We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents:

where d_w is the walk dimension, d_f is the fractal dimension, d_s is the spectral dimension, and \(\tilde{\zeta }\) 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 \(\tilde{\zeta } \geqslant 0\) exist, then d_w and d_s exist, and the aforementioned equalities hold. Moreover, our primary new estimate \(d_{w} \geqslant d_{f} + \tilde{\zeta }\) is established for all \(\tilde{\zeta } \in \mathbb{R}\).

For the uniform infinite planar triangulation (UIPT), this yields the consequence d_w=4 using d_f=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and \(\tilde{\zeta }=0\) (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 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.

我们研究了单模随机网络一般设置中“爱因斯坦关系”的有效性。这些是与缩放指数相关的等式：

其中d _w是行走维数，d _f是分形维数，d _s是谱维数，\(\tilde{\zeta }\)是阻力指数。粗略地说，这将随机游走器的平均位移和返回概率与底层介质的密度和电导率联系起来。我们证明，如果d _f和\(\tilde{\zeta } \geqslant 0\)存在，则d _w和d _s存在，并且上述等式成立。此外，我们的主要新估计\(d_{w} \geqslant d_{f} + \tilde{\zeta }\)是为所有\(\tilde{\zeta } \in \mathbb{R}\)建立的。

对于均匀无限平面三角剖分 (UIPT)，使用 d f =4 ( _{Angel in} Geom. Funct. Anal. 13(5):935–974, 2003) 和\ (\tilde{\ zeta }=0\)（根据 Gwynne-Miller 2020 和（Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020），根据刘维尔量子引力理论而建立）。Gwynne 和 Hutchcroft (2018) 之前使用更精细的方法得出了d _w =4的结论。一个新的结果是，对于均匀无限 Schnyder-wood 装饰三角剖分，d _w = d _f ，这意味着简单随机游走是次扩散的，因为d _f > 2。