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

TL;DR

This paper studies the asymptotic behavior of the myopic self-avoiding walk in dimensions three and higher, establishing diffusive bounds and a central limit theorem for certain interaction classes, confirming longstanding conjectures.

Contribution

It identifies stationary distributions and proves diffusive bounds and a CLT for the self-avoiding walk in higher dimensions, advancing understanding of its asymptotic behavior.

Findings

Established diffusive lower and upper bounds for the walk's displacement.

Proved a full CLT for specific interaction classes.

Confirmed parts of the conjectures from the original physics literature.

Abstract

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).