Since $L$ has degree $g$, the measure $c_{1}(\overline{L})_{v}$ on $X_{v}$ has total mass $g$ ; let us define a measure of total mass $1$ on $X_{v}$ by

$$\mu_{v}=\frac{1}{g}c_{1}(\overline{L})_{v}.$$

When $v$ is archimedean, the measure $\mu_{v}$ is the Arakelov measure on the Riemann surface $X_{v}({\mathbf{C}})$. Let us recall its definition. Consider an orthonormal basis $(\omega_{1},\dots,\omega_{g})$ of $H^{0}(X,\Omega^{1}_{X})$, i.e., a basis satisfying the relations

$$\int_{X_{v}({\mathbf{C}})}\omega_{j}\wedge\overline{\omega_{k}}=\delta_{j,k}=\begin{cases}1&\text{if $j=k$ ;}\\ 0&\text{otherwise.}\end{cases}$$

Then,

$$\mu_{v}=\frac{1}{g}\sum_{j=1}^{g}\omega_{j}\wedge\overline{\omega_{j}}.$$

Let us now assume that $v$ is ultrametric. By a theorem of Heinz [40], the metric on the line bundle $\overline{L}$ coincides with the canonical metric defined by Zhang [57] using the reduction graph of the minimal regular model of $X$. This allows in particular to compute the measure $\mu_{v}$ : the reader will find in [20, 57, 5]) a quite explicit formula for $\mu_{v}$, involving the physical interpretation of the graph as an electric network.