CS49: Analysis of Singular Diffusions and Related Areas
Organizer: Adina Oprisan (New Mexico State University)
A bridge between Random Matrix Theory and Schramm-Loewner Evolutions Theory
Vlad Margarint
I will describe a new method that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of modern techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with modern techniques from Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.
Bibliography
\([1]\) Campbell, A., Luh, K., Margarint, V. (2025). Rate of Convergence in Multiple SLE using Random Matrix Theory. Random Matrices Theory and Applications.
A bridge between Random Matrix Theory and Schramm-Loewner Evolutions Theory
Vlad Margarint
I will describe a new method that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of modern techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with modern techniques from Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. More details available at margarintvlad.com. This is a joint work with A. Campbell and K. Luh.
Bibliography
\([1]\) Campbell, A., Luh, K., Margarint, V. (2025). Rate of Convergence in Multiple SLE using Random Matrix Theory. Random Matrices Theory and Applications.
Asymptotic behavior of the Brownian motion with singular drifts
Adina Oprisan
We discuss the influence of a power law drift on the exit time of the Brownian motion from the half line when the order of the drift at 0 and infinity is different. We find that the first hitting time of 0 is finite almost surely, even when the drift near 0 is positive and unbounded, and study the asymptotic behavior of the process at the time of hitting. We find that the behavior of the process near the origin does not influence the time to hit 0 over long time periods and derive subexponential estimates for the tail distribution of the hitting time. Our results extend to the case in which the drifts are regularly varying functions at 0 respectively infinity. This talk is based on a joint work with Dante DeBlassie and Robert Smits.