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 prove the existence of infinite cycles in the interchange process on $\mathbb{Z}^d$ for all dimensions $d\geq 5$ and all large $\beta$, establishing a conjecture of Bálint Tóth from 1993 in these dimensions.

We also study the process on the torus of side length $L$ in dimension $d\geq 5$ 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.