Northwest Probability Seminar 2012

Talk abstracts

Jeff Steif (Chalmers University of Technology)

Title: Boolean Functions, Noise Sensitivity, Influences and Percolation

 
Abstract: Noise sensitivity concerns the phenomenon that certain types of events (Boolean functions) are sensitive to small noise. This topic is related to the notion of influence, which is a way to specify the importance of a particular variable on an event.  These concepts become especially interesting in the context of percolation theory. Some important tools in this area are discrete Fourier analysis and randomized algorithms in theoretical computer science.  In this lecture, I will give an overview of this subject.

Asaf Nachmias (University of British Columbia)

Title: Recurrence of planar graph limits

Abstract: We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite planar triangulation and quadrangulation (UIPT and UIPQ) are almost surely recurrent, resolving a conjecture of Angel, Benjamini and Schramm.
Joint work with Ori Gurel-Gurevich.

Douglas Rizzolo (University of Washington)

Title: Schroeder's problems and random trees

Abstract: In 1870 Schroeder introduced four problems concerning the enumeration of bracketings of words or sets of a given size. We will consider what uniform draws from these bracketings look like as the size of the word or set goes to infinity. Connections will be made to the recently developed theory of Markov branching trees as well as several types of conditioned Galton-Watson trees.
Joint work with Jim Pitman. 

Son Luu Nguyen (Oregon State University)

Title: Linear-Quadratic-Gaussian Mixed Game with Continuum-Parametrized Minor Players

Abstract: We consider a mean field linear-quadratic-Gaussian game with a major player and a large number of minor players parametrized by a continuum set. The mean field generated by the minor players is approximated by a random process depending only on the initial state and the Brownian motion of the major player, and this leads to two limiting optimal control problems with random coefficients, which are solved subject to a consistent requirement on the mean field approximation. The set of decentralized strategies constructed from the limiting control problems has an epsilon-Nash equilibrium property when applied to the large but finite population model.
Joint work with Minyi Huang.

Geoffrey Grimmett (Cambridge University)

Title: The star-triangle transformation in probability theory

Abstract: The star-triangle transformation was `discovered' in 1899. It has since become one of the basic tools for studying disordered systems in two dimensions, and it is known amongst physicists as the `Yang-Baxter equation'. We shall explain its harmony with de Bruijn's theory of tilings and isoradial graphs, as developed by Kenyon and co-authors. Then we outline its use in proving universality for percolation in two dimensions.