  We compute the persistence for the $2d$-diffusion equation with random
initial condition, i.e., the probability $p_0(t)$ that the diffusion field, at
a given point ${\bf x}$ in the plane, has not changed sign up to time $t$. For
large $t$, we show that $p_0(t) \sim t^{-\theta(2)}$ with $\theta(2) = 3/16$.
Using the connection between the $2d$-diffusion equation and Kac random
polynomials, we show that the probability $q_0(n)$ that Kac polynomials, of
(even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \sim
n^{-3/4}$. We obtain this result by using yet another connection with the
truncated orthogonal ensemble of random matrices. This allows us to compute
various properties of the zero-crossings of the diffusing field, equivalently
of the real roots of Kac polynomials. Finally, we unveil a precise connection
with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics
at zero temperature.
