Séminaire Calcul stochastique
organisé par l'équipe Probabilités
Coalescence des géodésiques en percolation de premier passage
17 janvier 2023 - 10:45Salle de séminaires IRMAConsidérons le modèle de percolation de premier passage sur Z^2: pour chaque arête, on associe indépendamment une variable aléatoire à valeur dans R_+ qui représente le temps pour traverser cette arête. On s'intéresse alors à la métrique aléatoire où le temps entre deux points correspond au temps du plus court chemin. Plus précisément, nous nous intéressons aux propriété de coalescence des géodésiques. Sous des hypothèses sur la loi des temps, nous prouvons que les géodésiques avec des extrémités proches ont une intersection significative. Nous verrons également le lien avec le problème du point milieu de BKS. Travail en commun avec Dor Elboim et Ron Peled.
Gaussian fields with logarithmically decaying correlations, such as branching Brownian motion and the two-dimensional Gaussian free field, are conjectured to form universality class of extreme value statistics (notably in the work of Carpentier & Le Doussal and Fyodorov & Bouchaud). This class is the borderline case between the class of IID random variables, and models where correlations start to affect the statistics. In this talk, I will describe a general approach based on rigorous works in spin glass theory to describe features of the Gibbs measure of these Gaussian fields. I will focus on the two-dimensional discrete Gaussian free field. At low temperature, we show that the normalized covariance of two points sampled from the Gibbs measure is either 0 or 1. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. (joint work with L.-P. Arguin, 2015). In a second work (with M. Pain, 2021), we prove absence of temperature chaos for the two-dimensional discrete Gaussian free field using the convergence of the full extremal process, which has been obtained by Biskup and Louidor. This means that the overlap of two points chosen under Gibbs measures at different temperatures has a nontrivial distribution. Whereas this distribution is the same as for the random energy model when the two points are sampled at the same temperature, we point out here that they are different when temperatures are distinct: more precisely, we prove that the mean overlap of two points chosen under Gibbs measures at different temperatures for the DGFF is strictly smaller than the REM's one. Therefore, although neither of these models exhibits temperature chaos, one could say that the DGFF is more chaotic in temperature than the REM. Finally, I will discuss ongoing works with B. Bonnefont (PhD student) and M. Pain (CR @ Toulouse), on questions suggested by B. Derrida.
Empirical Dynamic Modeling: Automatic Causal Inference & Forecasting for Time Series
7 février 2023 - 10:45Salle de séminaires IRMAHow can social and health researchers study complex dynamic systems that function in nonlinear and even chaotic ways? Common methods, such as experiments and equation-based models, may be ill-suited to this task. To address the limitations of existing methods and offer nonparametric tools for characterising and testing causality in nonlinear dynamic systems, we created empirical dynamic modeling (EDM) packages for Stata, R, and Python. The packages implement the key EDM methods for time series and panel data. In particular, it implements convergent cross-mapping, which offers a nonparametric approach to modeling causal effects. We can observe these algorithms in action on simulated data, and on real daily Chicago temperature and crime, showing an effect of temperature on crime but not the reverse. This is joint work with Jinjing Li, Michael Zyphur, and George Sugihara.
28 février 2023 - 10:45Salle de séminaires IRMAThe lighthouse result of mathematical finance is the Fundamental Theorem of Asset Pricing (FTAP) stating the equivalence between the no-arbitrage possibil- ity in a finacial market and the existence of an equivalent martingale measure. When it comes to insurance business linked to the financial market, two new aspects arise: First, insurance contracts are not traded in the market and, even more important: an insurance has a different information level than the market. Mathematically speaking, we are faced with (at least) two filtrations. By defining strategies on an insurance portfolio and combining them with financial trading strategies, we arrive at the notion of insurance-finance arbitrage (IFA) and give a fundamental theorem on the absence of IFA, leading to the existence of an insurance-finance-consistent probability. Joint work with Ph. Artzner and Th. Schmidt
RSK dynamics, TASEP, and the KPZ fixed point
21 mars 2023 - 10:45Salle de séminaires IRMAThe KPZ fixed point, constructed by Matetski-Quastel-Remenik, is a scaling invariant Markov process that is believed to be the universal scaling limit of a large family of models, known to form the Kardar-Parisi-Zhang universality class, which describes random interface growth in 1+1 dimensions. In this talk, I will discuss a new way of exactly solving (a discrete-time version of) the totally asymmetric simple exclusion process, a prototypical discrete model in the KPZ universality class. It is based on a classical combinatorial bijection known as the Robinson-Schensted-Knuth correspondence and standard non-intersecting path constructions. This allows a more systematic derivation for the transition probability formula compared to the original work of MQR and also leads to natural generalizations with particle and time inhomogeneity. Time permitting I will also discuss how to obtain the KPZ fixed point as a 1:2:3 scaling limit of TASEP and possible generalizations when there is spatial or temporal inhomogeneity. The talk is based on joint work with Elia Bisi, Axel Saenz and Nikos Zygouras.