The approximate inverse approach can be generalized to other Gibbs distributions beyond the monomer-dimer and hardcore models by adapting the method to suit the specific characteristics of the distribution under consideration. The key lies in identifying the properties of the distribution that allow for the construction of an approximate inverse matrix that simplifies the analysis of spectral independence. By understanding the structure of the inﬂuence matrix and leveraging the conditional independence properties of the distribution, it is possible to develop an approximate inverse that facilitates the analysis of spectral independence. This approach can be applied to a wide range of Gibbs distributions by ensuring that the conditions for the approximate inverse are met, such as the decay rate of total influence and the relationship between the inﬂuence matrix and its symmetric variant.