Central Limit Theorem for a Weakly Interacting Random Polymer

#### R. van der Hofstad, W. Konig, F. den Hollander

1997, v.3, Issue 1, 1-62

ABSTRACT

The Domb-Joyce model in one dimension is a transformed path measure for simple random walk on $Z$ in which an $n$-step path gets a penalty $e^{-2\beta_n}$ for every self-intersection. Here $\beta_n$ is the strength of repellence, which may depend on $n$. We prove a central limit theorem for the end-to-end distance of the path in the case where $\beta_n\rightarrow 0$ and $n^{3/2} \beta_n \rightarrow \infty$ as $n \rightarrow \infty$. It turns out that the mean grows like $b^* \beta_n^{1/3}n$ and the standard deviation like $c^*\sqrt{n}$, where $b^*$ and $c^*$ are constants that can be identified in terms of a Sturm-Liouville problem. The asymptotic mean shows an interpolation between ballistic behavior ($\beta_n\equiv \beta$) and diffusive behavior ($\beta_n = \beta n^{-3/2}$). Strikingly, the asymptotic standard deviation is independent of $\beta_n$. Our result is closely related to the central limit theorem for the Edwards model (the continuous space-time analogue of the Domb-Joyce model based on Brownian motion on $R$), which is proved in a separate paper.

Keywords: Domb-Joyce model,Knight's theorem for local times of simplerandom walk,time change and scaling,spectral analysis,central limit theorem