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

Diffusive limit for the myopic (or "true") self-avoiding random walk in three and more dimension