Theoretical Statistics and Mathematics Unit

Seminar and Colloquium of the week


Title : Limit shapes for groves
Speaker : Terrence George (Brown Univ)
Time : July 23, 2018 (Monday), 02:00 PM
Venue : Auditorium
Abstract : A grove is a spanning forest of a triangular portion of the triangular lattice with a prescribed boundary connectivity. A large random grove exhibits a limit shape i.e. there is a non-random algebraic curve outside which the grove is "frozen". TK Petersen and D Speyer proved that for the uniform measure on groves, the curve is the inscribed circle. I will talk about extensions of their results to probability measures on groves that are periodic in appropriate coordinates. These measures give interesting algebraic curves with higher genus and cusp singularities as limit shapes, as well as new "gaseous" phases.


Title : A hitting question for stochastic flows
Speaker : Carl Mueller (Univ Rochester)
Time : July 23, 2018 (Monday), 03:15 PM
Venue : Auditorium
Abstract : This is joint work with Jong Jun Lee and Eyal Neuman, in response to a question mentioned to me by Alesey Kuznetsov. Hitting questions play a central role in the theory of Markov processes. For example, it is well known that Brownian motion hits points in one dimension, but not in higher dimensions. For a general Markov process, we can determine whether the process hits a given set in terms of potential theory. However, for high-dimensional or infinite-dimensional processes, the potential theory becomes intractible, and we must fall back on less precise methods such as covering arguments. We study a stochastic flow in $\mathbf{R}^n$ driven by a single Brownian motion, and with an outward drift in the radial direction. Consider the image of an open set which is a positive distance from the origin. We find reasonably sharp conditions on the drift for which the image of this set can or cannot hit the origin, respectively.

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