Toth Balint
The Bond-True Self-Avoiding Walk on Z.
II: Local Limit Theorem
(21K, AmSTeX (ams preprint style))

ABSTRACT.  The bond-true self-avoiding walk is a nearest neighbour random
walk $X_n$ on $\Bbb Z$, for which the probability of jumping
along a bond of the lattice is proportional to
$\exp(-g\cdot\text{number of previous jumps along that bond})$.
This paper is a continuation of T\'oth (1993), where the local
time process and first hitting times of $X_{\cdot}$ were
investigated. For formal definitions and notation see that
paper. Here we prove a local limit theorem, as
$\alpha\to\infty$, for the distribution of
$\alpha^{-2/3}X_{\theta_{s/\alpha}}$, where $\theta_{s/\alpha}$
is a random time of geometric distribution with mean
$\left(1-\text{e}^{-s/\alpha}\right)^{-1}=\frac\alpha s+O(1)$.