  The myopic (or `true') self-avoiding walk model (MSAW) was introduced in the
physics literature by Amit, Parisi and Peliti (1983). It is a random motion in
Z^d pushed towards domains less visited in the past by a kind of negative
gradient of the occupation time measure. We investigate the asymptotic
behaviour of MSAW in the non-recurrent dimensions. For a wide class of
self-interaction functions, we identify a natural stationary (in time) and
ergodic distribution of the environment (the local time profile) as seen from
the moving particle and we establish diffusive lower and upper bounds for the
displacement of the random walk. For a particular, more restricted class of
interactions, we prove full CLT for the finite dimensional distributions of the
displacement. This result settles part of the conjectures (based on
non-rigorous renormalization group arguments) in Amit, Parisi and Peliti
(1983). The proof of the CLT follows the non-reversible version of
Kipnis-Varadhan-theory. On the way to the proof we slightly weaken the
so-called graded sector condition (that is: we slightly enhance the
corresponding statement).
