Lipschitz constant $\log{n}$ almost surely suffices for mapping $n$ grid points onto a cube

In 2018, the author, Kalu\v{z}a and Kopeck\'a showed that the best Lipschitz constant for mappings taking a given $n^{d}$-element set in the integer lattice $\mathbb{Z}^{d}$, with $n\in \mathbb{N}$, surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$ may be arbitrarily large. However, there remains no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with $n$. We approach this problem from a probabilistic point of view. More precisely, we consider the sequence space of all possible sequences in which the $n$th term is a configuration of $n^{d}$ points inside a given finite lattice. Equipping such spaces with their natural probability measure, we establish almost sure, asymptotic upper bounds of order $\log n$ on the best Lipschitz constant of mappings taking the $n$th element of the set sequence, that is an $n^{d}$-element subset of the given finite lattice, surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$.