  The dimer Mott insulator $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ can be tuned
into a metallic and superconducting state upon applying pressure of 1.5 kbar
and more. We have performed dielectric spectroscopy measurements (7 kHz to 5
MHz) on $\kappa$-(BEDT-TTF)$_2$Cu$_2$(CN)$_3$ single crystals as a function of
temperature (down to $T=8$ K) and pressure (up to $p=4$ kbar). At ambient
conditions, a relaxor-like dielectric behavior develops below 50 K that shifts
toward lower temperatures as the crystal is pressurized. Interestingly, a
second peak emerges in $\varepsilon_{1}(T)$ around $T=15$ K, which becomes
strongly enhanced with pressure and is attributed to a small volume fraction of
metallic puddles in the insulating host phase. When approaching the phase
boundary, this peak diverges rapidly reaching $\varepsilon_{1} \approx 10^{5}$.
Our dynamical mean-field theory calculations substantiate that the dielectric
catastrophe at the Mott transition is not caused by closing the energy gap, but
due to the spatial coexistence of correlated metallic and insulating regions.
We discuss the percolative nature of the first-order Mott insulator-to-metal
transition in all details.
