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We compute the persistence for the $2d$-diffusion equation with randominitial condition, i.e., the probability $p_0(t)$ that the diffusion field, ata given point ${\bf x}$ in the plane, has not changed sign up to time $t$. Forlarge $t$, 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 randompolynomials, we show that the probability $q_0(n)$ that Kac polynomials, of(even) degree $n$, have no real root decays, for large $n$, as $q_0(n) \simn^{-3/4}$. We obtain this result by using yet another connection with thetruncated orthogonal ensemble of random matrices. This allows us to computevarious properties of the zero-crossings of the diffusing field, equivalentlyof the real roots of Kac polynomials. Finally, we unveil a precise connectionwith a fourth model: the semi-infinite Ising spin chain with Glauber dynamicsat zero temperature.

Exact persistence exponent for the $2d$-diffusion equation and related Kac polynomials