Exponential Localization of Spatial Random Permutations in One Dimension

We consider a class of random permutations of the interval $[-n,n]$, in which points are typically displaced a distance $O(W)$. We show the cycles are localized on the scale $W^3$, with an exponentially decaying tail bound. Analogous to eigenfunctions of one dimensional random band matrices, the cycles are conjectured to be localized to the scale $W^2.$