Mayer expansions for gas of contours at low temperatures and in arbitrary external fields for the multicomponent ising model

Convergence of contour expansions is proved for $\operatorname{Re}\beta\ge\beta_1$ and arbitrary external fields. It is also shown that the cluster functions are holomorphic with respect to the external fields in regions in which the fields have constant sign. The results are based on the construction of uniform estimates for the considered expansions in the neighborhood of the physical region of variation of the external fields.