Particle systems and propagation of chaos for some kinetic models
In this talk we will make a quick historical review of some equations arising in the classical kinetic theory of gases and related models. We will start with the Boltzmann equation, which describes the evolution of the distribution of positions and velocities of infinitely many small particles of a gas in 3-dimensional space, subjected to elastic binary collisions. We consider a finite $N$-particle system and introduce the important concept of propagation of chaos: the convergence, as $N\to\infty$ and for each time $t\geq 0$, of the distribution of the particles towards the solution of the equation. We present some recent quantitative propagation of chaos results for the spatially homogeneous Boltzmann equation and Kac’s model. Lastly, we will introduce a relatively new class of one-dimensional kinetic equations modelling wealth redistribution in a population performing binary trades. When trades preserve wealth only on average, these models can exhibit an equilibrium distribution with heavy tails, as is seen in real-world economies. We focus on the corresponding finite $N$-particle system and study how the heaviness of the tails of its distribution relates to that of the limit kinetic equation. Unless wealth is preserved exactly in each trade, we find important qualitative differences between both cases.
A non-trivial bound for the critical threshold of a percolation model with columnar disorder.
A random perturbation with a tunable parameter in a regular media can sometimes produce a disruption on macroscopic observables. In this talk, we study a percolation model on a random lattice that features such a disruption. More specifically, we show that the Bernoulli percolation in a 3D lattice with columns randomly deleted has a uniform non-trivial threshold. In the course of the proof, we have to understand some geometrical properties of the supercritical 2D cluster. Joint work with M.V Sá and M.R. Hilário.
Two-time distribution for KPZ growth in one dimension
Consider the height fluctuations H(x,t) at spatial point x and time t of one-dimensional growth models in the Kardar-Parisi-Zhang (KPZ) class. The spatial point process at a single time is known to converge at large time to the Airy processes (depending on the initial data). The multi-time process however is less well understood. In this talk, I will discuss the result by Johansson on the two-time problem, namely the joint distribution of (H(x,t),H(x,at)) with a>0, in the case of droplet initial data. I also show how to adapt his approach to the flat initial case. This is based on joint work with Kurt Johansson.
Sharp Bounds for the Reliability of a k-out-of-n Sys- tem Under Dependent Failures Using Cutoff Phenomenon Techniques
In this work we consider the reliability of a network where link failures are correlated. We define the reliability as the probability of the network to be working at a given time instant. Our main contribution is a collection of results giving a detailed analysis of a non-trivial scaling regime for the probability of the network being working at a certain time, as the time and size of network scales. Here we consider that the network fails when there are no links working, or more generally when less than k-out-of-n edges are working (with k close to n) like in [2] and [5]. Our results allow to study the common-cause failure models describe in [3] on networks in a realistic, relevant, yet practical, fashion: it allows to capture correlated components in the network; it allows to estimate and give error bounds for the failure probabilities of the system; and at same time only needs to specify a reduced family of parameters. Moreover, our results for the k-out-of-n failure model allow to give new scaling regimes for the probabilistic behavior of the last-ordinals in the theory of extreme values for dependent tuples. The techniques are similar to those used to estimate the asymptotic convergence profile for ergodic Markov chains [1] or [4].
[1] J. Barrera and B. Ycart Bounds for Left and Right Window Cutoffs. ALEA, Lat. Am. J. Probab. Math. Stat., 11 (2): 445–458, 2014.
[2] I. Bayramoglu and M. Ozkut. The Reliability of Coherent Systems Subjected to Marshall?Olkin Type Shocks. EEE Transactions on Reliability,, 64 (1): 435–443, 2015.
[3] U. Cherubini, F. Durante, and S. Mulinacci. Marshall – Olkin Distributions-Advances in Theory and Applications: Bologna, Italy, October 2013, volume 141. Springer, 2015.
[4] B. Lachaud and B. Ycart Convergence Times for Parallel Markov Chains. Positive systems, 169–176, 2006.
[5] T. Yuge, M. Maruyama, and S. Yanagi Reliability of a k-out-of-n Systemwith Common-Cause Failures Using Multivariate Exponential Distribution. Procedia Computer Sci- ence, 96: 968?976, 2016.
Noise sensitivity for Voronoi Percolation
Pontificia Universidad Católica de Chile (PUC-Chile)
Av. Vicuña Mackenna 4860, Macul,
Santiago – Chile
(+56 2) 2354 5779
Facultad de Ciencias Físicas y Matemáticas (FCFM)
Universidad de Chile
Beauchef 851, Edificio Norte, Piso 7,
Santiago – Chile