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We investigate the microcanonical version of the complex induced Ginibreensemble, by introducing a fixed trace constraint for its second moment. Likefor the canonical Ginibre ensemble, its complex eigenvalues can be interpretedas a two-dimensional Coulomb gas, which are now subject to a constraint and amodified, collective confining potential. Despite the lack of determinantalstructure in this fixed trace ensemble, we compute all its density correlationfunctions at finite matrix size and compare to a fixed trace ensemble of normalmatrices, representing a different Coulomb gas. Our main tool of investigationis the Laplace transform, that maps back the fixed trace to the induced Ginibreensemble. Products of random matrices have been used to study the Lyapunov andstability exponents for chaotic dynamical systems, where the latter are basedon the complex eigenvalues of the product matrix. Because little is known aboutthe universality of the eigenvalue distribution of such product matrices, wethen study the product of $m$ induced Ginibre matrices with a fixed traceconstraint - which are clearly non-Gaussian - and $M-m$ such Ginibre matriceswithout constraint. Using an $m$-fold inverse Laplace transform, we obtain aconcise result for the spectral density of such a mixed product matrix atfinite matrix size, for arbitrary fixed $m$ and $M$. Very recently local andglobal universality was proven by the authors and their coworker for a moregeneral, single elliptic fixed trace ensemble in the bulk of the spectrum.Here, we argue that the spectral density of mixed products is in the sameuniversality class as the product of $M$ independent induced Ginibre ensembles.

Products of random matrices from fixed trace and induced Ginibre ensembles