The tropical-polynomial characterization of tropicalized local field Gaussians

Let $K$ be a local field with residue-field cardinality $q$, let $L$ be a lattice in $K^d$, and let $V$ be the tropicalization of a full-dimensional Gaussian in $K^d$ with lattice $L$. Define

$$\varphi_L(v)=-\log_q\bigl(\mathbb{P}(V\geq v)\bigr),\qquad v\in\mathbb{Z}^d.$$

A function $\phi:\mathbb{Z}^d\to\mathbb{R}$ equals $\varphi_L$ for some $L$ if and only if it is the restriction to lattice points of a tropical polynomial supported on the cube $\{0,1\}^d$ with integer supermodular coefficients. Tropical-polynomial characterization. More precisely,

$$P_L(v)=\max_{I\subseteq[d]}\left(\sum_{i\in I}v_i-c_I\right),$$

where $(c_I)_{I\subset[d]}$ is a sequence of integers satisfying

$$c_{\emptyset}=0$$

and

$$c_{I\cup J}+c_{I\cap J}\geq c_I+c_J,\qquad\text{for all }I,J\subset\{1,\dots,d\}.$$

This characterizes the possible tropicalized Gaussian measures by the integer supermodular coefficient data of their tropical polynomials. The supplied parser status is unknown, so whether this characterization remains open or has been established elsewhere should be checked.