  We study how to establish $\textit{spectral independence}$, a key concept in
sampling, without relying on total influence bounds, by applying an
$\textit{approximate inverse}$ of the influence matrix. Our method gives
constant upper bounds on spectral independence for two foundational Gibbs
distributions known to have unbounded total influences:
  $\bullet$ The monomer-dimer model on graphs with large girth (including
trees). Prior to our work, such results were only known for graphs with
constant maximum degrees or infinite regular trees, as shown by Chen, Liu, and
Vigoda (STOC '21).
  $\bullet$ The hardcore model on trees with fugacity $\lambda < \mathrm{e}^2$.
This remarkably surpasses the well-known $\lambda_r>\mathrm{e}-1$ lower bound
for the reconstruction threshold on trees, significantly improving upon the
current threshold $\lambda < 1.3$, established in a prior work by Efthymiou,
Hayes, \v{S}tefankovi\v{c}, and Vigoda (RANDOM '23).
  Consequently, we establish optimal $\Omega(n^{-1})$ spectral gaps of the
Glauber dynamics for these models on arbitrary trees, regardless of the maximum
degree $\Delta$.
