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. This paper derives novel results
on such integrals and applies these and other identities to correlation
functions of long-range random walk (LRRW) models consisting of hard-core
bosons. We generalize an identity due to Diaconis and Shahshahani which
computes unitary matrix integrals over products of power sum symmetric
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.



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