  We study the spectral statistics of spatially-extended many-body quantum
systems with on-site Abelian symmetries or local constraints, focusing
primarily on those with conserved dipole and higher moments. In the limit of
large local Hilbert space dimension, we find that the spectral form factor
$K(t)$ of Floquet random circuits can be mapped exactly to a classical Markov
circuit, and, at late times, is related to the partition function of a
frustration-free Rokhsar-Kivelson (RK) type Hamiltonian. Through this mapping,
we show that the inverse of the spectral gap of the RK-Hamiltonian lower bounds
the Thouless time $t_{\mathrm{Th}}$ of the underlying circuit. For systems with
conserved higher moments, we derive a field theory for the corresponding
RK-Hamiltonian by proposing a generalized height field representation for the
Hilbert space of the effective spin chain. Using the field theory formulation,
we obtain the dispersion of the low-lying excitations of the RK-Hamiltonian in
the continuum limit, which allows us to extract $t_{\mathrm{Th}}$. In
particular, we analytically argue that in a system of length $L$ that conserves
the $m^{th}$ multipole moment, $t_{\mathrm{Th}}$ scales subdiffusively as
$L^{2(m+1)}$. We also show that our formalism directly generalizes to higher
dimensional circuits, and that in systems that conserve any component of the
$m^{th}$ multipole moment, $t_{\mathrm{Th}}$ has the same scaling with the
linear size of the system. Our work therefore provides a general approach for
studying spectral statistics in constrained many-body chaotic systems.
