  We classify the ergodic invariant random subgroups of block-diagonal limits
of symmetric groups in the cases when the groups are simple and the associated
dimension groups have finite dimensional state spaces. These block-diagonal
limits arise as the transformation groups (full groups) of Bratteli diagrams
that preserve the cofinality of infinite paths in the diagram. Given a simple
full group $G$ admitting only a finite number of ergodic measures on the
path-space $X$ of the associated Bratteli digram, we prove that every non-Dirac
ergodic invariant random subgroup of $G$ arises as the stabilizer distribution
of the diagonal action on $X^n$ for some $n\geq 1$. As a corollary, we
establish that every group character $\chi$ of $G$ has the form $\chi(g) =
Prob(g\in K)$, where $K$ is a conjugation-invariant random subgroup of $G$.
