  We study a lattice sigma model which is expected to reflect Anderson
localization and delocalization transition for real symmetric band matrices in
3D, but describes the mixing measure for a vertex reinforced jump process too.
For this model we prove exponential localization at any temperature in a strip,
and more generally in any quasi-one dimensional graph, with pinning (mass) at
only one site. The proof uses a Mermin-Wagner type argument and a transfer
operator approach.
