Brydges D., Slade, G.
The diffusive phase of a model of self-interacting walks
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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.