Poisson Dirichlet distribution for the interchange process in five dimensions

Dor Elboim, IAS
Fine Hall 224

In the interchange process on a graph $G=(V,E)$, distinguished particles are placed on the vertices of $G$ with independent Poisson clocks on the edges. When the clock of an edge rings, the two particles on the two sides of the edge interchange. In this way, a random permutation $\pi _\beta:V\to V$ is formed for any time $\beta >0$. One of the main objects of study is the cycle structure of the random permutation and the emergence of long cycles. We consider the process on the torus of side length $L$ in dimension $d\ge 5$ and prove that macroscopic cycles emerge after a long time $\beta$. These are cycles whose length is proportional to the volume of the torus $L^d$. Moreover, we show that the cycle lengths converge to the Poisson-Dirichlet distribution. This is a joint work with Allan Sly.