Melting of two dimensional solids on disordered substrate

We study 2D solids with weak substrate disorder, using Coulomb gas renormalisation. The melting transition is found to be replaced by a sharp crossover between a high $T$ liquid with thermally induced dislocations, and a low $T$ glassy regime with disorder induced dislocations at scales larger than $\xi_{d}$ which we compute ($\xi_{d}\gg R_{c}\sim R_{a}$, the Larkin and translational correlation lengths). We discuss experimental consequences, reminiscent of melting, such as size effects in vortex flow and AC response in superconducting films.