The multi-wedge variational formula for the directed landscape

Let $Z\subset\mathbb{R}$ be a nonempty finite set, let $g:Z\to\mathbb{R}$, let $Y\subset\mathbb{R}$ be a nonempty finite set, and let $f:Y\to\mathbb{R}$ satisfy

$$f(y)>\max_{z\in Z}\{-(y-z)^2+g(z)\}\quad\text{for all }y\in Y.$$

For $z\in Z$, define

$$\mathscr{E}_z(c):=\{\phi\in C(\mathbb{R}):\phi|_{[-c,c]}\in H^1[-c,c],\ \phi(x)\geq -(x-z)^2+g(z)\ \text{on }\mathbb{R},\ \phi(x)=-(x-z)^2+g(z)\ \text{outside of }[-c,c]\}$$

and let $\mathscr{E}_z:=\bigcup_{c\in(0,\infty)}\mathscr{E}_z(c)$. The multi-wedge variational conjecture. The infimum in $I_{0,Z;1,Y}(g;f)$ is equal to

$$\inf\Big\{\sum_{z\in Z}\frac14\int_{\mathbb{R}}\mathrm{d}x\,\big((\partial_x\phi_z)^2-4(x-z)^2\big): (\phi_z)_{z\in Z}\in\prod_{z\in Z}\mathscr{E}_z,\ \big(\max_{z\in Z}\phi_z\big)|_Y=f\Big\}.$$

This conjecture would provide a variational decomposition of the finite-dimensional large-deviation rate for multi-wedge initial data. The preceding proposition establishes a decomposition of the rate-function infimum through an ordered partition, but incorporating the disjoint-support and domination conditions into the proposed formula requires further work.