Minimal inversion of a permuton sample

Given a permutation $\sigma$, its corresponding \textit{inversion graph} is obtained by adding an edge between $i<j$ if and only if $\sigma(i)>\sigma(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call \textit{minimal inversion} the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $\alpha>0$, we show that the probability that the minimal inversion rescaled by $n^{\alpha/(\alpha+1)}$ is larger than $t$ behaves like $\exp(-ct^{\alpha+1})$ for some constant $c>0$ depending on the distribution. We further show that every $\alpha>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^\beta$ for any $\beta\in[0,1]$ (the cases $\beta=0$ and $\beta=1$ being obtained via the identity and anti-identity permutations, among others).