Weak coloring numbers of minor-closed graph classes

We study the growth rate of weak coloring numbers of graphs excluding a fixed graph as a minor. Van den Heuvel et al. (European J. of Combinatorics, 2017) showed that for a fixed graph $X$, the maximum $r$-th weak coloring number of $X$-minor-free graphs is polynomial in $r$. We determine this polynomial up to a factor of $\mathcal{O}(r\log r)$. Moreover, we tie the exponent of the polynomial to a structural property of $X$, namely, $2$-treedepth. As a result, for a fixed graph $X$ and an $X$-minor-free graph $G$, we show that ${\mathrm{w}\mathrm{c}\mathrm{o}\mathrm{l}}_r(G)=\mathcal{O}(r^{\mathrm{t}\mathrm{d}(X)-1}\mathrm{l}\mathrm{o}\mathrm{g}r)$, which improves on the bound ${\mathrm{w}\mathrm{c}\mathrm{o}\mathrm{l}}_r(G)=\mathcal{O}(r^{g(\mathrm{t}\mathrm{d}(X))})$ given by Dujmović et al. (SODA, 2024), where $g$ is an exponential function. In the case of planar graphs of bounded treewidth, we show that the maximum $r$-th weak coloring number is in $\mathcal{O}(r^2\mathrm{l}\mathrm{o}\mathrm{g}r$), which is best possible.