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In this talk we present a model where particles interact via a coagulation mechanism and we focus our attention on some particular coagulation kernels that allows to write explicitly the probability to observe any given configuration. We draw connection with random graphs, in particular with the sparse Erd ̋os– R ́enyi random graph and its inhomogeneous generalization. Via this connection it is possible to derive a large deviation principle for the empirical measure of particle sizes at a fix time and to deduce the existence or not of a phase transition, which in this type of models depends on time and it is called gelation. This is an ongoing joint work with Wolfgang König (WIAS and TU Berlin), Tejas Iyer, Heide Langhammer, Elena Magnanini , Robert Patterson (WIAS Berlin).

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Broader Impacts (including but not limited to mentoring students, fostering diversity and inclusion in mathematics, and increased partnerships between academia and industry).

**Note that "broader impacts" are critical to any research proposal submitted to NSF.**

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We will discuss some new classes of matrix models for beta ensembles. For the special values of the parameter beta (1,2,4) these give rise to previously studied distributions and our aim is to provide a new perspective on those cases and better understand the behavior for general values of the parameter beta. Additionally, we will describe limiting relations among these models and the classical soft and hard edge ensembles. (joint work with B. Rider)

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.