It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.
"Multiple points of the Brownian sheet in critical dimensions." Ann. Probab. 43 (4) 1577 - 1593, July 2015. https://doi.org/10.1214/14-AOP912