Differential equations for singular values of products of Ginibre random matrices (Q2878629)

The topics discussed refer to a determinantal point process which arises in the asymptotic analysis of the squared singular values of the product of \(M\) complex Ginibre random matrices. The correlation kernels of such determinantal processes can be expressed in terms of Meijer's \(G\)-functions which enables to study the statistics of eigenvalues and of singular values for products by the usual methods of random matrix theory. As it is well known, at the edge of the spectrum, this correlation kernel has a scaling limit \(K_M(x,y)\) which can be understood as a generalization of the classical Bessel kernel of random matrix theory.NEWLINENEWLINEThe main results obtained in this paper relate to the Fredholm determinant of the operator \(K_M\) acting on \(L^2(0,\infty)\) with the kernel \(K_M(x,y)\chi_{_J}(y)\), where \(J\) is a disjoint union of intervals \((a_{2j-1},a_{2j})\), \(j=1,\dots,m\), with \(0\leq a_1<a_2\cdots<a_{2m}\) and \(\chi_{_J}\), the characteristic function of the set \(J\). The Fredholm determinant \(\det(1-K_M)\) gives the probability that no particles of the limiting determinantal point process lie in \(J\) (the gap probability).NEWLINENEWLINEThe intended purpose in the present work is to describe the gap probability in terms of solutions of nonlinear differential equations. Between the essential results obtained by the author in this new context, we emphasize the following: {\parindent=5,mm \begin{itemize}\item[--] the construction of the system of nonlinear partial differential equations associated to the Fredholm determinant \((1-K_M)\), where the end points of the intervals \((a_{2j-1},a_{2j})\), \(j=1,\dots,m\), serve as independent variables; \item[--] the construction of a Hamiltonian system, where the end points of the intervals play a role of multi-time independent variables; \item[--] the construction of the isomonodromy deformation equation. NEWLINENEWLINE\end{itemize}} The special case \(J=(0,s)\) is also analysed and a formula for the probability that no particles of the determinantal process lie in the interval \((0,s)\) is established.NEWLINENEWLINENew directions of investigation are suggested in the end.