Abstract. For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally nonnegative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Plücker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the nonplanar setting, we show that a similar formula holds for special choices of Plücker coordinates.

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. Submatrices with low rank correspond to generalizations of conditional independence constraints on collections of random variables. We give a precise graph-theoretic characterization of when submatrices of the covariance matrix have small rank for a general class of mixed graphs that includes directed acyclic and undirected graphs as special cases. Our new trek separation criterion generalizes the familiar $d$-separation criterion. Proofs are based on the trek rule, the resulting matrix factorizations and classical theorems of algebraic combinatorics on the expansions of determinants of path polynomials.Comment: Published in at http://dx.doi.org/10.1214/09-AOS760 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation, as shown by Kodama and the second author. Deodhar components R D of the Grassmannian are in bijection with certain tableaux D called Go-diagrams, and each component is isomorphic to (K * ) a × (K) b for some non-negative integers a and b.Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram D we construct a weighted network N D and its weight matrix W D , whose entries enumerate directed paths in N D . By letting the weights in the network vary over K or K * as appropriate, one gets a parameterization of the Deodhar component R D . One application of such a parameterization is that one may immediately determine which Plücker coordinates are vanishing and nonvanishing, by using the Lindstrom-Gessel-Viennot Lemma. We also give a (minimal) characterization of each Deodhar component in terms of Plücker coordinates. A main tool for us is the work of Marsh and Rietsch [8] on Deodhar components in the flag variety.

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Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient condition in terms of the graph for when a subdeterminant is zero for all covariance matrices that belong to the Gaussian graphical model. Here we extend this result to give explicit cancellation-free formulas for the expansions of nonzero subdeterminants.