Exceptional-time dimensions for Ornstein–Uhlenbeck Bargmann–Fock percolation

Let $f(t)$ be the Bargmann–Fock field on $\mathbb{R}^2$ evolving under Ornstein–Uhlenbeck dynamics, and define

$$\mathcal{C}_t:=\{x\in\mathbb{R}^2:f(t,x)>0\},\qquad \mathcal{C}_t^*:=\{x\in\mathbb{R}^2:f(t,x)=0\}.$$

Exceptional-time conjecture. Almost surely, exceptional times at which $\mathcal{C}_t$ contains an unbounded component exist, and the Hausdorff dimension of the random set of such times is

$$\frac{67}{72}.$$

Furthermore, exceptional times at which $\mathcal{C}_t^*$ contains an unbounded nodal line exist almost surely, and the corresponding Hausdorff dimension is

$$\frac{5}{6}.$$

This conjecture concerns the existence and dimensions of exceptional times for Bargmann–Fock percolation, by analogy with dynamical percolation on the triangular lattice. The analogous dimension for positive clusters on the triangular lattice is $31/36$, while the nodal-line analogue is known to be an almost-sure constant in $[1/9,2/3]$ and is conjectured there to equal $2/3$.