We report some progress on the old problem of estimating the probability,
P
n
{P_n}
, that a random
n
×
n
±
1
n \times n \pm 1
-matrix is singular: Theorem. There is a positive constant
ε
\varepsilon
for which
P
n
>
(
1
−
ε
)
n
{P_n} > {(1 - \varepsilon )^n}
. This is a considerable improvement on the best previous bound,
P
n
=
O
(
1
/
n
)
{P_n} = O(1/\sqrt n )
, given by Komlós in 1977, but still falls short of the often-conjectured asymptotical formula
P
n
=
(
1
+
o
(
1
)
)
n
2
2
1
−
n
{P_n} = (1 + o(1)){n^2}{2^{1 - n}}
. The proof combines ideas from combinatorial number theory, Fourier analysis and combinatorics, and some probabilistic constructions. A key ingredient, based on a Fourier-analytic idea of Halász, is an inequality (Theorem 2) relating the probability that
a
_
∈
R
n
\underline a \in {{\mathbf {R}}^n}
is orthogonal to a random
ε
_
∈
{
±
1
}
n
\underline \varepsilon \in {\{ \pm 1\} ^n}
to the corresponding probability when
ε
_
\underline \varepsilon
is random from
{
−
1
,
0
,
1
}
n
{\{ - 1,0,1\} ^n}
with
P
r
(
ε
i
=
−
1
)
=
P
r
(
ε
i
=
1
)
=
p
Pr({\varepsilon _i} = - 1) = Pr({\varepsilon _i} = 1) = p
and
ε
i
{\varepsilon _i}
’s chosen independently.
With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random
graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$)
$$\Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}].$$
This is tight up to the value of $C_{\eta}$.Comment: 10 page
It is often required to find the probability of the union of given n events A1,...,An.The answer is provided, of course, by the inclusion-exclusion formula: Pr(UAi) = ~i Pr(Ai) -~i
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