We consider the interchange process with $k$ particles (${ \rm IP}(k)$) on $n$-vertex hypergraphs in which each hyperedge $e$ rings at rate $r_e$. When $e$ rings, the particles occupying it are permuted according to a random permutation from some arbitrary law, where our only assumption is that ${ \rm IP}(2)$ has uniform stationary distribution. We show that $t_{ \rm mix}^{ { \rm IP}(k)}(\epsilon)=O_{ b}(t_{ \rm mix}^{ { \rm IP}(2)}(\epsilon/k))$, where $t_{ \rm mix}^{ { \rm IP}(i)}(\epsilon)$ is the $\epsilon$ total-variation mixing time of ${ \rm IP}(i)$, provided that $kn^{ -2}Rt_{ \rm mix}^{ { \rm IP}(2)}(\epsilon/k)=O((\epsilon/k)^b)$ for some $b>0$, where $R=\sum_e r_e|e|(|e|-1)$ is $n(n-1)$ times the particle-particle interaction rate at equilibrium. This has some consequences concerning the validity in this regime of conjectures of Oliveira about comparison of the $\epsilon$ mixing time of ${ \rm IP}(k)$ to that of $k$ independent particles, each evolving according to ${ \rm IP}(1)$, denoted ${ \rm RW}(k)$, and of Caputo about comparison of the spectral-gap of ${ \rm IP}(k)$ to that of a single particle ${ \rm IP}(1)={ \rm RW}(1)$. We also show that $t_{ \rm mix}^{ \mathrm{ IP}(k)}(\epsilon) \asymp t_{ \rm mix}^{ { \rm RW}(1)}(\epsilon)\asymp t_{ { \rm mix}}^{ { \rm RW}(k)}(\epsilon k/4)$ for all $k\lesssim n^{ 1-\Omega(1)}$ and all $\epsilon\le\frac 1k\wedge\frac 14$ for vertex-transitive graphs of constant degree, as well as for general graphs satisfying a mild ("transience-like") heat-kernel condition. In the case where the particles occupying a hyperedge $e$ are permuted uniformly at random when $e$ rings we obtain results bounding the spectral gap of ${ \rm IP}(k)$ in terms of that ${ \rm RW}(1)$. The proof does not use Morris' chameleon process. It can be seen as a rigorous and direct way of arguing that when the number of particles is fairly small, the system behaves similarly to $k$ independent particles.
Categories: math.PR