We study the effects of boundary conditions in two-dimensional
rigidity percolation. Specifically, we consider generic rigidity in the bond
percolation model on the triangular lattice. We introduce a theory of boundary
conditions and define two different notions of “rigid clusters,”
called $\mathrm{r}^0$-clusters and $\mathrm{r}^1$-clusters, which correspond to
free boundary conditions and wired boundary conditions respectively. The
definition of an $\mathrm{r}^ 0$-cluster turns out to be equivalent to the
definition of a rigid component used in earlier papers by Holroyd and
Häggström. We define two critical probabilities, associated with
the appearance of infinite $\mathrm{r}^0$-clusters and infinite
$\mathrm{r}^1$-clusters respectively, and we prove that these two critical
probabilities are in fact equal. Furthermore, we prove that for all parameter
values $p$ except possibly this unique critical probability, the set of
$\mathrm{r}^ 0$-clusters equals the set of $\mathrm{r}^ 1$-clusters almost
surely. It is an open problem to determine what happens at the critical
probability.