  We study bootstrap percolation processes on random simplicial complexes of
some fixed dimension $d \geq 3$. Starting from a single simplex of dimension
$d$, we build our complex dynamically in the following fashion. We introduce
new vertices one by one, all equipped with a random weight from a fixed
distribution $\mu$. The newly arriving vertex selects an existing
$(d-1)$-dimensional face at random, with probability proportional to some
positive and symmetric function $f$ of the weights of its vertices, and
attaches to it by forming a $d$-dimensional simplex. After a complex on $n$
vertices is constructed, we infect every vertex independently at random with
some probability $p = p(n)$. Then, in consecutive rounds, we infect every
healthy vertex the neighbourhood of which contains at least $r$ disjoint
$(k-1)$-dimensional, fully infected faces. Using a reduction to the generalised
P\'olya urn schemes, we determine the value of critical probability $p_c = p_c
(n; \mu, f)$, such that if $p \gg p_c$ then, with probability tending to 1 as
$n \to \infty$, the infection spreads to the whole vertex set of the complex,
while if $p \ll p_c$ then the infection process stops with healthy vertices
remaining in the complex.
