Unitary matrix integrals, long-range random walks, and spectral statistics

Unitary matrix integrals play an important role in a wide variety of fields, ranging from gauge theory, enumerative combinatorics, number theory, and quantum chaos, to areas of telecommunication and quantitative finance. This thesis considers unitary matrix integrals over symmetric polynomials and presents novel, recursive expansions for these objects. These expressions generalize various long-standing results in ways that allow for a greater range and ease of application.
These results are then applied to the study of non-intersecting long-range random walkers (LRRW’s), that is, hard-core bosons on a 1D lattice which can move over large distances, whose correlation functions are given by weighted unitary matrix integrals over Schur polynomials. These are remarkably rich systems, which have gained increasing attention in the last 15 years due in part to their experimental realizability in trapped ion systems. Various basic aspects of symmetric function theory lead directly to certain surprising results on LRRW’s, including physical dualities between certain LRRW systems, as well as between LRRW’s and long-range fermionic systems. The recursive expressions derived in this thesis can be directly applied to LRRW’s as well, leading to an expansion in powers of the time parameter which can be truncated at the desired order, with coefficients given explicitly in terms of the hopping parameters of the hamiltonian. These results are directly amenable to experimental checks, and further include procedures for benchmarking experimental setups such as trapped ion systems.
We then consider the computation of the spectral form factor of a matrix model description of Chern-Simons theory, which was found by previous authors to describe the ‘intermediate’ statistics of quantum systems in between order and chaos. Despite a large collection of previous publications on this application, our calculations display an absence of intermediate statistics in this model, reproducing the statistics of fully chaotic quantum systems instead. These findings have precipitated a numerical follow-up study to clarify this apparent contradiction.