  We discuss higher dimensional generalizations of the 0+1-dimensional
Sachdev-Ye-Kitaev (SYK) model that has recently become the focus of intensive
interdisciplinary studies by, both, the condensed matter and field-theoretical
communities. Unlike the previous constructions where multiple SYK copies would
be coupled to each other and/or hybridized with itinerant fermions via
spatially short-ranged random hopping processes, we study algebraically varying
long-range (spatially and/or temporally) correlated random couplings in the
general d+1 dimensions. Such pertinent topics as translationally-invariant
strong-coupling solutions, emergent reparametrization symmetry, effective
action for fluctuations, chaotic behavior, and diffusive transport (or a lack
thereof) are all addressed. We find that the most appealing properties of the
original SYK model that suggest the existence of its 1+1-dimensional
holographic gravity dual do not survive the aforementioned generalizations,
thus lending no additional support to the hypothetical broad (including
'non-AdS/non-CFT') holographic correspondence.
