Ron Peled (UC Berkeley)

Given a translation invariant point process in R^d of intensity 1, an allocation rule is a translation-equivariant mapping that allocates to each point in the process a set in R^d of unit volume, such that the sets allocated to different points are disjoint and their union covers almost all of R^d. In other words, we partition R^d to sets of volume 1 and match them with the point process in a translation equivariant way. Allocation rules can give a better understanding of the underlying point process, they measure in some sense how uniformly the mass is spread over space. They can also be used for obtaining so called extra head rules. In this talk we will consider the standard Poisson point process in R^d, allocation rules for this process were constructed by Hoffman, Holroyd and Peres using the Gale-Shapley stable marriage algorithm. I will describe a new allocation rule in dimensions 3 and higher, inspired by recent work of Nazarov, Tsirelson, Sodin and Volberg, that is defined by flow along the integral curves of a gravitational force field induced by the Poisson points. The main result is that this allocation is 'efficient', in the sense that the diameter of the cell allocated to a given point is a random variable with exponentially decaying tails. This is the first deterministic allocation with this property. Time permitting, I will tell also of matching lower bounds for the tail of the diameter, and bounds for other parameters of the process. This is joint work with Sourav Chatterjee, Yuval Peres and Dan Romik.
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Sandrine Péché (Institut Fourier Grenoble)

This talk is concerned with the asymptotic distribution of the largest eigenvalues of random matrices when the size of the matrices goes to infinity. It is well-known that the asymptotic distribution of the rescaled largest eigenvalue of the GUE (i.e. a Hermitian random matrix with i.i.d. complex Gaussian entries) is the Tracy-Widom law. We will review some other possible limiting distributions for the largest eigenvalue of random matrices. Some connections with directed percolation, random growth processes are also discussed.
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Asaf Nachmias (UC Berkeley)

We will explore recent progress in two topics in percolation theory: the critical Erdos-Renyi random graph G(n,1/n) and percolation on Z^d at the critical probability p_c, for large d. Our results are:
1. The incipient infinite cluster (IIC) of critical percolation on Z^d (which can be though of as the critical percolation cluster condition on being infinite) has spectral dimension 4/3 when d>18; that is, where the lace expansion estimates hold. This affirms a conjecture of Alexander and Orbach (1982) in high dimensions. (Joint work with Gady Kozma)
2. The largest component of G(n,1/n) has diameter of order n^{1/3} and the mixing time of the lazy simple random walk on it is of order n. The latter answers a question of Benjamini, Kozma and Wormald (2005). (Joint work with Yuval Peres)
The key ingredient in both results is the usage of a new critical exponent related to the classical "arm" exponent. No previous knowledge of percolation will be assumed.
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Jason Fulman (USC)

It is shown that the combinatorics of commutation relations can be used to give sharp convergence rate results for certain Markov chains. The main example described in this talk will be a random walk on partitions whose stationary distribution is the Ewens distribution from population genetics.
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Gady Kozma (Weizmann Institute)

We survey the loop-erased random walk and the closely related uniform spanning tree, and some features of the proof that the scaling limit exists in 2 and 3 dimensions.
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Harrie Willems

Harrie Willems will present his film with Agnes Handwerk: Wolfgang Doeblin, A Mathematician Rediscovered.
Wolfgang Doeblin, one of the great probabilists of the 20th century, was already widely known in the 1950s for his fundamental contributions to the theory of Markov chains. His coupling method became a key tool in later developments at the interface of probability and statistical mechanics. But the full measure of his mathematical stature became apparent only in 2000 when the sealed envelope containing his construction of diffusion processes in terms of a time change of Brownian motion was finally opened, 60 years after it was sent to the Academy of Sciences in Paris. This film documents scientific and human aspects of this amazing discovery and throws new light on the startling circumstances of Doeblin's death at the age of 25.
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Michel Mandjes

In the analysis of stochastic systems, estimates for the speed of convergence to stationarity play a crucial role. Consider for example the situation in which one is interested in the distribution of $M_\infty$, where $M_t := \sup_{s\in [0,t]}X(t) - t$, for some centered stochastic process X(.) -- for instance fractional Brownian motion (fBm). In order to estimate ${\mathbb P}(M_\infty>x)$ by simulation, one needs to determine a simulation horizon $T$ such that the difference between the distributions of $M_\infty$ and $M_T$ is, in some metric, negligible.
In the first part of my talk I present results on the decay rate (in $T$) of several metrics for the special case of fBm. More concretely, I show that the distance behaves as $\exp(-\gamma T^{2H-2})$, where $\gamma$ can be translated in terms of the asymptotics of long busy periods.
These busy-period asymptotics are non-trivial, and are essentially determined by the most likely way in which a long busy period occurs; we show that this path has a rather unexpected shape.
Time permitting, I'll conclude by focusing on the correlation structure of reflected fBm, and corresponding transient characteristics. The main result is that certain correlation measures decay in the same as the input process. This means that, in this respect, the queueing process inherits the long-range dependent properties of the input process.
This talk is based on joint work with several others, including K. Debicki, A. Es-Saghouani, P. Glynn, and I. Norros.
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