If $L$ is a line bundle on $X$, a model ${\mathcal{L}}$ of $L$ is a ${\mathbb{Q}}$-line bundle ${\mathcal{L}}$ on a proper model ${\mathcal{X}}$, together with an identification ${\mathcal{L}}|_{X}=L$. It defines a model metric $\phi_{\mathcal{L}}$ on the Berkovich analytification $L^{\mathrm{an}}$ of $L$. If ${\mathcal{L}}^{\prime}$ is another model of $L$, determined on a proper model ${\mathcal{X}}^{\prime}$ of $X$, then $\phi_{\mathcal{L}}=\phi_{{\mathcal{L}}^{\prime}}$ if and only if the pull-backs of ${\mathcal{L}}$ and ${\mathcal{L}}^{\prime}$ to some higher model ${\mathcal{X}}^{\prime\prime}$ coincide.

A model of ${\mathcal{O}}_{X}$ is given by a ${\mathbb{Q}}$-Cartier divisor $D$ supported on the central fiber of a proper model ${\mathcal{X}}$; the corresponding model metric will then be identified with the model function $\phi_{D}\colon X^{\mathrm{an}}\to{\mathbb{R}}$ defined by $\phi_{D}(v)=v(D)$. It satisfies

$$\inf_{X^{\mathrm{an}}}\phi_{D}=\min_{E}\phi_{D}(v_{E})$$

(5.1)

where $E$ runs over the irreducible components of ${\mathcal{X}}_{0}$.