  The low temperature properties of fermionic solids are governed by spin
exchanges. Near the melting transition, the spin-exchange energy increases as
well as the relative contribution of large loops. In this paper, we check the
convergence of the multi-spin exchange model and the validity of the Thouless
theory near the melting of the Wigner crystal in two dimensions. Exchange
energies are computed using Path Integral Monte Carlo for loop sizes up to 8,
at $r_s=40$, 50 and 75. The data are then fitted to a geometric model. Then the
exchange energies are extrapolated to larger sizes in order to evaluate their
contributions to the leading term of the magnetic susceptibility and the
specific heat. These results are used to check the convergence of the
multi-spin exchange model.
