  The rigidity transition occurs when, as the density of microscopic components
is increased, a disordered medium becomes able to transmit and ensure
macroscopic mechanical stability, owing to the appearance of a space-spanning
rigid connected component, or cluster. As a continuous phase transition it
exhibits a scale invariant critical point, at which the rigid clusters are
random fractals. We show, using numerical analysis, that these clusters are
also conformally invariant, and we use conformal field theory to predict the
form of universal finite size effects. Furthermore, although connectivity and
rigidity percolation are usually though to belong to different universality
classes and thus be of fundamentally different natures, we provide evidence of
unexpected similarities between the statistical properties of their random
clusters at criticality. Our work opens a new research avenue through the
application of the powerful 2D conformal field theory tools to understand the
critical behavior of a wide range of physical and biological materials
exhibiting such a mechanical transition.
