  In the vast amount of results linking gravity with thermodynamics,
statistics, information, a path is described which tries to explore this
connection from the point of view of (non)locality of the gravitational field.
First the emphasis is put on that well-known thermodynamic results related to
null hypersurfaces (i.e. to lightsheets and to generalized covariant entropy
bound) can be interpreted as implying an irreducible intrinsic nonlocality of
gravity. This nonlocality even if possibly concealed at ordinary
scales(depending on which matter is source of the gravitational field, and
which matter we use to probe the latter) unavoidably shows up at the smallest
scales, read the Planck length $l_p$, whichever are the circumstances we are
considering. Some consequences are then explored of this nonlocality when
embodied in the fabric itself of spacetime by endowing the latter with a
minimum length $L$, in particular the well-known and intriguing fact that this
brings to get the field equations, and all of gravity with it, as a
statistical-mechanical result. This is done here probing the neighborhood of a
would-be (in ordinary spacetime) generic event through lightsheets (instead of
spacelike or timelike geodesic congruences as in other accounts) from it. The
tools for these derivations are nonlocal quantities, and among them the
minimum-length Ricci scalar stands out both for providing micro degrees of
freedom for gravity in the statistical account and for the fact that
intriguingly the ordinary, or `classical', Ricci scalar can not be recovered
from it in the $L\to 0$ limit. Emphasis is put on that classical gravity is
generically obtained this way for $\hbar\ne 0$, but not in the $\hbar\to 0$
limit (the statistically derived field equations become singular in this
limit), adding to previous results in this sense. (truncated Abstract; see the
paper for full Abstract)
