Georgia Institute of Technology College of Sciences

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.

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.

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.

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.