Random matrices, non-backtracking walks, and orthogonal polynomials

Several well-known results from the random matrix theory, such as Wigner's law and the Marchenko--Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a role in this approach.Comment: (more) minor change

Resolvent of Large Random Graphs

We analyze the convergence of the spectrum of large random graphs to the spectrum of a limit infinite graph. We apply these results to graphs converging locally to trees and derive a new formula for the Stieljes transform of the spectral measure of such graphs. We illustrate our results on the uniform regular graphs, Erdos-Renyi graphs and preferential attachment graphs. We sketch examples of application for weighted graphs, bipartite graphs and the uniform spanning tree of n vertices.Comment: 21 pages, 1 figur

Moment curves and cyclic symmetry for positive Grassmannians

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map.Comment: 18 pages. v2: Minor change

Exact Reconstruction using Beurling Minimal Extrapolation

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc). Generalized minimal extrapolation shares related geometric properties with basis pursuit of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points,can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.Comment: 27 pages, 3 figures version 2 : minor changes and new titl

The totally nonnegative Grassmannian is a ball

We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.Comment: 19 pages. v2: Exposition improved in many place