Related papers: Asymptotic enumeration of incidence matrices
We examine the asymptotics of the moments of characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as $N\to\infty$. We focus in particular on the Gaussian Unitary…
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations.…
We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical…
We consider the statistical inference for noisy incomplete binary (or 1-bit) matrix. Despite the importance of uncertainty quantification to matrix completion, most of the categorical matrix completion literature focuses on point estimation…
Based on the work of Green, Tao and Ziegler, we give asymptotics when $N \to \infty$ for the number of $n \times n$ magic squares with their entries being prime numbers in $[0,N]$. For every $n \ge 3$ we give appropriate systems of linear…
We derive asymptotic formulas for the number of integer partitions with given sums of $j$th powers of the parts for $j$ belonging to a finite, non-empty set $J \subset \mathbb N$. The method we use is based on the `principle of maximum…
We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive…
This is a review of the Riemann-Hilbert approach to the large $N$ asymptotics in random matrix models and its applications. We discuss the following topics: random matrix models and orthogonal polynomials, the Riemann-Hilbert approach to…
A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…
We extend the asymptotic formula for counting integral matrices with a given irreducible characteristic polynomial by Eskin, Mozes and Shah in 1996 to the case of counting elements in a maximal order of certain central simple algebra with a…
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system.…
In the sufficiently sparse case, we find the probability that a uniformly random bipartite graph with given degree sequence contains no edge from a specified set of edges. This enables us to enumerate loop-free digraphs and oriented graphs…
A certain class of directed metric graphs is considered. Asymptotics for a number of possible endpoints of a random walk at large times is found.
Does a given system of linear equations with nonnegative constraints have an integer solution? This is a fundamental question in many areas. In statistics this problem arises in data security problems for contingency table data and also is…
A problem from thermodynamic formalism for countable symbolic Markov chains is considered. It concerns asymptotic behavior of the equilibrium measures corresponding to increasing sequences of finite sub-matrices of an infinite nonnegative…
In this paper, we count all non-isomorphic lattices on $n$ elements, containing four reducible elements and having nullity three. This work is in respect of Birkhoff's open problem (which is NP-complete) of counting all finite lattices on…
Let $A_kA_{k-1}\cdots A_1$ be product of some nonnegative 2-by-2 matrices. In general, its elements are hard to evaluate. Under some conditions, we show that $\forall i,j\in\{1,2\},$ $(A_kA_{k-1}\cdots A_1)_{i,j}\sim…
We consider the problem of detecting the presence of a submatrix with larger-than-usual values in a large data matrix. This problem was considered in (Butucea and Ingster, 2013) under a one-parameter exponential family, and one of the test…
In a finite undirected simple graph, a {\it chordless cycle} is an induced subgraph which is a cycle. We propose two algorithms to enumerate all chordless cycles of such a graph. Compared to other similar algorithms, the proposed algorithms…
We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform…