Fixed-energy harmonic functions
Fixed-energy harmonic functions
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固定能量谐波函数
DOI：
10.19086/da.2730
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
R. Kenyon
中科院分区：
文献类型：
--
作者：
Aaron Abrams;R. Kenyon
研究了具有Dirichlet边界条件的有限图上调和函数的电导到边能的映射。我们证明了对于任何相容的无环取向和能量选择，存在一个唯一的电导选择，使得相关的谐波函数实现了这些取向和能量。我们称相关函数为简谐函数。对于有理能和边界数据，${\mathbb Q}$上的伽罗瓦群${\mathbb Q}^{tr}$（全实数代数）排列作用于相容无环取向集的调和函数。一个结果是某些多边形不能被有理面积矩形平铺。
对于平面图，有一个共轭函数，它们一起构成了一个“定能”解析函数的实部和虚部。在${\mathbb Z}^2$（和固定的南/西方向）的平面缩放极限中，这些函数满足柯西-黎曼方程的非线性模拟，即\begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*}。我们给出了这些函数的黎曼映射定理的模拟，以及在离散和连续设置中寻找解的变分方法。
We study the map from conductances to edge energies for harmonic functions on finite graphs with Dirichlet boundary conditions. We prove that for any compatible acyclic orientation and choice of energies there is a unique choice of conductances such that the associated harmonic function realizes those orientations and energies. We call the associated function enharmonic. For rational energies and boundary data the Galois group of ${\mathbb Q}^{tr}$ (the totally real algebraic numbers) over ${\mathbb Q}$ permutes the enharmonic functions, acting on the set of compatible acyclic orientations. A consequence is the non-tileability of certain polygons by rational-area rectangles.
For planar graphs there is an enharmonic conjugate function, together these form the real and imaginary parts of a "fixed energy" analytic function. In the planar scaling limit for ${\mathbb Z}^2$ (and the fixed south/west orientation), these functions satisfy a nonlinear analog of the Cauchy-Riemann equations, namely \begin{eqnarray*}u_xv_y &=& 1\\u_yv_x&=&-1.\end{eqnarray*} We give an analog of the Riemann mapping theorem for these functions, as well as a variational approach to finding solutions in both the discrete and continuous settings.