Stable Invariants of Words from Random Matrices II: Formulas and Extensions

Let $w$ be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients of $w$-random unitary matrices. In the first part of the current work [arXiv:2311.17733], we demonstrated how this phenomenon is much broader: we proved more instances of such results and conjectured others. These new results and conjectures involved other topological invariants (relatives of scl) and different families of groups. In the current paper we further extend and support this theory. We provide another instance of the theory and prove that the stable primitivity rank, too, can be expressed in terms of stable Fourier coefficients of $w$-random elements of groups. We introduce concrete formulas for stable Fourier coefficients of $w$-random elements in the symmetric group $S_N$ and its generalizations in the form of the wreath products $G\wr S_N$ where $G$ is an arbitrary compact group. We also define new stable invariants related to these groups, and prove they give bounds to many of the stable Fourier coefficients. As an aside, we generalize to tuples of words a result of Puder and Parzanchevski [arXiv:1202.3269] about the expected number of fixed points of $w$-random permutations.