  Motivated by recent free-standing graphene experiments, we show how thermal
fluctuations affect the mechanical properties of microscopically thin solid
ribbons. A renormalization group analysis of flexural phonons reveals that
elongated ribbons behave like highly anisotropic polymers, where the two
dimensional nature of ribbons is reflected in non-trivial power law scalings of
the persistence length and effective bending and twisting rigidities with the
ribbon width. With a coarse-grained transfer matrix approach, we then show how
thermalized ribbons respond to pulling and bending forces over a wide spectrum
of temperatures, forces and ribbon lengths.
