Probabilistic properties of systems of random linear equations over finite algebraic structures
This is a simple review of the former Soviet Union's probabilists' works on the limiting properties of random matrices and systems of random linear equations over finite algebraic structures. The first research works in this field were initiated by D. Slepian, P. Erdös and A. Rényi in 1955 and 1963, then followed by G. V. Balakin, V. F. Kolchin and the authors. Let \({\mathbf A} = (a_{ij})_{1 \leq i \leq N,1 \leq j \leq n}\), \(N \leq n\), be a random matrix, where \(a_{ij}\) are jointly independent random variables taking values in a finite field or ring. Then the asymptotic distributions of the linear rank of \({\mathbf A}\) and the number of solutions of (homogeneous or non-homogeneous) systems of linear equations with random coefficient matrix \({\mathbf A}\) are considered. The invariance theory has been developed. It means under the restrictions that the probabilities of \(a_{ij}\) are bounded away from zero in some manner the limit properties are invariant to a distribution of the matrix \({\mathbf A}\). Many results have been collected in ``Selected problems of probabilistic combinatorics'' (Kiev, 1986) by the authors and \textit{M. N. Sovchuk}.
scientific article; zbMATH DE number 780307
scientific article