Dynamics and charge fluctuations in large- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>q</mml:mi></mml:math> Sachdev-Ye-Kitaev lattices

It is known that the large-$q$ complex Sachdev-Ye-Kitaev (SYK) dot thermalizes instantaneously under rather general dynamical protocols. We consider a lattice of such dots coupled together, allowing for $r/2$ body hopping particles between nearest neighbors. develop analytical framework to study dynamics leading order in $1/q$ on lattice, arbitrary time dependent couplings, hence find physics diffusive case $r>2$ effectively same as kinetic $r=2$, assuming $r=\mathcal{O}(q^0)$. Remarkably, we local charge densities $\mathcal{Q}_i$ form closed set equations. They however only show fluctuations $\mathcal{O}(\mathcal{Q}_i/q)$, remaining constant limit $q\rightarrow \infty$. Despite this effective lack dynamics, do not fact behave isolated sites which would thermalize instantaneously. Indeed, via proof by contradiction generally possible connected lattice. Importantly, results are shown be independent dimensionality