  Some aspects of the functional RG (FRG) approach to pinned elastic manifolds
(of internal dimension $d$) at finite temperature $T>0$ are reviewed and
reexamined in this much expanded version of [Europhys. Lett. {\bf 76} 457
(2006)]. The particle limit $d=0$ provides a test for the theory: there the FRG
is equivalent to the decaying Burgers equation, with viscosity $\nu \sim T$ -
both being formally irrelevant. Analogy between Kolmogorov scaling and FRG
cumulant scaling is discussed. Next we prove that the FRG function $R(u)$ and
higher cumulants defined from the field theory can be obtained {\it for any
$d$} from moments of a renormalized potential defined in an sliding harmonic
well. This allows to measure the fixed point function $R(u)$ in numerics and
experiments. In $d=0$ the beta function (of the inviscid limit) is obtained
from first principles to four loop. For Sinai model (uncorrelated Burgers
initial velocities) the ERG hierarchy can be solved and the exact function
$R(u)$ is obtained. Connections to exact solutions for the statistics of shocks
in Burgers and to ballistic aggregation is detailed. A relation is established
between the size distribution of shocks and the one for droplets. A droplet
solution to the ERG functional hierarchy is found for any $d$, and the form of
$R(u)$ in the thermal boundary layer is related to droplet probabilities. These
being known for the $d=0$ Sinai model the function $R(u)$ is obtained there at
any $T$. Consistency of the $\epsilon=4-d$ expansion in one and two loop FRG is
studied from first principles, and connected to shock and droplet relations
which could be tested in numerics.
