论文标题
高维圆环上弱避免自我的步行的缩放限制
The scaling limit of the weakly self-avoiding walk on a high-dimensional torus
论文作者
论文摘要
我们证明，如果$ d $维离散的圆环在$ d $ d $ d的范围内的缩放限制是连续曲线上的布朗尼运动，那么如果重新缩放的步行的长度为$ o（v^{1/2}）$，其中$ v $是the $ d $ d> 4 $，则$ v $是$ o（$ v $的数量（点数）。我们还证明，由此产生的圆环布朗运动的扩散常数与在$ \ mathbb {z}^d $上的通常弱自我避免自我避免行走的缩放限制的扩散常数相同。这进一步表明了这一事实，即圆环上弱的避免自我的步行模型不会感觉到它在圆环上，直到它达到$ v^{1/2} $步骤，我们认为这是锋利的。
We prove that the scaling limit of the weakly self-avoiding walk on a $d$-dimensional discrete torus is Brownian motion on the continuum torus if the length of the rescaled walk is $o(V^{1/2})$ where $V$ is the volume (number of points) of the torus and if $d>4$. We also prove that the diffusion constant of the resulting torus Brownian motion is the same as the diffusion constant of the scaling limit of the usual weakly self-avoiding walk on $\mathbb{Z}^d$. This provides further manifestation of the fact that the weakly self-avoiding walk model on the torus does not feel that it is on the torus up until it reaches about $V^{1/2}$ steps which we believe is sharp.
