On the Analyticity of the Pressure in the Hierarchical Dipole Gas

In ret: 1 the analyticity of the pressure in the dipole gas was proved by a somewhat indirect method, i.e., by methods not based on a direct C" bound on the nth order coefficient of the Mayer expansion. On the other hand, recently success has been met in finding direct C" bounds on the nth-order coefficient of the Mayer series for the twodimensional Yukawa gas in the region fie(0, 6~z) [here fl_~2 will be symbols denoting the inverse temperature; see refs. 2 and 3 for the unit conventions; in ref. 2 the result is stated only for fl < 4n, but it is obvious that the bounds obtained in fact imply that it holds for fie (0, 6n); in ref. 3 the result is obtained for fie (0, 16n/3) by a similar method]. With the above results in mind, we have tried to prove by direct methods also the convergence of the Mayer expansion for the dipole gas, with the aim of developing techniques eventually suitable to prove the often conjectured convergence of the Mayer expansion for the two-dimensional Coulomb gas at low temperature [i.e., for fie (8g, ~ ) ] . So far we have been unable to understand the delicate cancellation mechanism necessary to obtain the desired bounds: we have, however, been