Unitary matrix integrals, symmetric polynomials, and long-range random walks
arXiv:2209.06538 · doi:10.1088/1751-8121/acc21f
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 and fermionic particles hopping by 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.
65 pages, 2 figures; Moved background sections to appendices. Added context to section 1.2. Inverted order of presentation of section 2.3.2. Clarified the variable dependence in various expressions. Changed the color of certain border strips. Added further references. Corrected typos. Accepted for publication by J. Phys. A
References in corpus (11)
- Dynamics of Loschmidt echoes and fidelity decay
- Probing quantum information propagation with out-of-time-ordered correlators
- On the complete perturbative solution of one-matrix models
- Probing Operator Spreading via Floquet Engineering in a Superconducting Circuit
- New Insights into Superintegrability from Unitary Matrix Models
- Migdal-Eliashberg theory as a classical spin chain
- Tau Functions and Matrix Integrals
- Statistical properties of structured random matrices
- Scaling of many-particle correlations in a dissipative sandpile
- Unitary matrix models, free fermion ensembles, and the giant graviton expansion
- Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions