  Disorder-free localization is a novel mechanism for ergodicity breaking which
can occur in interacting quantum many-body systems such as lattice gauge
theories (LGTs). While the nature of the quantum localization transition (QLT)
is still debated for conventional many-body localization, here we provide the
first comprehensive characterization of the QLT in two dimensions (2D) for a
disorder-free case. Disorder-free localization can appear in homogeneous 2D
LGTs such as the U(1) quantum link model (QLM) due to an underlying classical
percolation transition fragmenting the system into disconnected real-space
clusters. Building on the percolation model, we characterize the QLT in the
U(1) QLM through a detailed study of the ergodicity properties of finite-size
real-space clusters via level-spacing statistics and localization in
configuration space. We argue for the presence of two regimes - one in which
large finite-size clusters effectively behave non-ergodically, a result
naturally accounted for as an interference phenomenon in configuration space
and the other in which all large clusters behave ergodically. As one central
result, in the latter regime we claim that the QLT is equivalent to the
classical percolation transition and is hence continuous. Utilizing this
equivalence we determine the universality class and critical behaviour of the
QLT from a finite-size scaling analysis of the percolation problem.
