论文标题
临界2D渗透的化学亚施用性
Chemical subdiffusivity of critical 2D percolation
论文作者
论文摘要
我们表明，在化学距离（即，在固有的图指标中），在二维临界渗透的初期无限簇（IIC）上随机行走是替代的。 Kesten（1986）著名地表明，对于欧几里得距离是正确的，但众所周知，化学距离通常在渐近上更大。更普遍地，我们表明，只要有多尺度覆盖图的方式，因此，化学距离中的亚物质性具有多项式体积生长的固定随机图，因此“深斑块”具有“薄骨架”。我们的估计值是定量的，并根据一臂和两臂指数的明确界限$η_2>η_1> 0 $：对于$ d $维二维模型，随机步行量表的$ t $ t $ thept缩放后的平均化学位移比$ t^{1/p.1/p.1/ps \ [ β<2 + \ \ frac {η_2-η_1} {d-η_1} \，.。 \]使用$η_2=η_1 + 1/4 $和$η_1= 5/48 $的猜想值，对于2D晶格，后一个数量为$ 2 + 12/91 $。
We show that random walk on the incipient infinite cluster (IIC) of two-dimensional critical percolation is subdiffusive in the chemical distance (i.e., in the intrinsic graph metric). Kesten (1986) famously showed that this is true for the Euclidean distance, but it is known that the chemical distance is typically asymptotically larger. More generally, we show that subdiffusivity in the chemical distance holds for stationary random graphs of polynomial volume growth, as long as there is a multi-scale way of covering the graph so that "deep patches" have "thin backbones". Our estimates are quantitative and give explicit bounds in terms of the one and two-arm exponents $η_2 > η_1 > 0$: For $d$-dimensional models, the mean chemical displacement after $T$ steps of random walk scales asymptotically slower than $T^{1/β}$, whenever \[ β< 2 + \frac{η_2-η_1}{d-η_1}\,. \] Using the conjectured values of $η_2 = η_1 + 1/4$ and $η_1 = 5/48$ for 2D lattices, the latter quantity is $2+12/91$.
