  Despite great progress in the study of critical percolation on $\mathbb{Z}^d$
for $d$ large, properties of critical clusters in high-dimensional fractional
spaces and boxes remain poorly understood, unlike the situation in two
dimensions. Closely related models such as critical branching random walk give
natural conjectures for the value of the relevant high-dimensional critical
exponents; see in particular the conjecture by Kozma-Nachmias that the
probability that $0$ and $(n, n, n, \ldots)$ are connected within $[-n,n]^d$
scales as $n^{-2-2d}$.
  In this paper, we study the properties of critical clusters in
high-dimensional half-spaces and boxes. In half-spaces, we show that the
probability of an open connection ("arm") from $0$ to the boundary of a
sidelength $n$ box scales as $n^{-3}$. We also find the scaling of the
half-space two-point function (the probability of an open connection between
two vertices) and the tail of the cluster size distribution. In boxes, we
obtain the scaling of the two-point function between vertices which are any
macroscopic distance away from the boundary.
