  We present a single crystal Cu NMR study of the cuprate superconductor
La$_{2-x}$Sr$_{x}$CuO$_4$ with hole doping levels between $x=2$\% and $8$\%.
Measurements with short spin echo times enable us to systematically study the
local properties of the electronic spin system in the region of the phase
diagram where the material evolves from the insulating antiferromagnetic parent
phase to the superconducting state. We find evidence for qualitative changes as
the Sr concentration increases through $x=5$\% in both NMR spectra and
relaxation times, which we interpret as signatures of a low-temperature
transition from a state with disconnected metallic islands to a granular metal
with tunneling between the grains. These results provide microscopic insight
into the physics of a doped charge-transfer insulator in the presence of
quenched substitutional disorder, indicate that a connectivity transition
occurs, and demonstrate that complex nanoscale electronic phase separation is
ubiquitous in LSCO around the doping level where superconductivity first
appears in the phase diagram.
