2 problems

Let $\mathcal{L} = (\mathcal{L}_1, \mathcal{L}_2, \ldots)$ be a line ensemble such that $\mathcal{L}(x)+x^2$ is stationary under deterministic horizontal shifts, $\mathcal{L}$ has…

Let $\mathcal{A}$ be an $\mathbb{N}$-indexed line ensemble. Say that $\mathcal{A}$ is $x$-invariant if $\mathcal{A}(s+cdot)$ has the same distribution as $\mathcal{A}$ for every…