
We consider the problem of colouring the planar map given by the Voronoi tessellation corresponding to a Poisson process in R^{2}. We seek colouring rules that are isometry invariant and are factors of the Poisson process. We prove that six colours suffice.
With Benjamini, GurelGurevich, Meyerovitch, and Peled.
We consider the stochastic heat equation u_{t}Du = B¢ in (0,T) ×R^{d}, with additive noise. The noise B¢ is a Gaussian process, which is fractional in time, with Hurst index H Î (1/2,1), and colored in space, with spatial covariance kernel f. Our main result gives the necessary and sufficient condition for the existence of the solution. When f is the Riesz or the Bessel kernel of order a, this condition is H > (da)/4. This is a relaxation of the condition H > d/4 encountered when the noise is white in space. When f is the heat or the Poisson kernel, the solution exists for any d and H. The case of the equation with multiplicative noise is examined in the second part of the talk.
Based on joint work with Ciprian Tudor.
In five or more dimensions the scaling limit of selfavoiding walk is proven to be Brownian motion. This is conjectured to also hold in four dimensions. I will describe progress in a program to prove this conjecture.
In this talk, I would like to introduce longrange oriented percolation with index a > 0 and present the Fourier transform of the properlyscaled normalized twopoint function converges to e^{CkaÙ2} for some C Î (0,¥) above the uppercritical dimension dc º 2(aÙ2). Moreover, the constant C exhibits crossover at a = 2, which is a result of interactions among occupied paths.
Let S(0)=0, S(i), i=1,2,..., be a simple symmetric random walk on the line, and let X(k,n) : = # {i: 1 £ i £ n, S(i)=k}, k=0,±1,±2,... be its local time process. Let {W(t), t ³ 0} be a standard Brownian motion, and let {L(x,t), ¥ < x < ¥, t ³ 0} be its local time process. The study of the asymptotic behaviour of the centered local time processes {X(k,n)  X(0,n)} and {L(x,t)  L(0,t)} has played a significant role in the development of the local time theory of random walks and that of Brownian local times. A glimpse of these developments will be attempted in their historical context, leading up to a strong approximation of the local time difference {X(k,n)  X(0,n)} by a Wiener sheet and an independent Brownian motion, time changed by an independent Brownian local time. The latter is based on E. Csáki, M. Csörgö, A. Földes and P. Révész (2008), Annales de l'Institut Henri PoincaréProbabilités et Statistiques, to appear.
We consider the question of percolation on a class of infinite random graphs based on a hierarchical structure and related meanfield limits.
This is joint work with Luis Gorostiza.
The critical, nonuniversal properties of the Eight Vertex, AshkinTeller and XYZ models are widely expected to be described by the quantum field theory obtained as formal scaling limit. On the basis of this assumption, Kadanoff, Luther and Peschel conjectured universal scaling formulas that relate nonuniversal critical indexes. So far these conjectures had remained unproven. We present a constructive, renormalizationgroup approach that allows us to prove some of them under the condition of small coupling.
Work in collaboration with G. Benfatto and V. Mastropietro.
Several results are presented involving the asymptotic behaviour of the twoparameter PoissonDirichlet distribution. Some dynamical models will also be discussed.
We consider linearquadraticGaussian (LQG) games with a major player and a large number of minor players. The major player has significant influence on others. The minor players individually have negligible impact, but they collectively contribute mean field coupling terms in the individual dynamics and costs. To overcome the dimensionality difficulty and obtain decentralized strategies, the socalled Nash certainty equivalence methodology is applied. The control synthesis is preceded by a state space augmentation via a set of aggregate quantities giving mean field approximation. Subsequently, within the population limit the original game is decomposed into a family of twoplayer limiting games as each locally seen by a representative minor player. Next, when solving these twoplayer limiting games, we impose certain interaction consistency conditions such that the aggregate quantities initially assumed coincide with the ones replicated by the closedloop of a large number of minor players. This procedure leads to decentralized strategies for the original LQG game, and it is shown that the set of strategies is a decentralized eNash equilibrium.
We generalize the classic changepoint problem to a "changeset" framework: a spatial Poisson process changes its intensity on an unobservable random set. Optimal detection of the set is defined by maximizing the expected value of a gain function. In the case that the unknown changeset is defined by a locally finite set of incomparable points, we present a sufficient condition for optimal detection of the set using multiparameter martingale techniques. Two examples are discussed.
The SchrammLoewner evolution (SLE) is a oneparameter family of random growth processes that has been successfully used to analyze a number of models from twodimensional statistical mechanics. Currently there is interest in trying to formalize our understanding of conformal field theory (CFT) using SLE. S. Smirnov recently showed that the scaling limit of interfaces of the 2d critical Ising model can be described by SLE(3). The goal of this talk is to explain how a certain nonlocal observable of the 2d critical Ising model studied by L.P. Arguin and Y. SaintAubin can be rigorously described using SLE(3) and Smirnov's result.
We introduce the notion of singular points of random matrixvalued analytic functions and present some exact results as well as asymptotic results on the distribution of singular points. The former lead to some determinantal processes in the hyperbolic plane, while the latter lead to certain generalizations of the circular law.
We investigate global performance of nonlinear wavelet estimation in randomdesign regression models with long memory errors. Convergence properties are studied over a wide range of Besov classes and for a variety of L^{p} error measures. The setting is as follows. We observe Y_{i} = f(X_{i}) + s(X_{i}) e_{i}, i = 1,...,n, where X_{i},i ³ 1, are (observed) independent identically distributed (i.i.d.) random variables with a distribution function G, e_{i}, i ³ 1 is a stationary Gaussian dependent sequence with a covariance function r(m) ~ m^{a}, a Î (0,1) and s(·) is a deterministic function.
For nonlinear wavelet estimator we obtain the rates under L_{p} risk. Furthermore, we construct an estimator for fòf. This estimator has better convergence rates than the estimator of f.
Our obtained rates of convergence agree (up to the log term) with the minimax rates of Yang, 2001. Results reveal a dense, an intermediate and a sparse zone. In particular, in the latter two zones nonlinear estimators are better than linear ones. This phenomena was observed before in i.i.d. setting (Donoho, Johnstone, Kerkyacharian, Picard, ...).
From a probabilistic point of view the main new ingredient of our proof is a large deviation result for long memory sequences. The idea comes from martingale approximation as in Wu and Mielniczuk, 2002. It is also based on a smoothing dichotomy heuristic. Estimators of highfrequency coefficients should behave as if the random variables e_{i} were independent. Estimators for lowresolution levels are influenced by longmemory. This has immediate consequences for the estimator of f. The dichotomous effect is suppressed when we consider the estimator of fòf.
When studying the rate of convergence of an ergodic Markov chain to its equilibrium distribution, the usual metric of "convergence" is total variation; however, for continuous state spaces, it is sometimes easier to work with the (typically weaker) Wasserstein metric. We show how one can convert bounds on Wasserstein convergence rates into bounds on total variation convergence rates (under certain checkable assumptions). We illustrate using two examples:
This is joint work with Deniz Sezer (Calgary).
The WiFi protocol allows a varying and unknown number of users to access a base station without any centralized coordination. We will discuss the exponential backoff algorithm which makes this possible. We will also consider a mean field approximation to a system with N users which allows us to predict the performance of this system. We'll finish with a list of drawbacks of this protocol.
We show there exist parameters for which coexistence of the two species in the twodimensional LotkaVolterra model holds. The proof borrows many ideas from earlier results obtained recently by Cox, Durrett and Perkins. However, the proof of coexistence in the planar case is more involved than that of greater dimensions. In particular it requires a new convergence theorem for a wellchosen sequence of rescaled LotkaVolterra models.
This is joint work with Ted Cox, Rick Durrett and Ed Perkins.
We prove pathwise uniqueness for solutions of parabolic stochastic PDEs with multiplicative white noise if the coefficient is Hölder continuous of index g > 3/4. The method of proof is an infinitedimensional version of the YamadaWatanabe argument for ordinary stochastic differential equations.
This is joint work with Leonid Mytnik.
Stochastic models of cellular chemical reaction networks typically involve chemical species numbers and reaction rates varying over several orders of magnitude. In order to reduce the analytical and computational complexity of the model one can exploit the "multiscale" nature of these models arriving at approximate asymptotic models. In general, approximations will be "hybrid" in the sense that some components will be discrete, some diffusive, and some absolutely continuous. Systematic approaches to model reduction for systems on two time scales will be discussed.
We study existence, uniqueness and mass conservation of signed measure valued solutions of a class of stochastic evolution equations with respect to the Wiener sheet, including as particular cases the stochastic versions of the regularized twodimensional NavierStokes equations in vorticity form introduced by Kotelenez.
This is joint work with Jean Vaillancourt.
We present some structure results on nonsymmetric Dirichlet forms. These include BeurlingDeny formula, an analogue of LeJan's transformation rule for the diffusion parts, and a LevyKhintchine type formula for regular nonsymmetric Dirichlet forms on R^{d}.
This is joint work with ZeChun Hu and ZhiMing Ma.
Weighted approximations in probability of selfnormalized and studentized partial sums processes will be reviewed and applied to studying the problem of change in the mean of random variables in the domain of attraction of the normal law.
The talk will be based on joint works by Miklós Csörgö, Barbara Szyszkowicz and Qiying Wang.
Let X be the ddimensional (a,b)superprocess with Lebesgue initial measure, i.e., X is a superprocess with astable spatial movement and (1+b)stable branching. For a = 2 (super Brownian motion) Iscoe proved that its total occupation time ò_{0}^{¥} X_{t}(B) dt is finite a.s. if and only if db < 2, where B denotes the unit ball. He further conjectured that the similar result should hold for a < 2 with db < 2 replaced by db < a. In this talk we want to give a partial answer to Iscoe's conjecture.