prof. Bertola's seminar announcement

[Emanuele Tuillier Illingworth]

Mathematical Physics Seminar

SISSA room 136  Thursday 17th of July 11.00-12.00am

Speaker: Prof Marco Bertola, Concordia University,  Montreal

Title: Gap probabilities and Riemann-Hilbert problems in determinantal 
random point processes with or without outliers

Abstract: 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.