|XXXIII Workshop on Geometric Methods in Physics||29.06-5.07.2014|
Participants of Workshop
Participants of School
Gap probabilities and Riemann-Hilbert problems in determinantal random point processes with or without outliers
It is well known that the gap probabilities for determinantal random point processes are computed by suitable Fredholm determinants of integral operators. For special type of kernels known as "integrable" (Its-Izergin-Korepin-Slavnov) the connection with a Riemann--Hilbert problem is also well known. On the other hand, in the case of processes, the kernels do not have this property but we will show how to still connect an appropriate RHP. The approach also yields a more straightforward proof of the well-known Tracy Widom distribution expressed in terms of Painleve' II solutions (a matrix version of the result produces solutions of the non-commutative PII equation).
We will also show how gap probabilities of processes with outliers (Airy and other examples) relate to the notion of Schlesinger discrete transformations, a notion that originates in the theory of ODEs but can be extended to RHPs as well.