As in §3.1, we introduce the logarithmic form

$$\Omega:=\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n},$$

and the corresponding local trivialization $\Omega^{\mathrm{rel}}=\Omega\otimes(dt/t)^{-1}$ of $K^{\mathrm{log}}_{{\mathcal{X}}/{\mathbb{D}}}$. The restriction $\Omega_{t}$ of $\Omega^{\mathrm{rel}}$ to the fiber $U_{t}:={\mathcal{X}}_{t}\cap{\mathcal{U}}$ is a trivializing section of $K_{U_{t}}$, explicitly given by

$$\Omega_{t}=\frac{1}{p+1}\sum_{j=0}^{p}\frac{(-1)^{j}}{b_{j}}\frac{dz_{0}}{z_{0}}\wedge\dots\wedge\widehat{\frac{dz_{j}}{z_{j}}}\wedge\dots\wedge\frac{dz_{p}}{z_{p}}\wedge dz_{p+1}\wedge\dots\wedge dz_{n}\bigg|_{U_{t}}.$$

For $t\in{\mathbb{D}}^{*}$ close to 0, consider the map $\operatorname{Log}_{t}\colon U_{t}\to\sigma\times(Y\cap{\mathcal{U}})$ defined by

$$\operatorname{Log}_{t}=(\operatorname{Log}_{{\mathcal{U}}},y)=\left(\frac{\log|z_{0}|}{\log|t|},\dots,\frac{\log|z_{p}|}{\log|t|},z_{p+1},\dots,z_{n}\right).$$

Note the similarity to the situation considered in §1.4. More precisely, view $U:={\mathcal{U}}\cap X$ as embedded in $T\times{\mathbb{C}}^{n-p}$, where $T=({\mathbb{C}}^{*})^{p+1}$, and consider the character $\chi=\prod_{i=0}^{p}z_{i}^{b_{i}}$ on $T$. If $L\colon T\to{\mathbb{R}}^{p+1}$ is the tropicalization map, then

$$\operatorname{Log}_{t}=(\lambda(t)^{-1}L(z^{\prime},z^{\prime\prime}),y).$$

Each fiber $\operatorname{Log}_{t}^{-1}(w,y)$ is a torsor for the (possibly disconnected) compact Lie group

$$K=\left\{\theta\in({\mathbb{R}}/{\mathbb{Z}})^{p+1}\mid\sum_{i}b_{i}\theta_{i}=0\right\};$$

hence carries a unique $K$-invariant probability measure $\rho_{t,w,y}$.

The analysis in §1.4 now gives the following expression for the volume form $|\Omega_{t}|^{2}$ on $U_{t}$ in logarithmic polar coordinates:

As before, view $\tau:=\prod_{i=0}^{p}z_{i}^{a_{i}}\Omega^{\mathrm{rel}}$ as a local ${\mathbb{Q}}$-generator of ${\mathcal{L}}$, and set

$$g:=-\log|\tau|_{\psi}\in C^{0}({\mathcal{U}}).$$

By definition, we have $\mu_{t}=(2\pi)^{-d}\lambda(t)^{d}|\Omega_{t}|^{2}/|\Omega_{t}|^{2}_{\psi_{t}}$, and hence

$$(2\pi)^{d-p}\lambda(t)^{p-d}|t|^{-2\kappa_{0}}\int\limits_{U_{t}}h\mu_{t}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}|t|^{2\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}\\ =\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}b_{\sigma}^{-1}\lambda_{\sigma}\otimes|dy|^{2}\int he^{2g}\,\rho_{t,w,y}$$

(3.3)

for every $h\in C^{0}_{c}({\mathcal{U}})$, thanks to Lemma 3.7.

We use the following change of variables. For $t\in{\mathbb{D}}^{*}$, consider the polytope

$$\sigma_{t}:=\{(w^{\prime},x^{\prime\prime})\in{\mathbb{R}}_{+}^{q+1}\times{\mathbb{R}}_{+}^{p-q}\mid b^{\prime}\cdot w^{\prime}=1,\ b^{\prime\prime}\cdot x^{\prime\prime}\leq\lambda(t)^{-1}\}\subset\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-q}\subset{\mathbb{R}}_{+}^{p+1},$$

where $b^{\prime}=(b_{0},\dots,b_{q})$ and $b^{\prime\prime}=(b_{q+1},\dots,b_{p})$.

Using the map $Q_{t}$ and the fact that $\kappa_{i}-\kappa_{0}>0$ for $i>q$, it is easy to see that

$$\int_{\sigma}e^{-2\lambda(t)^{-1}\sum_{i=q+1}^{p}b_{i}w_{i}(\kappa_{i}-\kappa_{0})}\lambda_{\sigma}(dw)=O(\lambda(t)^{p-q}).$$

By (3.3), it follows that

$$\mu_{t}(U_{t})=O(\lambda(t)^{d-q}|t|^{2\kappa_{0}}),$$

(3.4)

and hence $\mu_{t}(U_{t})\to 0$ unless $\kappa_{0}=0$ and $q=d$, which we henceforth assume. Given $\varphi\in C^{0}(\sigma)$, our goal is now to show

$$\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\to\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right).\left(\int_{Y^{\prime}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)\right).$$

(3.5)

Let us first express both sides of (3.5) in logarithmic polar coordinates. We start by the left-hand side. Set $f:=\chi e^{2g}\in C^{0}({\mathcal{U}})$. By (3.3) and Lemma 3.8 we have

$$(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,\mu_{t}\\ =\lambda(t)^{d-p}\int\limits_{\sigma\times(Y\cap{\mathcal{U}})}\varphi(w)e^{-2\lambda(t)^{-1}a^{\prime\prime}\cdot w^{\prime\prime}}b_{\sigma}^{-1}\lambda_{\sigma}(dw)\otimes|dy|^{2}\int f\,\rho_{t,w,y}\\ =\int\limits_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}H_{t}(w^{\prime},x^{\prime\prime})b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{t,w^{\prime},x^{\prime\prime},y},$$

(3.6)

where

$$H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma_{t}}\varphi(Q_{t}(w^{\prime},x^{\prime\prime}))e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}(1-\lambda(t)b^{\prime\prime}\cdot x^{\prime\prime})^{d},$$

and $\rho_{t,w^{\prime},x^{\prime\prime},y}$ is the same measure as $\rho_{t,w,y}$ via the identification $Q_{t}(w^{\prime},x^{\prime\prime})=w$.

Note that $\lim_{t\to 0}Q_{t}(w^{\prime},x^{\prime\prime})=(w^{\prime},0)$, so

$$\lim_{t\to 0}H_{t}(w^{\prime},x^{\prime\prime})=\mathbf{1}_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}.$$

Consider the tropicalization map

$$S\colon Y^{\prime}\cap{\mathcal{U}}\to{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})$$

given by $S=(-\log|z_{d+1}|,\dots,-\log|z_{p}|,y)$. Each fiber $S^{-1}(x^{\prime\prime},y)$ is a torsor for the compact torus $({\mathbb{R}}/{\mathbb{Z}})^{p-d}$ and hence carries a unique invariant probability measure $\rho_{x^{\prime\prime},y}$. As $t\to 0$, the probability measure $\rho_{t,w^{\prime},x^{\prime\prime},y}$ converges weakly to $\rho_{x^{\prime\prime},y}$ for any $w^{\prime}\in\sigma^{\prime}$.

By dominated convergence it follows that

$$\lim_{t\to 0}(2\pi)^{d-p}\int_{U_{t}}(\varphi\circ\operatorname{Log}_{{\mathcal{U}}})\chi\,d\mu_{t}\\ =\int_{\sigma^{\prime}\times{\mathbb{R}}_{+}^{p-d}\times(Y\cap{\mathcal{U}})}\varphi(w^{\prime},0)e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}(dw^{\prime})\otimes|dx^{\prime\prime}|\otimes|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\\ =\left(\int_{\sigma^{\prime}}\varphi b_{\sigma^{\prime}}^{-1}\lambda_{\sigma^{\prime}}\right)\left(\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y}\right).$$

(3.7)

It only remains to compare the second factor of (3.7) to the second factor in (3.5). To this end, we again use logarithmic polar coordinates. We have

$$\chi\operatorname{Res}_{Y^{\prime}}(\psi)=f\prod_{i=d+1}^{p}|z_{i}|^{2a_{i}-2}|dz^{\prime\prime}|^{2}\otimes|dy|^{2}.$$

(3.8)

For $d<j\leq p$, set $z_{j}=e^{-x_{j}+2\pi i\theta_{j}}$ with $x^{\prime\prime}\in{\mathbb{R}}_{+}^{p-d}$ and $\theta^{\prime\prime}\in({\mathbb{R}}/{\mathbb{Z}})^{p-d}$. Then

$$\int_{Y^{\prime}\cap{\mathcal{U}}}\chi\operatorname{Res}_{Y^{\prime}}(\psi)=(2\pi)^{p-d}\int_{{\mathbb{R}}_{+}^{p-d}}e^{-2a^{\prime\prime}\cdot x^{\prime\prime}}|dx^{\prime\prime}|\int_{Y\cap{\mathcal{U}}}|dy|^{2}\int f\,\rho_{x^{\prime\prime},y},$$

(3.9)

which completes the proof of (3.5), and hence of Theorem 3.4.