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
Noise sensitivity for Voronoi Percolation
Harnack inequality for degenerate balanced random walks.
We consider an i.i.d. balanced environment $\omega(x,e)=\omega(x,-e)$,genuinely d dimensional on the lattice and show that there exist a
positive constant $C$ and a random radius $R(\omega)$ with streched
exponential tail such that every non negative
$\omega$ harmonic function $u$ on the ball $B_{2r}$ of radius
$2r>R(\omega)$,
we have $\max_{B_r} u <= C \min_{B_r} u$. Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martins Barlow’s Harnack’s inequality for i.i.d. bond percolation to the directed case.
This is joint work with N.Berger M. Cohen and X. Guo.
Sala 3, Facultad de Matemáticas, PUC
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