Adding a column of numbers produces “carries” along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a one-dependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group the...

We consider the problem of writing real polynomials as determinants of symmetric linear matrix polynomials. This problem of algebraic geometry, whose roots go back to the nineteenth century, has recently received new attention from the viewpoint of convex optimization. We relate the question to sums of squares decompositions of a certain Hermite matrix. If some power of a polynomial admits a de...

We distinguish a class of random point processes which we call Giambelli compatible point processes. Our definition was partly inspired by determinantal identities for averages of products and ratios of characteristic polynomials for random matrices found earlier by Fyodorov and Strahov. It is closely related to the classical Giambelli formula for Schur symmetric functions. We show that orthogo...

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vande...

The determinantal identities of Hamel and Goulden have recently been shown to apply a tableau-based ninth variation skew Schur functions. Here we extend this approach its results the analogous supersymmetric These tableaux are built on entries taken from an alphabet unprimed primed numbers that may be ordered in myriad different ways, each leading identity. At level corresponding all distinct b...

In this paper, we provide combinatorial interpretations for some determinantal identities involving Fibonacci numbers. We use the method due to Lindström-Gessel-Viennot in which we count nonintersecting n-routes in carefully chosen digraphs in order to gain insight into the nature of some well-known determinantal identities while allowing room to generalize and discover new ones.

We present some obvious physical requirements on gravitational avatars of non-linear elec-trodynamics and illustrate them with explicit determinantal Born–Infeld–Einstein models. A related procedure, using compensating Weyl scalars, permits us to formulate conformally invariant versions of these systems as well. Born–Infeld (BI) electrodynamics [1] has earned its longevity through its elegant, ...

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.

The determinantal conjecture of M. Marcus and G. N. de Oliveira is known in many special cases. The case of 3 × 3 matrices was settled by N. Bebiano, J. K. Merikoski and J. da Providência. The case n = 4 remains open. In this article a technical conjecture is established implying a weakened form of the determinantal conjecture for n = 4.