Consider a unit-resistive plate in the shape of a regular polygon with 2n sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2n-gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests. 1

The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on

A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for “tripartite ” pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the double-dimer model on bipartite planar graphs. 1

network approaches to electrical impedance tomography

Topology and static response of interaction networks in molecular biology

Abstract. The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally linked with the study of the totally nonnegative Grassmannian. We investigate its cell decomposition, where the cells are the totally nonnegative parts of the matroid strata. The boundary measurements of networks give parametrizations of the cells. We present several different combinatorial descriptions of the cells, study the partial order on the cells, and describe how they are glued to each other.

We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition σ, Pr[grove has type σ] / Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type σ] / Pr[grove has maximal number of trees] is an integer-coefficient polynomial in the entries of the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient polynomial formulas for the pairings of chains in the double-dimer model. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the double-dimer model. These partition probabilities are relevant to multichordal SLE2, SLE4, and SLE8. 1

We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition σ, Pr[grove has type σ] / Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type σ] / Pr[grove has maximal number of trees] is an integer-coefficient polynomial in the entries of the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient polynomial formulas for the pairings of chains in the double-dimer model. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the double-dimer model. These partition probabilities are relevant to multichordal SLE2, SLE4, and SLE8. 1

Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure, i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy’s percolation crossing probabilities, and generalize Kirchhoff’s formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural combinatorial interpretation. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have direct application to connection probabilities for multiple-strand SLE2, SLE8, and SLE4.

Consider a unit-resistive plate in the shape of a regular polygon with 2n sides, in which even-numbered sides are wired to electrodes and odd-numbered sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows one to compute the currents flowing through the electrodes when they are held at specified voltages. We show that the entries of the response matrix of the regular 2n-gon are given by the differences of cotangents of evenly spaced angles, and we describe some connections with the limiting distributions of certain random spanning forests. 1