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Determinantal random point fields, gap probabilities and the Riemann-Hilbert approach

Speaker: 
Marco Bertola
Institution: 
Concordia University, Montreal
Schedule: 
Wednesday, April 17, 2013 - 14:00 to 15:30
Location: 
A-136
Abstract: 

I will start by reviewing the notion of (determinantal) random point fields. Famous examples of their beautiful applications are:
1) the eigenvalues of certain type of random matrices;
2) mutually avoiding one-dimensional Brownian motions (Dyson process) and certain limits thereof;
3) the Frobenius coordinates of Young Diagrams associated to the Poissonized Plancherel measure on the set of representations of the permutation groups.

The expectations of occupation numbers of subsets of the configuration space can be expressed as Fredholm (infinite dimensional) determinants, the evaluation of which leads very often to a Riemann-Hilbert problem (a notion that I will review briefly).
This in turns allows to connect these probabilities to the ever-expanding area of Painlevé like equations and isomonodromic deformations.
I will try to touch upon these topics and review some results of my collaborators and myself.

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