  We consider the problem of minimizing cost among one-to-one assignments of
$n$ jobs onto $n$ machines. The random assignment problem refers to the case
when the cost associated with performing jobs on machines are random variables.
Aldous established the expected value of the smallest cost, $A_n$, in the
limiting $n$ regime. However the distribution of the minimum cost has not been
established yet. In this paper we conjecture some distributional properties of
matchings in matrices. If this conjecture is proved, this will establish that
$\sqrt{n}(A_n - E(A_n)) \overset{w}{\Rightarrow} N(0,2)$. We also establish the
limiting distribution for a special case of the Random Assignment Problem.
