización del juego de campo medio. Esta última puede interpretarse como un juego de campo medio en tiempo discreto y espacio de estados finitos introducido por Gomes, Mohr y Souza en el año 2010. Luego de enunciar el teorema de convergencia principal, terminaremos la charla con algunos ejemplos numéricos. 

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We consider particles in the torus, subject to binary, smooth interactions and Brownian perturbation, in the mean-field setting. Recently, Delarue and Tse (2021) obtained, using the master equation, a new proof of uniform in time propagation of chaos for this model. We will present how the combination of those techniques, and of the Glauber calculus introduced by Duerinckx (2021) to treat deterministic particle systems, allows one to control, uniformly in time, the cumulants of this system. We recover the expected order for the N-particle correlation functions, in some weak norm, which allows us to correct the mean-field limit and describe the fluctuations around it.

Joint work with Mitia Duerinckx (ULB-FNRS)

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We revisit the well known duality relation between integer valued height functions and spin systems with O(2) symmetry. 

We will apply some simple corollaries from this duality to transfer information from one model to the other.

First, we deduce a type of Gaussian domination for the height-function variance. 

Second, we deduce the monotonicity of this variance with respect to a natural temperature parameter. 

Finally, we show how ``delocalisation'' of planar height functions implies a so-called BKT phase transition in the dual (planar) spin models. 

This is based on joint work with Marcin Lis https://arxiv.org/abs/2303.08596 and perhaps https://link.springer.com/article/10.1007/s00220-022-04550-3.

 En esta charla, estudiaremos métodos de suavizamiento para estimar funciones no-paramétricas. Veremos cómo elegir los parámetros de suavizamiento con el objetivo de tener estimadores óptimos. Más específicamente, introduciremos el método de Goldenshluger-Lepski (2011) que permite obtener estimadores que se adaptan a la regularidad de la función. Mostraremos cómo extender este método en el caso de la regresión funcional donde la variable regresora es un proceso de Wiener. Definiremos una familia de estimadores para la función de regresión basándose en la descomposición de Wiener Itô de la función de regresión evaluada en el proceso de Wiener. Los estimadores obtenidos satisfacen una desigualdad de oráculo, son adaptativos y alcanzan velocidades de convergencia polinomial sobre clases de función específicas. 

 Video  

 In this talk we will study the evolution of the graph distance between two fixed vertices in dynamically growing random graph models. More precisely, we consider preferential attachment models with parameters such that the asymptotic degree distribution has infinite second moment. First, we grow the graph until it contains $t$ vertices, then we sample $u_t, v_t$ uniformly at random from the largest component and study the evolution of the graph distance as the surrounding graph grows. This yields a stochastic process in $t'\ge t$ that we call the distance evolution. We identify a function $f(t,t')$ such that there exists a tight strip around this function that the distance evolution never leaves with high probability as $t$ tends to infinity.

If time permits, we will consider the generalization of graph distance to weighted distance, in which every edge is equipped with an i.i.d. copy of a non-negative random variable $L$. For any such edge-weight distribution $L$, we obtain explicit asymptotic results: either the typical weighted distance at time $t$ tends to an almost surely finite random variable as $t$ tends to infinity, or the typical weighted distance at time $t$ diverges, in which case we identify a function $f_L(t,t')$ that describes the weighted-distance evolution for $t'>t$.

Based on joint work with Julia Komjathy. 

 Video 

The talk is about two types of interacting particle systems related to the classical ensembles of random matrices. The prototypical examples are Dyson Brownian motion (also called non-intersecting Brownian motions) and Brownian TASEP (also called Brownian motions with one-sided collisions/reflections) respectively. I will discuss explicit formulae for their distributions, their correlation functions and some non-trivial connections between the two types of interacting particle systems. The bulk of the talk will be mainly focussed on surveying results for the Gaussian/Brownian case.

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Motivated by the stochastic Lotka-Volterra model, we introduce continuous-time discrete-state interacting multitype branching processes (both through intratype and intertype competition or cooperation). We show that these processes can be obtained as the sum of a multidimensional random walk with a Lamperti-type change proportional to the population size; and a multidimensional Poisson process with a time-change proportional to the pairwise interactions. We define the analogous continuous-state process as the unique strong solution of a multidimensional stochastic differential equation. Finally, we prove that a large population scaling limits of the discrete-state process correspond to its continuous counterpart. In addition, we show that the continuous-state model can be constructed as a generalized Lamperti-type transformation of multidimensional Lévy processes. Joint work with Sandra Palau (IIMAS-UNAM, México).

A transition matrix U on ℕ is said to be almost upper triangular if U(i,j)≥0⇒j≥i−1, so that the increments of the corresponding Markov chains are at least −1; a transition matrix L on ℕ is said to be almost lower triangular if L(i,j)≥0⇒j≤i+1, and then, the increments of the corresponding Markov chains are at most +1. In this talk I will characterise the recurrence, positive recurrence and invariant distribution for the class of almost triangular transition matrices. These results encompass the case of birth and death processes (BDP), which are famous Markov chains being simultaneously almost upper and almost lower triangular. Their properties were studied in 50's by Karlin & McGregor whose approach relies on some profound connections between the theory of BDP, the spectral properties of their transition matrices, the moment problem, and the theory of orthogonal polynomials. Our approach is mainly combinatorial and uses elementary algebraic methods. Joint work with J.F. Marckert. 

Video 

Multiplicative functions play an important role in Analytic Number Theory. By multiplicative we mean that whenever positive integers and are coprime. Since the prime powers are the building blocks of multiplication, we have that a multiplicative function is completely determined by its values at these powers.

An important example that will be discussed in this talk is the Möbius function which is defined to be at primes and at powers bigger than of primes. For example, while . When we consider the partial sums of the Möbius function, we can see this as an “arithmetic random walk”, and naturally we can ask about the size of the fluctuations of this walk. Just as in the simple random walk in Probability theory, we expect square-root cancellation for these sums, but it turns out that to prove this is so difficult as to establish the Riemann Hypothesis (RH). Inspired by the square-root cancellation in the i.i.d. simple random walk, this equivalence of the sums of Möbius with RH led many authors to investigate statistical properties of . An important work is that of Wintner in the 40’s. He considered a question proposed by Lévy, and investigated the model of a random multiplicative function , which is nothing more than a multiplicative function where at primes, is an i.i.d. sequence of with probability 1/2 each. Wintner proved almost sure square-root cancelation for this multiplicative random walk, and later many authors investigated this problem from different perspectives. In this talk, my plan is to present what is known in the literature about these random functions and also to present my joint contribution at different papers with Vladas Sidoravicius, and with Winston Heap and Jing Zhao. 

 We derive rigorous estimates on the speed of invasion of an advantageous trait in a spatially advancing population in the context of a system of one-dimensional coupled F-KPP equations. The model was introduced and studied heuristically and numerically in a paper by Venegas-Ortiz et al. In that paper, it was noted that the speed of invasion by the mutant trait is faster faster when the resident population ist expanding in space compared to the speed when the resident population is already present everywhere. We use probabilistic methods, in particular  the Feynman-Kac representation, to provide rigorous estimates that confirm these predictions. Based on joint work in progress with A. Bovier. 

Video  

The polynuclear growth model (PNG) is a model for crystal growth in one dimension. It is one of the most basic models in the KPZ universality class, and in the droplet geometry, it can be recast in terms of a Poissonized version of the longest increasing subsequence problem for a uniformly random permutation. In this talk, we will show how the multipoint distributions of the model can be expressed through solutions of a classical integrable system, the two-dimensional non-Abelian Toda lattice. In the appropriate scaling limit, these solutions become solutions of the KP equation, an integrable dispersive PDE which arises in a similar way for the KPZ fixed point. The proof is based on a new Fredholm determinant formula for the transition probabilities of PNG with general initial data, which is built out of hitting probabilities of a continuous-time simple random walk, the invariant measure of the process. 

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Langevin dynamics for gradient interface models are important in statistical physics due to their connection with random surfaces. It is of particular interest to understand their behavior over large-scales. In this direction a number of results have been established in the last 20 years (including the hydrodynamic limit of Funaki-Spohn and the scaling limit of Naddaf-Spencer and Giacomin-Olla-Spohn). In this talk, we will present the model, its motivations and main results. We will study a connection with the stochastic homogenization of nonlinear equations, discuss some new results that can be deduced from this approach and as well as possible extension to degenerate potentials. This is joint work with S. Armstrong.

Video

We determine the distributions of some random variables related to a simple model of an epidemic with contact tracing and cluster isolation, which is inspired by a recent work of Bansaye, Gu and Yuan. 

Notably,  we compute explicitly  the asymptotic proportion of isolated clusters with a given size amongst all isolated clusters, conditionally on survival of the epidemic.  Somewhat surprisingly, the latter differs from the distribution of the size of a typical cluster at the time of its detection; and we explain the reasons behind this seeming paradox.