  We study the Integrated Density of States (IDS) of the random Schr\"odinger
operator appearing in the study of certain reinforced random processes in
connection with a supersymmetric sigma-model. We rely on previous results on
the supersymmetric sigma-model to obtain lower and upper bounds on the
asymptotic behavior of the IDS near the bottom of the spectrum in all
dimension. We show a phase transition for the IDS between weak and strong
disorder regime in dimension larger or equal to three, that follows from a
phase transition in the corresponding random process and supersymmetric
sigma-model. In particular, we show that the IDS does not exhibit Lifshitz
tails in the strong disorder regime, confirming a recent conjecture. This is in
stark contrast with other disordered systems, like the Anderson model. A Wegner
type estimate is also derived, giving an upper bound on the IDS and showing the
regularity of the function.
