  It is well known that the metal-insulator transition in two dimensions for
non-interacting fermions takes place at infinitesimal disorder. In contrast,
the superconductor-to-insulator transition takes place at a finite critical
disorder (on the order of V_c ~ 2t), where V is the typical width of the
distribution of random site energies and t is the hopping scale. In this
article we compare the localization/delocalization properties of one and two
particles. Whereas the metal-insulator transition is a consequence of
single-particle Anderson localization, the superconductor-insulator transition
(SIT) is due to pair localization - or, alternatively, fluctuations of the
phase conjugate to pair density. The central question we address is how
superconductivity emerges from localized single-particle states. We address
this question using inhomogeneous mean field theory and quantum Monte Carlo
techniques and make several testable predictions for local spectroscopic probes
across the SIT. We show that with increasing disorder, the system forms
superconducting blobs on the scale of the coherence length embedded in an
insulating matrix. In the superconducting state, the phases on the different
blobs are coherent across the system whereas in the insulator long-range phase
coherence is disrupted by quantum fluctuations. As a consequence of this
emergent granularity, we show that the single-particle energy gap in the
density of states survives across the transition, but coherence peaks exist
only in the superconductor. A characteristic pseudogap persists above the
critical disorder and critical temperature, in contrast to conventional
theories. Surprisingly, the insulator has a two-particle gap scale that
vanishes at the SIT despite a robust single-particle gap.
