Probability Day
(9 Sep 2011)

~ Abstracts ~


On the range, local times and periodicity of random walk on an interval
Siva R. Athreya, ISI, Bangalore and National University of Singapore

We shall discuss some elementary (yet surprising) results of the range, local times, and periodicity of symmetric, weakly asymmetric and asymmetric random walks at the time of exit from a strip with N locations. We shall obtain asymptotic results as N approaches infinity.
This is joint work with Sunder Sethuraman and Balint Toth.

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Polynomial ballisticity conditions for RWRE
Alexander Drewitz, ETH Zurich, Switzerland

We consider random walk in random environment (RWRE) in dimension larger than or equal to four. A couple of years ago Sznitman introduced a certain class of conditions denoted (T)_\gamma, \gamma \in (0,1], where the parameter \gamma governs the (stretched) exponential decay of certain slab exit probabilities of the RWRE. The importance of these conditions stems, among others, from the fact that they imply ballistic and diffusive behavior of the RWRE. They are known equivalent for parameters \gamma \in (0, 1).

We propose here a new class of conditions (M), which are similar in nature to (T)_\gammma but only require polynomial decay of the corresponding exit probabilities.

Our main result states that these newly introduced conditions already imply (T)_\gamma, \gamma \in (0,1), and hence all the results deduced from that. (Work in progress with Noam Berger and Alejandro Ramirez)

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Degree asymptotics with rates for preferential attachment random graphs
Adrian Roellin, National University of Singapore

We provide rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of a certain distributional transformation that allows us to obtain rates of convergence using a new variation of Stein's method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind. (Joint work with E. Peköz and N. Ross)

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Brownian web in the scaling limit of supercritical oriented percolation in dimension 1+1
Rongfeng Sun, National University of Singapore

We show that, after suitable centering and diffusive rescaling of space and time, the collection of rightmost infinite open paths in a supercritical oriented percolation configuration in dimension 1+1 converges in distribution to the Brownian web. This proves a conjecture of Wu and Zhang. Loosely speaking, the Brownian web is a collection of coalescing Brownian motions starting from every point in the space-time plane R2. Our key observation is that each rightmost infinite open path can be approximated by a percolation exploration cluster, and different exploration clusters evolve independently before they intersect. This is joint work with Anish Sarkar.

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Contact processes on some random graphs
Qiang Yao, East China Normal University, China

In this talk, a brief review of some basic facts on random graph theory and contact processes followed by an introduction to some existing models on contact processes on random graphs will be given at the beginning. Then, a result of our recent work on this topic will be reported. In our work, the random graphs were chosen according to the power law model of Newman, Strogatz and Watts(2001). Our results showed that there were three distinct regimes for the limiting density which depended on the tail of the degree law. (The talk is based on a joint work with Mountford Thomas and Valesin Daniel.)

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