We derive enumeration formulas for families of nonintersecting lattice paths with given starting and end points and a given total number of North-East turns. These formulas are important for the computation of Hilbert series for determinantal and pfaffian rings.

To any positive contraction Q on `2(W ), there is associated a determinantal probability measure PQ on 2W , where W is a denumerable set. Let 0 be a countable sofic finitely generated group and G = (0, E) be a Cayley graph of 0. We show that if Q1 and Q2 are two 0-equivariant positive contractions on `2(0) or on `2(E) with Q1 ≤ Q2, then there exists a 0-invariant monotone coupling of the corres...

A natural and efficient method for producing numerous examples of interesting schemes is to consider the vanishing locus of the minors of a homogeneous polynomial matrix. If the matrix satisfies certain genericity conditions then the resulting schemes have a number of well described properties. These objects have been studied in both a classical context and a modern context and go by the name o...

Let (X1, . . . , Xd) be a random nonnegative integer vector. Many conditions are known to imply a central limit theorem for a sequence of such random vectors, for example, independence and convergence of the normalized covariances, or various combinatorial conditions allowing the application of Stein’s method, couplings, etc. Here, we prove a central limit theorem directly from hypotheses on th...

We propose the first algebraic determinantal formula to enumerate random rhombus tilings filling a centro-symmetric octagon of any size. This result uses the GesselViennot technique and generalizes to any octagon a former specialized formula by Elnitsky.

A determinantal approximation is obtained for the permanent of a doubly stochastic matrix. For moderate-deviation matrix sequences, the asymptotic relative error is of order O(n−1). keywords: Doubly stochastic Dirichlet distribution; Maximum-likelihood projection; Sinkhorn projection

Some inequalities in 2-inner product spaces generalizing Bessel’s result that are similar to the Boas-Bellman inequality from inner product spaces, are given. Applications for determinantal integral inequalities are also provided.

We show how the Riemann–Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete Bessel kernel are computed as examples.

This research is aimed to give a determinantal definition for the q-Appell polynomials and show some classical properties as well as find some interesting properties of the mentioned polynomials in the light of the new definition.