Probabilistic properties about permanent of boolean matrices

Abstract

It is well known that for random n by m matrices $w=(w_{ij})(1≤i≤n, 1≤j≤m; n≤m)$ such that the $w_{ij}$ are independent random variables which take on value 1 with probability 1/2, the probability of (w:per(w)>0) tends to 1 as m≥n→∞. In this paper, by using this method, we obtain that the probability of (w:per(w)>0) tends to 1 as m≥n→∞ if the number of 1``s in n by m Boolean matrix is more than or equal to nm/2, and by using the existence of 0-rows or 0-columns case, we investigated the limiting distribution of the number of 0-rows or 0-columns with independent components. Moreover we deal with some interpretations of our result to the graph Theory and some applications related to this problems.