The local universality of Muttalib-Borodin biorthogonal ensembles with parameter theta = 1/2 Articles uri icon

publication date

  • July 2019

start page

  • 3023

end page

  • 3081

issue

  • 8

volume

  • 32

International Standard Serial Number (ISSN)

  • 0951-7715

Electronic International Standard Serial Number (EISSN)

  • 1361-6544

abstract

  • The Muttalib-Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter ... and an external field V. For ... we find the large n behavior of the associated correlation kernel with only few restrictions on V. The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a 3 × 3 Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.

subjects

  • Mathematics

keywords

  • riemann-hilbert problem; biorthogonal ensemble; steepest descent analysis; meijer g-functions