Unitary matrix integrals, symmetric polynomials, and long-range random walks

TL;DR

This paper develops new mathematical tools for unitary matrix integrals over symmetric polynomials and applies them to analyze long-range random walks of hard-core bosons, revealing dualities and fermionic representations with potential experimental relevance.

Contribution

It generalizes identities for unitary matrix integrals over Schur and power sum polynomials and introduces a duality in long-range random walk models, connecting bosonic and fermionic systems.

Findings

Derived new formulas for unitary matrix integrals over Schur polynomials.

Established a quasi-local particle-hole duality in long-range random walks.

Connected long-range bosonic models to fermionic systems for easier analysis.

Abstract

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.