  In this study, we compare the radio, optical and environmental properties of
GRGs with those of a control sample of smaller RGs we found in the three
LOw-Frequency ARray (LOFAR) deep fields, namely the Bootes, ELAIS-N1, Lockman
Hole, for a total area of about 95 deg^2. We inspected the LOFAR deep fields
and created a catalogue of 1609 extended radio galaxies (ERGs). By visual
inspection, we identified their host galaxies and spectroscopically or
photometrically classified 280 of these as GRGs. We studied their properties,
such as their accretion state, stellar mass and star formation rate (SFR) using
deep optical and infrared survey data. Moreover, we explored the environment in
terms of the surface number density of neighbouring galaxies within these
surveys. Integrated flux densities and radio luminosities were also determined
for a subset of ERGs through available survey images at 50, 150, 610, and 1400
MHz to compute integrated spectral indices. Considering the fraction of GRGs
displaying an FRII morphology alongside the host galaxy properties, we suggest
that GRGs consistently possess sufficient power to overcome jet frustration
caused by the interstellar medium. Moreover, clear differences emerge in the
environmental densities between GRGs and smaller RGs, using the number of
neighbouring galaxies within 10 Mpc from the host galaxy as a proxy. GRGs
preferentially reside in sparser environments compared to their smaller
counterparts. In particular, only 3.6% of the GRGs reside within a 3D comoving
distance of 5 Mpc from a previously reported galaxy cluster. We found that
larger sources exhibit steeper integrated spectral indices, suggesting that
GRGs are late-stage versions of RGs. These results suggest that GRGs are
amongst the oldest radio sources with the most stable nuclear activity that
reside in sparse environments.
