电阻网络的算子理论
Operator theory of electrical resistance networks
摘要：
电阻网络是具有本征（电阻）度量$R$的加权图$(G,c)$。我们将电阻网络嵌入到有限能量函数的Hilbert空间${\mathcal H}_{\mathcal E}$中。我们使用的阻力度量研究${\mathcal H}_{\mathcal E}$，反之亦然，并表明嵌入图像的顶点$\{v_x\}$形成再生核的希尔伯特空间。我们还得到了一个离散版本的高斯-格林公式的电阻网络，并表明，电阻网络，支持有限能量的非常数调和函数具有一定类型的\emph{boundary}。我们得到一个解析边界表示的调和函数的有限能量在某种意义上类似于泊松或马丁边界表示，但不同的假设，并为不同的功能。在这个过程中，我们构造了一个有限能量的“光滑”函数的稠密空间，并得到了${\mathcal H}_{\mathcal E}$的Gel'fand三元组。这使我们能够将电阻网络表示为由顶点索引的高斯随机变量系统。我们还研究了${\mathcal H}_{\mathcal E}$上$Δ$的谱表示，并展示了非零缺陷如何导致非平凡边界。以上所有的结果都是由${\mathcal H}_{\mathcal E}$的算子理论检测到的，而不是由$\ell^2$检测到的。我们的结果适用于各向同性铁磁体的海森堡模型，改进了R。T.权力的问题，长程秩序（在参考KMS国家的$C^\ast$ -代数的模型）。
英文摘要：
A resistance network is a weighted graph $(G,c)$ with intrinsic (resistance) metric $R$. We embed the resistance network into the Hilbert space ${\mathcal H}_{\mathcal E}$ of functions of finite energy. We use the resistance metric to study ${\mathcal H}_{\mathcal E}$, and vice versa and show that the embedded images of the vertices $\{v_x\}$ form a reproducing kernel for this Hilbert space. We also obtain a discrete version of the Gauss-Green formula for resistance networks and show that resistance networks which support nonconstant harmonic functions of finite energy have a certain type of \emph{boundary}. We obtain an analytic boundary representation for the harmonic functions of finite energy in a sense analogous to the Poisson or Martin boundary representations, but with different hypotheses, and for a different class of functions. In the process, we construct a dense space of "smooth" functions of finite energy and obtain a Gel'fand triple for ${\mathcal H}_{\mathcal E}$. This allows us to represent the resistance network as a system of Gaussian random variables indexed by vertices. We also study the spectral representation for $Δ$ on ${\mathcal H}_{\mathcal E}$ and show how nonzero defect entails a nontrivial boundary. All of the above are are detected by the operator theory of ${\mathcal H}_{\mathcal E}$ but not $\ell^2$. Our results apply to the Heisenberg model for the isotropic ferromagnet, improving earlier results of R. T. Powers on the problem of long-range order (in reference to KMS states on the $C^\ast$-algebra of the model).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
