94-145 Brydges D., Slade, G.

The diffusive phase of a model of self-interacting walks (102K, LaTeX) May 26, 94

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Abstract. We consider simple random walk on ${\bf Z}^d$ perturbed by a factor $\exp[\beta T^{-p} J_T]$, where $T$ is the length of the walk and $J_T = \sum_{0 \leq i < j \leq T} \delta_{\omega(i),\omega(j)}$. For $p=1$ and dimensions $d \geq 2$, we prove that this walk behaves diffusively for all $-\infty < \beta < \beta_0$, with $\beta_0 >0$. For $d>2$ the diffusion constant is equal to $1$, but for $d=2$ it is renormalized. For $d=1$ and $p=3/2$, we prove diffusion for all real $\beta$ (positive or negative). For $d>2$ the scaling limit is Brownian motion, but for $d \leq 2$ it is the Edwards model (with the ``wrong'' sign of the coupling when $\beta >0$) which governs the limiting behaviour; the latter arises since for $p=\frac{4-d}{2}$, $T^{-p}J_T$ is the discrete self-intersection local time. This establishes existence of a diffusive phase for this model. Existence of a collapsed phase for a very closely related model has been proven in work of Bolthausen and Schmock.

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