Probability Seminar, Statistical Laboratory

Probability Seminar, Statistical Laboratory

Tuesdays 2-3 pm, MR 12.
Organisers: Nathanaël Berestycki, Peter Friz, and Mike Tehranchi.

Michaelmas 2008

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Date Speaker Title Notes
22 September. Vladas Sidoravicius (CWI Amsterdam and IMPA, Rio de Janeiro). "Sensibility for localized disorder"
7 October. Dan Romik (Hebrew University). "The oriented swap process". Room: MR4
14 October. Tom Spencer (IAS, Princeton). "Transience in a Statistical Mechanics model of Random Band Matrices."
20 October Evarist Gine (U. of Connecticut) "Uniform Limit Theorems for Wavelet Density Estimators" Joint seminar with statistics.
23 October John Cardy (Oxford). "Quantum network models and classical trails" Newton Institute, 2pm
28 October. Nadia Sidorova (UCL). "Phase transitions for dilute particle systems with potentials of Lennard-Jones type"
4 November. Hansjoerg Albrecher (RICAM, Linz) "On refracted stochastic processes and the analysis of insurance risk."
11 November. Serge Cohen (Toulouse). "Some stationary fields and their spectral measure on hyperbolic plane."
18 November. Ariel Yadin (Weizmann Institute). "Loop-erased random walk on planar graphs.".
25 November. Jean Picard (U. Blaise Pascal, Clermont-Ferrand). "Trees and rough paths".
2 December. Richard Nickl (Cambridge). "Talagrand's concentration inequality for empirical processes"

22 September. Vladas Sidoravicius (CWI Amsterdam and IMPA, Rio de Janeiro). "Sensibility for localized disorder"

In my talk I will discuss the following question: how small local disorder can affect global behaviour of large complex systems. Why some systems are very sensitive to small disorders and others are not? I will present several paradigm examples of such behaviour (Random Walks, Directed Polymers, Random Permutations) and several recent theorems which shade some new light on the mechanisms governing these processes.

7 October. Dan Romik (Hebrew University). "The oriented swap process". (Room: MR4, exceptionally)

The oriented swap process is a random walk on the symmetric group of order N. Starting from the identity permutation, at each step an adjacent swap is chosen uniformly and applied to the current permutation, but only if it increases the number of inversions. Eventually the walk terminates when it reaches the permutation with maximal number of inversions. In recent work with Omer Angel and Alexander Holroyd, we analyzed the asymptotic behavior of the oriented swap process when N tends to infinity using the theory of totally asymmetric exclusion processes, deriving formulas for the limiting trajectories of individual numbers ("particles") in the permutation and for the flow of particles en masse. An interesting connection to random matrix theory also makes an appearance. I will explain these results and show computer simulations.

14 October. Tom Spencer (IAS, Princeton). "Transience in a Statistical Mechanics model of Random Band Matrices."

We describe a statistical mechanics model on a d-dimensional lattice with a hyperbolic symmetry. This model is expected to reflect spectral properties of random band matrices such as localization and diffusion. The model has a probabilistic formulation as a random walk in a highly correlated random environment. In three dimensions, we prove that this model has a "diffusive phase". This is joint work with M. Disertori and M. Zirnbauer.

20 October (note unusual day). Evarist Gine (U. of Connecticut) "Uniform Limit Theorems for Wavelet Density Estimators" (joint seminar with statistics).

The linear wavelet density estimator of a bounded density $f$ consists of a truncated wavelet expansion with the coefficients for the expansion of $f$ replaced by their empirical counterparts. The optimal number of terms of the expansion, obtained by balancing bias and variance, depends on the degree of smoothness of $f$, typically unknown. Donoho-Johnstone-Kerkyacharian-Picard (1996) introduced the `hard thresholding' wavelet density estimator -where part of the empirical coefficients are set equal to zero if they are smaller than a certain threshold- in order to obtain an estimator which is rate adaptive in $L_p$ norm loss to the smoothness of $f$, up to a logarithmic factor. The sup-norm behavior of wavelet density estimators (thresholded or not) had not been considered before, and we use empirical process theory to close this gap, thus deriving optimal results first for the linear and then for the thresholded estimator. This is joint work with Richard Nickl.

23 October (2pm): talk of interest at the Newton Institute by John Cardy (Oxford). "Quantum network models and classical trails"

28 October. Nadia Sidorova (UCL). "Phase transitions for dilute particle systems with potentials of Lennard-Jones type"

We consider a dilute stationary system of N particles uniformly distributed in space and interacting pairwise according to a compactly supported potential, which is repellent at short distances and attractive at moderate distances. We are interested in the large-N behaviour of the system. We show that at a certain scale there are phase transitions in the temperature parameter and describe the energy and ground states explicitly in terms of a variational problem. This is a joint work with Andrea Collevecchio, Wolfgang Koenig and Peter Moerters.

30 October (2pm): talk of interest at the Newton Institute by Michael Aizenman (Princeton). "Localisation bounds for multiparticle systems"

4 November. Hansjoerg Albrecher (RICAM, Linz) "On refracted stochastic processes and the analysis of insurance risk."

We show a somewhat surprising identity for first passage probabilities of spectrally-negative Levy processes that are refracted at their running maximum. The result has applications in the study of insurance risk processes in the presence of tax payments. Extensions and related questions are discussed. Finally, we investigate certain statistics that help to identify heavy-tailed claim data situations in which the first or second moment of the underlying distribution does not exist and propose an alternative method to estimate the extreme value index of Pareto-type tails.

11 November. Serge Cohen (Toulouse). "Some stationary fields and their spectral measure on hyperbolic plane."

Brownian field on hyperbolic plane is a Gaussian field with stationary increments, and a Kintchine's theorem associates with the variance of the increments a spectral measure. In this talk a formula for this spectral measure will be introduced. I will also consider some sationary fields associated with spherical functions. I will make comparisons with the Euclidean case and recall basic notions of hyperbolic geometry, to make the talk as self-contained as possible.

18 November. Ariel Yadin (Weizmann Institute)."Loop-erased random walk on planar graphs.".

This talk focuses on loop-erased random walk, or LERW. LERW is a random self-avoiding curve obtained by erasing the loop in the trajectory of a random walk in chronological order. Lawler, Schramm, and Werner proved that LERW on the Euclidean plane converges to SLE(2) as the mesh goes to 0. SLE, or Schramm-Loewner Evolution, is a fascinating random process discovered by Oded Schramm in 1999. SLE arises as the scaling limit of many models in mathematical physics. It has many wonderful properties, perhaps the most important is "conformal invariance". Lawler, Schramm, and Werner's proof for the scaling limit of LERW essentially uses the symmetry of the lattice structure. The question arises whether a similar result holds even under perturbed lattices; for example, if only a small portion of edges are removed from the original lattice. We extend Lawler, Schramm and Werner's result: For any planar Markov chain (that is a Markov chain embedded into the complex plane so that edges do not cross one another), if the scaling limit of the Markov chain is planar Brownian motion, then the scaling limit of the loop erasure of the Markov chain is SLE(2). One main example, is loop-erased random walk on the super-critical percolation cluster; that is, the infinite component after super-critical percolation on Z^2. Berger and Biskup showed that the random walk on the super-critical percolation cluster converges to Brownian motion. Thus, our result implies that the loop-erased random walk on the super-critical percolation cluster converges to SLE(2). Joint work with Amir Yehudayoff.

25 November. Jean Picard (U. Blaise Pascal, Clermont-Ferrand). "Trees and rough paths".

Each real-valued continuous path defined on [0,1] can be associated to a tree. Our aim is to discuss the relationship between these two objects. It turns out that some properties of the path, such that the fact that it has finite p-variation, can be translated into a geometric property of the tree. Moreover, integrals with respect to the path can be written as integrals on the tree; this can be done for the Young integral but also for some integrals of the rough paths theory. This approach can be applied to stochastic paths such as fractional Brownian motions in order to construct stochastic integrals.

2 December. Richard Nickl (Cambridge). "Talagrand's concentration inequality for empirical processes"

This is an expository talk about a deep probability inequality due to Talagrand (Invent. Math. 1996, cf. Ledoux's book in 2001), which gives a Prohorov- (and then also Bernstein-) type exponential bound for the concentration of the supremum of an empirical process around its mean. I will try to discuss the following points: A) Some ideas of the proof, in particular the proof due to Ledoux using logarithmic Sobolev inequalities, which is related to the more general "concentration of measure" phenomenon. B) Discuss a variety of probabilistic applications, which should show how versatile this inequality is, in particular that it reproduces most known exp. inequalities for i.i.d. sums of (possibly Banach-)valued random variables. C) Discuss recent statistical applications to adaptive estimation, model selection problems, Rademacher processes, and almost sure limit laws (LIL-type results).