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Series: Stochastics Seminar

Random balls models are collections of Euclidean balls whose centers and radii are generated by a Poisson point process. Such collections model various contexts ranging from imaging to communication network. When the distributions driving the centers and the radii are heavy-tailed, interesting interference phenomena occurs when the model is properly zoomed-out. The talk aims to illustrate such phenomena and to give an overview of the asymptotic behavior of functionals of interest. The limits obtained include in particular stable fields, (fractional) Gaussian fields and Poissonian bridges. Related questions will also be discussed.

Series: Stochastics Seminar

A Markov intertwining relation between two Markov processes X and Y is a weak similitude relation G\Lambda = \Lambda L between their generators L and G, where \Lambda is a transition kernel between the underlying state spaces. This notion is an important tool to deduce quantitative estimates on the speed of convergence to equilibrium of X via strong stationary times when Y is absorbed, as shown by the theory of Diaconis and Fill for finite state spaces. In this talk we will only consider processes Y taking as values some subsets of the state space of X. Our goal is to present extensions of the above method to elliptic
diffusion processes on differentiable manifolds, via stochastic
modifications of mean curvature flows. We will see that Pitman's theorem about the
intertwining relation between the Brownian motion and the Bessel-3
process is curiously ubiquitous in this approach. It even serves as an inspiring guide to construct couplings associated to finite
Markov intertwining relations via random mappings, in the spirit of the
coupling-from-the-past algorithm of Propp and Wilson and of the evolving sets of Morris and Peres.

Series: Stochastics Seminar

We shall prove that a certain stochastic ordering defined in terms of
convex symmetric sets is inherited by sums of independent symmetric
random vectors. Joint work with W. Bednorz.

Series: Stochastics Seminar

In this talk I will explore the subject of Bernoulli percolation on
Galton-Watson trees. Letting $g(T,p)$ represent the probability a tree
$T$ survives Bernoulli percolation with parameter $p$, we establish
several results relating to the behavior of $g$ in the supercritical
region. These include an expression for the right derivative of $g$ at
criticality in terms of the martingale limit of $T$, a proof that $g$ is
infinitely continuously differentiable in the supercritical region, and
a proof that $g'$ extends continuously to the boundary of the
supercritical region. Allowing for some mild moment constraints on the
offspring distribution, each of these results is shown to hold for
almost surely every Galton-Watson tree. This is based on joint work
with Marcus Michelen and Robin Pemantle.

Series: Stochastics Seminar

How should one estimate a signal, given only access to noisy versions of the signal corrupted by unknown cyclic shifts? This simple problem has surprisingly broad applications, in fields from aircraft radar imaging to structural biology with the ultimate goal of understanding the sample complexity of Cryo-EM. We describe how this model can be viewed as a multivariate Gaussian mixture model whose centers belong to an orbit of a group of orthogonal transformations. This enables us to derive matching lower and upper bounds for the optimal rate of statistical estimation for the underlying signal. These bounds show a striking dependence on the signal-to-noise ratio of the problem. We also show how a tensor based method of moments can solve the problem efficiently. Based on joint work with Afonso Bandeira (NYU), Amelia Perry (MIT), Amit Singer (Princeton) and Jonathan Weed (MIT).

Series: Stochastics Seminar

We obtain an extension of
the Ito-Nisio theorem to certain non separable Banach spaces and apply
it to the continuity of the Ito map and Levy processes. The Ito map
assigns a rough path input of an ODE to its solution (output).
Continuity of this map usually
requires strong, non separable, Banach space norms on the path space.
We consider as an input to this map a series expansion a Levy process
and study the mode of convergence of the corresponding series of
outputs. The key to this approach is the validity of
Ito-Nisio theorem in non separable Wiener spaces of certain functions
of bounded p-variation.
This talk is based on a joint work with Andreas Basse-O’Connor and Jorgen Hoffmann-Jorgensen.

Series: Stochastics Seminar

Place Poi(m) particles at each site of a d-ary tree of height n. The particle at the root does a simple random walk. When it visits a site, it wakes up all the particles there, which start their own random walks, waking up more particles in turn. What is the cover time for this process, i.e., the time to visit every site? We show that when m is large, the cover time is O(n log(n)) with high probability, and when m is small, the cover time is at least exp(c sqrt(n)) with high probability. Both bounds are sharp by previous results of Jonathan Hermon's. This is the first result proving that the cover time is polynomial or proving that it's nonpolymial, for any value of m. Joint work with Christopher Hoffman and Matthew Junge.

Series: Stochastics Seminar

I will discuss two projects concerning Mallows permutations, with Ander
Holroyd, Tom Hutchcroft and Avi Levy. First, we relate the Mallows
permutation to stable matchings, and percolation on bipartite graphs.
Second, we study the scaling limit of the cycles in the Mallows
permutation, and relate it to diffusions and continuous trees.

Series: Stochastics Seminar

It has been conjectured that phenomena as diverse as the behavior of large "self-organizing" neural networks, and causality in standard model particle physics, can be explained by suitably rich algebras acting on themselves. In this talk I discuss the asymptotics of large causal tree diagrams that combine freely independent elements of such algebras. The Marchenko-Pastur law and Wigner's semicircle law are shown to emerge as limits of a normalized sum-over-paths of non-negative elements assigned to the edges of causal trees. The results are established in the setting of non-commutative probability. Trees with classically independent positive edge weights (random multiplicative cascades) were originally proposed by Mandelbrot as a model displaying the fractal features of turbulence. The novelty of the present work is the use of non-commutative (free) probability to allow the edge weights to take values in an algebra.

Series: Stochastics Seminar

Today's era of cloud computing is powered by massive data centers. A data center network enables the exchange of data in the form of packets among the servers within these data centers. Given the size of today's data centers, it is desirable to design low-complexity scheduling algorithms which result in a fixed average packet delay, independent of the size of the data center. We consider the scheduling problem in an input-queued switch, which is a good abstraction for a data center network. In particular, we study the queue length (equivalently, delay) behavior under the so-called MaxWeight scheduling algorithm, which has low computational complexity. Under various traffic patterns, we show that the algorithm achieves optimal scaling of the heavy-traffic scaled queue length with respect to the size of the switch. This settles one version of an open conjecture that has been a central question in the area of stochastic networks. We obtain this result by using a Lyapunov-type drift technique to characterize the heavy-traffic behavior of the expected total queue length in the network, in steady-state.