  We develop a general method for constructing random manifolds and
submanifolds in arbitrary dimensions. The method is based on associating colors
to the vertices of a triangulated manifold, as in recent work for curves in
3-dimensional space by Sheffield and Yadin (2014). We determine conditions on
which submanifolds can arise, in terms of Stiefel-Whitney classes and other
properties. We then consider the random submanifolds that arise from randomly
coloring the vertices. Since this model generates submanifolds, it allows for
studying properties and using tools that are not available in processes that
produce general random subcomplexes. The case of 3 colors in a triangulated
3-ball gives rise to random knots and links. In this setting, we answer a
question raised by de Crouy-Chanel and Simon (2019), showing that the
probability of generating an unknot decays exponentially. In the general case
of k colors in d-dimensional manifolds, we investigate the random submanifolds
of different codimensions, as the number of vertices in the triangulation
grows. We compute the expected Euler characteristic, and discuss relations to
homological percolation and other topological properties. Finally, we explore a
method to search for solutions to topological problems by generating random
submanifolds. We describe computer experiments that search for a low-genus
surface in the 4-dimensional ball whose boundary is a given knot in the
3-dimensional sphere.
