  Ever since the discovery of high-temperature superconductivity in cuprates,
gaining microscopic insights into the nature of pairing in strongly correlated,
repulsively interacting fermionic systems has remained one of the greatest
challenges in modern condensed matter physics. Following recent experiments
reporting superconductivity in the bilayer nickelate La$_3$Ni$_2$O$_7$ (LNO)
with remarkably high critical temperatures of $T_c = 80$ K, it has been argued
that the low-energy physics of LNO can be described by the strongly correlated
mixed-dimensional (mixD) bilayer $t-J$ model. Here we investigate this bilayer
system and utilize density matrix renormalization group techniques to establish
an intricate understanding of the model and the magnetically induced pairing
through comparison to the perturbative limit of dominating inter-layer spin
couplings. In particular, this allows us to explain appearing finite-size
effects, firmly establishing the existence of long-range pairing order in the
thermodynamic limit. As the effective model in the perturbative limit is known
to show linear resistivity above the superconducting transition temperature, we
propose a pair-based interpretation of the extended strange metal phase
observed in LNO. By analyzing binding energies, we predict a BEC-BCS crossover
as a function of the Hamiltonian parameters, whereas LNO is anticipated to lie
on the BCS side in vicinity of the transition. We find large binding energies
of the order of the inter-layer coupling that suggest strikingly high critical
temperatures of the Berezinskii-Kosterlitz-Thouless transition, raising the
question whether nickelate superconductors possibly facilitate critical
temperatures above room temperature.
