An expansion in a small parameter of the probability that a random determinant in the field GF(2) equals 1. (English. Ukrainian original) Zbl 1001.60007

Masol, V. V.

Theory Probab. Math. Stat. 64, 117-121 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 102-105 (2001).

Consider the random \(n\)th order determinant \(\Delta_n=|a_{ij}|_{i,j\in I}\), \(I={1,2,\dots,n}\), where \(a_{ij}\) are independent random variables with \(p= P(a_{ij}=0)= 1-P(a_{ij}=1)= (1+\varepsilon x_{ij})/2\) if \((i,j)\in D\) and \(p= (1-\varepsilon x_{ij})/2\) if \((i,j)\in T\); \(D\) and \(T\) are disjoint subsets of \(I\times I\) and \(D\cup T=I\times I\). Using a recurrence formula the author finds an expansion of the probability \(P(\Delta_n=1)\) with respect to a small parameter \(\varepsilon\).

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60C05

Combinatorial probability

Keywords:

random determinant; expansion; small parameter