  We introduce a new class of large structured random matrices characterized by
four fundamental properties which we discuss. We prove that this class is
stable under matrix-valued and pointwise non-linear operations. We then
formulate an efficient method, based on an extremization problem, for computing
the spectrum of subblocks of such large structured random matrices. We present
different proofs -- combinatorial or algebraic -- of the validity of this
method, which all have some connection with free probability. We illustrate
this method with well known examples of unstructured matrices, including Haar
randomly rotated matrices, as well as with the example of structured random
matrices arising in the quantum symmetric simple exclusion process. tured
random matrices arising in the quantum symmetric simple exclusion process.
