\sum_{(x_{1},\ldots,x_{d}) \subset \{u_{1},\ldots,u_{n} \} } \mathbb{H}^{0}(\cap_{i=1}^{d}x_{i}) \right)\lambda^n({\text{\rm d}}(u_{1}, \ldots , u_{n})).$$ It holds $\tilde{A}(a)/\tilde{B}(a)=A(a)/B(a),$ where using (\[rdf\]) we write $A(a)= e^{-\frac{aT}{d}(d-1)}\times $ $$\label{ab}\times \sum_{n=0}^{\infty} \left( \frac{aT}{d} \right)^n \sum_{n_{1}+ \ldots + n_{d}=n} \frac{1}{n_{1}! \ldots n_{d}!} \exp \left(   \nu_{d} \sum_{D\subset [d-k]} \prod_{l \in [d]\setminus D} n_{l} \right),$$ $$B(a)=
\nonumber e^{-\frac{aT}{d}(d-1)} \sum_{n=0}^{\infty} \left( \frac{aT}{d} \right)^n \sum_{n_{1}+ \ldots + n_{d}=n} \frac{1}{n_{1}! \ldots n_{d}!} \exp \left(   \nu_{d}\prod_{l = 1}^{d} n_{l} \right).$$ From Lemmas \[lemB2\] and \[lemA2\] below, (\[dals\]) follows.

b\) We need to compute $\rho_{2(d-k)}(y_{1},\ldots,y_{2(d-k)};\mu_{a}^{(d)})$ for all sets of arguments which fullfil that each facet of $\{ y_{1},\ldots,y_{d-k} \}$ has different orientation and the same applies to $\{ y_{d-k+1},\ldots,y_{2(d-k)} \}$. Without loss of generality we need to consider only situations where facets $ y_{1},\ldots,y_{d-k}$ have orientations $e_{1},\ldots,e_{d-k}$ and facets $y_{d-k+1},\ldots,y_{2(d-k)}$ have orientations $e_{1},\ldots,e_{l},e_{d-k+1},\ldots e_{2d-2k-l}$, where $l =\max(d-2k,0),\ldots,d-k $, probability of both sets of facets having $l$ common orientations can be expressed by multinomial coefficient $$\label{ddbb}
\frac{(d-k)!^{2}}{d^{2(d-k)}} \binom{d}{l,d-k-l,d-k-l,2k-d+l}.$$ It holds $$\begin{gathered}
\nonumber \lim_{a\rightarrow\infty}\rho_{2(d-k)}(y_{1},\ldots,y_{2(d-k)};\mu_{a}^{(d)}) = \lim_{a\rightarrow\infty}\frac{e^{-a(d-1)}}{d} \sum_{n=0}^{\infty} \frac{a^{n}}{n!}\times  \\
\nonumber \times    \int_{Y^n} \exp \left( \frac{\nu_{d}}{d!} \sum_{ (x_{1},\ldots,x_{d} ) \subset \{u_{1},\ldots,u_{n},y_{1},\ldots,y_{2(d-k)} \} } \mathbb{H}^{0}(\cap_{i=1}^{d}x_{i}) \right)  \lambda^n({\text{\rm d}}(u_{1}, \ldots,u_{n})),\end{gathered}$$ where we get $\frac{e^{-a(d-1)}}{d}$ from calculations of the limit $B(a)$ in Lemma \[lemB2\]. For selected $l$ the limit of the 