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We study the singularity probability of random integer matrices. Concretely, the probability that a random $n times n$ matrix, with integer entries chosen uniformly from ${-m,ldots,m}$, is singular. This problem has been well studied in two regimes: large $n$ and constant $m$; or large $m$ and constant $n$. In this paper, we extend previous techniques to handle the regime where both $n,m$ are large. We show that the probability that such a matrix is singular is $m^{-cn}$ for some absolute constant $c>0$. We also provide some connections of our result to coding theory.
We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrice
We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matr
Let $xi$ be a non-constant real-valued random variable with finite support, and let $M_{n}(xi)$ denote an $ntimes n$ random matrix with entries that are independent copies of $xi$. For $xi$ which is not uniform on its support, we show that begin{alig
We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the
We introduce a simple logical inference structure we call a $textsf{spanoid}$ (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry, algebra (arrangements of hypersurfaces