  We propose a model of quasiparticle tunneling across the high-$T_c$
superconductor-insulator-normal metal junction based on the different
mechanisms for tunneling of electrons at positive bias and dissociating
polaronic Cooper pairs and large polarons at negative bias, and the gap
inhomogeneity (i.e., multi-gap) picture. We show that the main features of the
tunneling spectra such as low-bias U- and V-shaped features, asymmetry and
high-bias dip-hump features, their temperature and doping dependences, and
shoulders inside the conductance peaks observed in high-$T_c$ cuprates arise
naturally from the model. The experimental tunneling spectra of
$\rm{Ba_2Sr_2CaCu_2O_{8+\delta}}$ are fitted quite well by taking into account
the distribution of BCS and polaronic gap values.
