  The Dirac operator in finite temperature QCD is equivalent to the Hamiltonian
of an unconventional Anderson model, with on-site noise provided by the
fluctuations of the Polyakov lines. The main features of its spectrum and
eigenvectors, concerning the density of low modes and their localization
properties, are qualitatively reproduced by a toy-model random Hamiltonian,
based on an Ising-type spin model mimicking the dynamics of the Polyakov lines.
Here we study the low modes of this toy model in the vicinity of the ordering
transition of the spin model, and show that at the critical point the spectral
density at the origin has a singularity, and the localization properties of the
lowest modes change. This provides further evidence of the close relation
between deconfinement, chiral transition and localization of the low modes.
