93-195 Toth Balint

The Bond-True Self-Avoiding Walk on Z. II: Local Limit Theorem (21K, AmSTeX (ams preprint style)) Jun 29, 93

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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 $\left(1-\text{e}^{-s/\alpha}\right)^{-1}=\frac\alpha s+O(1)$.

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