**Originator(s):** Van Vu (UCSD, vanvu@ucsd.edu)

**Definitions:**

**Question:**
What is the "usual" magnitude of the determinant of an *n*-by-*n*
symmetric random matrix? That is, is there a function *f(n)* such that
*|det M(n)|* is almost always close to *f(n)*? Here the entries
*a _{i,j}* of

**Background/motivation:**
This is a variant of the well known question of estimating *det M(n)*,
where *M(n)* is a random Bernoulli matrix not required to be symmetric.
Komlos (1967) proved that almost surely *M(n)* is not singular. Thus
*det M(n) > 0*, but *det M(n)* is divisible by
*2 ^{n-1}*, so in fact

**Comments/Partial results:**
Tao, Vu, and Costello [in preparation] proved Komlos's statement for the
symmetric problem: almost surely the matrix is non singular. This
answers a question of Weiss. It does not seem that the apporoach is
appropriate for the determinant.

**References:**

[TV] Tao T. and Vu V. On random (-1,1) matrices: Singularity and Determinant,
submitted (extended abstract in STOC 2005). See also
http://www.math.ucsd.edu/~vanvu/papers.html.

Posted 04/06/05; Last update 04/06/05