\left\{ 
\begin{aligned}
& -\operatorname{tr}\left( A(Dm_\mu,x) D^2m_\mu\right) + H(Dm_\mu,x) = \mu & \mbox{in} & \ {\mathcal{H}}_e^+, \\
& m_\mu = 0 & \mbox{on} & \ \partial {\mathcal{H}}_e^+,
\end{aligned}
\right.$$ where $\mu>0$ and, for a unit direction $e\in\partial B_1$, we define ${\mathcal{H}}_e^+$ and ${\mathcal{H}}_e^-$ to be the half-spaces $$\label{e.He}
{\mathcal{H}}_e^+: = \left\{ x\in{\mathbb{R}^d}\,:\, x\cdot e > 0\right\} \qquad \mbox{and} \qquad  {\mathcal{H}}_e^-: = \left\{ x\in{\mathbb{R}^d}\,:\, x\cdot e < 0\right\}.$$ In Appendix \[s.metricproblem\], we show that this problem is well-posed and there exists a unique Lipschitz solution which we denote by $m_\mu(\cdot,{\mathcal{H}}_e^-)$. See Theorem \[t.LipschitzAppenA1\]. The main step in the proof of Theorem \[t.homogenization\] is the demonstration that  homogenizes, that is, that for every $\mu>0$ and $e\in \partial B_1$, there exists a deterministic quantity $\overline m_\mu(e)>0$ such that $$\label{e.MPlim}
\lim_{t\to \infty} \frac {m_\mu(te,{\mathcal{H}}_e^-)}{t} = \overline m_\mu(e), \quad \mbox{${\mathbb{P}}$--a.s.}$$ We actually prove more, namely the quantitative version of  given in the following proposition.

\[p.planarquantitative\] Fix $L\geq 1$, $e \in \partial B_1$ and $\mu\in (0,L]$, there exist $\overline m_\mu(e)\geq 0$, $C({\mathrm{data}},L)\geq1$ and $q({\mathrm{data}})<\infty$ such that, for every $x\in {\mathcal{H}}_e^+$ and $\lambda>0$, we have the estimate $$\label{e.Pm-barm}
{\mathbb{P}}\left[ \left| m_\mu(x,{\mathcal{H}}_e^-) - \overline m_\mu (e)(x\cdot e) \right| > \lambda \right]  \leq C \exp\left( \frac{-\lambda^2}{(1+x\cdot e)^{\frac85}} \right).$$ Moreover, the 