  We report on tunneling experiment between two quantum Hall droplets separated
by a nearly ideal tunnel barrier. The device is produced by cleaved edge
overgrowth that laterally juxtaposes two two-dimensional electron systems
across a high quality semiconductor barrier. The dramatic evolution of the
tunneling characteristics is consistent with the magnetic field-dependent
tunneling between the coupled edge states of the quantum Hall droplets. We
identify a series of quantum critical points between successive strong and weak
tunneling regimes that are reminiscent of the plateau-transitions in quantum
Hall effect. Scaling analysis shows that the conductance near the critical
magnetic fields $B_{c}$ is a function of a single scaling argument
$|B-B_{c}|T^{-\kappa}$, where the exponent $\kappa = 0.42$. This puzzling
resemblance to a quantum Hall-insulator transition points to the significance
of interedge correlation in the lateral tunneling of quantum Hall droplets.
