  We show that for an $n\times n$ random symmetric matrix $A_n$, whose entries
on and above the diagonal are independent copies of a sub-Gaussian random
variable $\xi$ with mean $0$ and variance $1$, \[\mathbb{P}[s_n(A_n) \le
\epsilon/\sqrt{n}] \le O_{\xi}(\epsilon^{1/8} + \exp(-\Omega_{\xi}(n^{1/2})))
\quad \text{for all } \epsilon \ge 0.\] This improves a result of Vershynin,
who obtained such a bound with $n^{1/2}$ replaced by $n^{c}$ for a small
constant $c$, and $1/8$ replaced by $(1/8) + \eta$ (with implicit constants
also depending on $\eta > 0$). Furthermore, when $\xi$ is a Rademacher random
variable, we prove that \[\mathbb{P}[s_n(A_n) \le \epsilon/\sqrt{n}] \le
O(\epsilon^{1/8} + \exp(-\Omega((\log{n})^{1/4}n^{1/2}))) \quad \text{for all }
\epsilon \ge 0.\] The special case $\epsilon = 0$ improves a recent result of
Campos, Mattos, Morris, and Morrison, which showed that $\mathbb{P}[s_n(A_n) =
0] \le O(\exp(-\Omega(n^{1/2}))).$
  The main innovation in our work are new notions of arithmetic structure --
the Median Regularized Least Common Denominator and the Median Threshold, which
we believe should be more generally useful in contexts where one needs to
combine anticoncentration information of different parts of a vector.
