论文标题
单型随机图中的缩放指数之间的关系
Relations between scaling exponents in unimodular random graphs
论文作者
论文摘要
我们研究了“爱因斯坦关系”在单模型随机网络的一般环境中的有效性。这些是与缩放指数相关的平等性：$ d_w = d_f + \tildeζ$和$ d_s = 2 d_f/d_w $，其中$ d_w $是步行维度，$ d_f $是分形维度，$ d_s $是光谱维度，$ \ \ \ \ \ \ tilde eilde面$是电阻。粗略地说，这将随机步行者的平均位移和返回概率与基础介质的密度和电导率联系起来。我们表明，如果存在$ d_f $和$ \tildeζ\ geq 0 $，则存在$ d_w $和$ d_s $，并且上述平等性。此外，我们的主要新估计是关系$ d_w \ geq d_f + \tildeζ$，它是针对所有$ \tildeζ\ in \ mathbb {r} $建立的。 对于统一的无限平面三角剖分（UIPT），这将产生后果$ d_w = 4 $，使用$ d_f = 4 $（Angel 2003）和$ \ \tildeζ= 0 $（在此建立在Liouville量子重力理论的结果，在Gwynne-Miller 2017和Ding-Gwynne 2020之后，在此建立。结论$ d_w = 4 $以前是Gwynne and Hutchcroft（2018）使用更多精心制作的方法建立的。一个新的结果是，$ d_w = d_f $用于统一的无限施尼德木材装饰的三角剖分，这意味着简单的随机步行是不合适的，因为$ d_f> 2 $（ding and Gwynne 2020）。对于$ \ m athbb {z}^2 $的随机步行，由带有指数$γ> 0 $的高斯自由场的导电驱动，一个$ d_f = d_f = d_f（γ）$和$ \tildeζ= 0 $（biskup，ding，ding和goswami 2020）。这会产生$ d_s = 2 $和$ d_w = d_f $，确认了这些作者的两个预测。
We investigate the validity of the "Einstein relations" in the general setting of unimodular random networks. These are equalities relating scaling exponents: $d_w = d_f + \tildeζ$ and $d_s = 2 d_f/d_w$, where $d_w$ is the walk dimension, $d_f$ is the fractal dimension, $d_s$ is the spectral dimension, and $\tildeζ$ is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if $d_f$ and $\tildeζ \geq 0$ exist, then $d_w$ and $d_s$ exist, and the aforementioned equalities hold. Moreover, our primary new estimate is the relation $d_w \geq d_f + \tildeζ$, which is established for all $\tildeζ \in \mathbb{R}$. For the uniform infinite planar triangulation (UIPT), this yields the consequence $d_w=4$ using $d_f=4$ (Angel 2003) and $\tildeζ=0$ (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2017 and Ding-Gwynne 2020). The conclusion $d_w=4$ had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that $d_w = d_f$ for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since $d_f > 2$ (Ding and Gwynne 2020). For the random walk on $\mathbb{Z}^2$ driven by conductances from an exponentiated Gaussian free field with exponent $γ> 0$, one has $d_f = d_f(γ)$ and $\tildeζ=0$ (Biskup, Ding, and Goswami 2020). This yields $d_s=2$ and $d_w = d_f$, confirming two predictions of those authors.
