An M-dependent process X(n) on the integers, is a process for which every event concerning with X(-1),X(-2),... is independent from every event concerning with X(M),X(M+1),...
Such processes play an important role both as scaling limits of physical systems and as a tool in approximating other processes.
A question that has risen independently in several contexts is:
"is there an M dependent proper colouring of the integer lattice for some finite M?"
Abstract: It was noticed in the 30's by Doeblin & Forte that Markov
operators with "chains with complete connections"
act quasi-compactly on the Lipschitz functions. These are operators
like the transfer operators of certain expanding
C^2 interval maps (e.g. the square of Gauss map).
It is folklore that stochastic processes generated by smooth
observables under these maps satisfy many of the results
of "classical probability theory" (e.g. CLT, Chernoff inequality).
I'll try to explain some of this in a "lunchtime" mode.
Let G be an infinite connected graph. For each vertex of G we decide randomly and independently: with probability p we paint it blue and with probability 1-p we paint it yellow. Now, consider the subgraph of blue vertices: does it contain an infinite connected component?
There is a critical probability p_c(G), such that if p>p_c then almost surely there is a blue infinite connected component and if pp_c or p<p_c.
We will focus on planar graphs, specifically on the triangular
Abstract: This talk will describe joint work with Aravind Asok
and Jean Fasel using the methods of homotopy theory to construct new
algebraic vector bundles. I will describe a natural conjecture
true, implies that over the complex numbers the classification
vector bundles over smooth affine varieties admitting an
decomposition coincides with the classification of topological
complex vector bundles.
Abstract: We will discuss the question: for a random walk in a random environment, when should one expect a central limit theorem, i.e. that after appropriate scaling, the random walk converges to Brownian motion? The answer will turn out to involve the spectral theory of unbounded operators. All notions will be defined in the talk. Joint work with Balint Toth.
I will present an overview of some recent progress on the study of the nodal sets of Steklov eigenfunctions. In particular, I will discuss sharp estimates on the nodal length of Steklov eigenfunctions on real-analytic Riemannian surfaces with boundary obtained in my joint
work with D. Sher and J. Toth.
Abstract: Consider a random matrix with i.i.d. normal entries. Since its distribution is invariant under rotations, any normalized eigenvector is uniformly distributed over the unit sphere. For a general distribution of the entries, this is no longer true. Yet, if the size of the matrix is large, the eigenvectors are distributed approximately uniformly. This property, called delocalization, can be quantified in various senses. In these lectures, we will discuss recent results on delocalization for general random matrices.