Deconfinement transition and localization of Dirac modes in finite-temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math> gauge theory on the lattice

We study the localization properties of eigenmodes staggered Dirac operator across deconfinement transition in finite-temperature ${\mathbb{Z}}_{3}$ pure gauge theory on lattice $2+1$ dimensions. This allows for nontrivial tests sea-islands picture localization, according to which low modes should localize favorable Polyakov-loop fluctuations deconfined phase a theory. observe localized theory, both real sector, where they are expected, and complex sectors, not. Our findings expose limitations standard call its refinement. An improved picture, spatial hopping terms play more prominent role, is proposed found be excellent agreement with numerical results.