  We study the effects of Coulomb interaction and inter-grain quantum
tunnelling in an array of metallic grains using the phase-functional approach
for temperatures $T$ well below the charging energy $E_{c}$ of individual
grains yet large compared to the level spacing in the grains. When the
inter-grain tunnelling conductance $g\gg1$, the conductivity $\sigma$ in $d$
dimensions decreases logarithmically with temperature
($\sigma/\sigma_{0}\sim1-\frac{1}{2\pi gd}\ln(gE_{c}/T)$), while for $g\to0$,
the conductivity shows simple activated behaviour ($\sigma \sim \exp(-E_c/T)$).
We show, for bare tunnelling conductance $g \gtrsim 1$, that the parameter
$\gamma \equiv g(1-2/(g\pi)\ln(gE_{c}/T))$ determines the competition between
charging and tunnelling effects. At low enough temperatures in the regime
$1\gtrsim \gamma \gg 1/\sqrt{\beta E_{c}}$, a charge is shared among a finite
number $N=\sqrt{(E_{c}/T)/\ln(\pi/2\gamma z)}$ of grains, and we find a soft
activation behaviour of the conductivity, $\sigma\sim
z^{-1}\exp(-2\sqrt{(E_{c}/T)\ln(\pi/2\gamma z)})$, where $z$ is the effective
coordination number of a grain.
