A formula for Plücker coordinates associated with a planar network

条件简单随机游走返回坐标超平面的渐进行为

Asymptotics of returns to the coordinate axes for conditioned planar simple random walks

本文研究了条件简单随机游走返回坐标超平面的渐进行为，通过不同条件下的极限分布分析，揭示了不同情况下分布的多样性及坐标独立性。

Monotone drawings of planar graphs

Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments.

Spectra of biperiodic planar networks

A biperiodic planar network is a pair $(G,c)$ where $G$ is a graph embedded on the torus and $c$ is a function from the edges of $G$ to non-zero complex numbers. Associated to the discrete Laplacian on a biperiodic planar network is its spectrum: a triple $(C,S,ν)$, where $C$ is a curve and $S$ is a divisor on it. We give a complete classification of networks (modulo a natural equivalence) in terms of their spectral data. The space of networks has a large group of cluster automorphisms arising from the $Y-Δ$ transformations. We show that the spectrum provides action-angle coordinates for the d…

Planar flows and Plücker's type quadratic relations over semirings

It is well known, due to Lindström, that the minors of a (real or complex) matrix can be expressed in terms of weights of flows in a planar directed graph. Another classical fact is that there are plenty of homogeneous quadratic relations involving flag minors, or Plücker coordinates of the corresponding flag manifold. Generalizing and unifying these facts and their tropical counterparts, we consider a wide class of functions on $2^{[n]}$ that are generated by flows in a planar graph and take values in an arbitrary commutative semiring, where $[n]=\{1,2,\ldots,n\}$. We show that the ``universa…

The space of circular planar electrical networks

We discuss several parametrizations of the space of circular planar electrical networks. For any circular planar network we associate a canonical minimal network with the same response matrix, called a "standard" network. The conductances of edges in a standard network can be computed as a biratio of Pfaffians constructed from the response matrix. The conductances serve as coordinates that are compatible with the cell structure of circular planar networks in the sense that one conductance degenerates to 0 or infinity when moving from a cell to a boundary cell. We also show how to test if a net…

Generic 3-connected planar constraint systems are not soluble by radicals

We show that planar embeddable 3-connected CAD graphs are generically non-soluble. A CAD graph represents a configuration of points on the Euclidean plane with just enough distance dimensions between them to ensure rigidity. Formally, a CAD graph is a maximally independent graph, that is, one that satisfies the vertex-edge count 2v - 3 = e together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen in the planar case. Let G be a maximally independent 3-connected planar graph, with more than 3 vertices, together with a realis…