Permutations from Random Walk

Abstract

Xavier and Yushi run a "random race" as follows. An atomless probability distribution $\mu$ on the real line is chosen. The runners begin at zero. At time $i$ Xavier draws ${\mathbf{X}}_i$ from $\mu$ and advances that distance, while Yushi advances by an independent drawing ${\mathbf{Y}}_i$. After $n$ such moves, what is the probability that Yushi led all the way? That the answer (namely, $4^{-n}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$) is independent of $\mu$ follows from a classical theorem of Darling, stating that for symmetric atomless increments, the distribution of each individual rank in the permutation obtained by ranking the partial sums is independent of the step law. We give a self-contained proof and extend the result to the permutations generated by partial sums of uniformly random signed permutations of any fixed, finite, generic set of reals. For atomless increments with mean zero and finite variance, without assuming symmetry, we show that random-walk permutations approach a random object that we call the "Wiener permuton," whose expected pattern densities equal the probabilities of the corresponding permutations generated by finite random walks with centered Laplace increments. Finally, we exhibit an infinite family of constructions whose limiting permutons interpolate between the Wiener permuton and the recursive separable permuton; each has the same intensity permuton, providing a single two-dimensional extension of the classical arcsine law for all of them.