  Unitary matrix integrals over symmetric polynomials play an important role in
a wide variety of applications, including random matrix theory, gauge theory,
number theory, and enumerative combinatorics. We derive novel results on such
integrals and apply these and other identities to correlation functions of
long-range random walks (LRRW) consisting of hard-core bosons. We generalize an
identity due to Diaconis and Shahshahani which computes unitary matrix
integrals over products of power sum polynomials. This allows us to derive two
expressions for unitary matrix integrals over Schur polynomials, which can be
directly applied to LRRW correlation functions. We then demonstrate a duality
between distinct LRRW models, which we refer to as quasi-local particle-hole
duality. We note a relation between the multiplication properties of power sum
polynomials of degree $n$ and fermionic particles hopping by $n$ sites. This
allows us to compute LRRW correlation functions in terms of auxiliary fermionic
rather than hard-core bosonic systems. Inverting this reasoning leads to
various results on long-range fermionic models as well. In principle, all
results derived in this work can be implemented in experimental setups such as
trapped ion systems, where LRRW models appear as an effective description. We
further suggest specific correlation functions which may be applied to the
benchmarking of such experimental setups.
