Probability seminar

• Title: BROWNIAN MOTION AND AN INERT PARTICLE
• Speaker: Clayton Barnes (University of Washington)
• Time: 2:30 p.m., Monday, October 12, 2015
• Room: THO 325
• Abstract: In his 2001 paper, Frank Knight constructed a process that moves with inertia and is impinged upon by a Brownian motion. Later, Bass, Burdzy, Chen and Hairer found the stationary distribution for the process extended to multi-dimensions. In this talk we give a discrete version of the inert particle, showing it converges in distribution after interpolation. In addition, we characterize the distribution of the maximum, and show that the sequence of local maximums forms a submartingale when a reflected boundary is introduced.

• Title: THE SMALLEST SINGULAR VALUE OF RANDOM MATRICES WITH INDEPENDENT ROWS
• Speaker: Konstantin Tikhomirov (University of Alberta)
• Time: 2:30 p.m., Monday, October 5, 2015
• Room: THO 325
• Abstract: We consider a classical problem of estimating the least singular value of random rectangular and square matrices with independent identically distributed entries as well as a more general model of random matrices with i.i.d. rows. The novelty of our results consists primarily in imposing very weak, or nonexisting, moment assumptions on the distribution of the entries. In particular, we show that the smallest singular value of a sufficiently tall $N\times n$ rectangular matrix with i.i.d. entries with certain condition on the Levy concentration function is of order $\sqrt{N}$ with a large probability. Further, we extend a fundamental result of Bai and Yin from early 1990-es on the limiting behaviour of the smallest singular value of rectangular matrices, by dropping the assumption of bounded 4th moment (a problem whether the assumption was necessary was discussed, in particular, in a book of Bai and Silverstein). Finally, we will discuss very recent joint results with Elizaveta Rebrova (University of Michigan) and Djalil Chafai (Paris Dauphine) regarding invertibility of random square matrices with i.i.d. heavy-tailed entries and Bai--Yin type convergence for matrices with i.i.d. log-concave rows, respectively. To obtain the results, we have used various techniques including special versions of the $\varepsilon$-net argument and rank one update approach of Batson--Spielman--Srivastava.

• Title: INVARIANT TRANSPORTS AND MATCHINGS OF RANDOM MEASURES
• Speaker: Guenter Last (Karlsruhe Institute of Technology)
• Time: 2:30 p.m., Monday, September 21, 2015
• Room: THO 134
• Abstract: The extra head problem formulated and solved by Liggett (2001) requires finding a head in an infinite sequence of independent fair coin tosses, without changing the (joint) distribution of the rest of the sequence. Holroyd and Peres (2005) constructed a a stable and shift invariant transport (marriage) of Lebesgue measure and an ergodic point process with unit intensity. Invariant matchings of stationary point processes were studied by Holroyd, Pemantle, Peres and Schramm (2008). In this talk we shall discuss matching and embedding problems for a two-sided standard Brownian motion and explain the close relationship of all these matching problems with shift invariant transports and Palm measures of stationary random measures. This talk is based on joint work with Peter Morters (Bath) and Hermann Thorisson (Reykjavik).