  In 1927 P\'olya proved that the Riemann Hypothesis is equivalent to the
hyperbolicity of Jensen polynomials for the Riemann zeta function $\zeta(s)$ at
its point of symmetry. This hyperbolicity has been proved for degrees $d\leq
3$. We obtain an asymptotic formula for the central derivatives
$\zeta^{(2n)}(1/2)$ that is accurate to all orders, which allows us to prove
the hyperbolicity of a density $1$ subset of the Jensen polynomials of each
degree. Moreover, we establish hyperbolicity for all $d\leq 8$. These results
follow from a general theorem which models such polynomials by Hermite
polynomials. In the case of the Riemann zeta function, this proves the GUE
random matrix model prediction in derivative aspect. The general theorem also
allows us to prove a conjecture of Chen, Jia, and Wang on the partition
function.
