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Title:
Singular Values of Products of Ginibre Random Matrices and a generalisation
of Painlev\'e III
Speaker:
Abstract:
Recently Strahov has extended Tracy and Widom's Fredholm theory of the hard edge of single ($ M=1 $) random matrices with unitary symmetry to products of matrices ($ M>1 $ in number). This advance was preceded by the work of many authors who showed that the singular values of such products of standard Gaussian random matrices could be described in terms of a determinantal point process with a kernel involving Meijer G-functions. The particular Meijer G-functions satisfy a linear differential equation of order M+1, generalising the Bessel functions. In the earlier Fredholm theory it was discovered that certain solutions to Painlev\'e's third transcendent were central in determining the distribution of the lowest singular value of the random matrix ensemble. In the recent work a generalisation of this integrable system plays the same role and we explore some of the properties of the simplest extension, i.e the $ M=2 $ case, in some detail. This is joint work with Peter Forrester.
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