Similar recursion relations can be derived for the shear modulus and the bulk modulus. I 0 and I 1 are Bessel functions, respectively. Depending on the starting point, the recursion relation can run into two directions. y → → 0 implies no defects, the ensemble is crystalline. y → → ∞ ∞ , implies arbitrary many defects, the ensemble is fluid. The recursion relation have a fix-point at y = 0 with E R / k B T = 16 π π . Now, E R is the renormalized value instead of the bare one. Figure 2 shows Youngs’modulus as function of the dimensionless control parameter Γ Γ . It measures the ratio of the repelling energy between two particles and the thermal energy (which was constant in this experiment). It can be interpreted as pressure or inverse temperature. The black curve is a thermodynamic calculation of a perfect hexagonal crystal at T = 0 . The blue curve is from computer simulations and shows a reduced elasticity due to lattice vibrations at T > 0 . The red curve is the renormalization following the recursion relations, Young's modulus disappears discontinuously to zero at 16 π π . Turquoise symbols are from measurements of elasticity in a colloidal monolayer, and confirm the melting point at Y R = 16 π π .