  We present novel results for fast mixing of Glauber dynamics using the newly
introduced and powerful Spectral Independence method from [Anari, Liu,
Oveis-Gharan: FOCS 2020]. In our results, the parameters of the Gibbs
distribution are expressed in terms of the spectral radius of the adjacency
matrix of $G$, or that of the Hashimoto non-backtracking matrix. The analysis
relies on new techniques that we introduce to bound the maximum eigenvalue of
the pairwise influence matrix $\mathcal{I}^{\Lambda,\tau}_{G}$ for the two spin
Gibbs distribution $\mu$. There is a common framework that underlies these
techniques which we call the topological method. The idea is to systematically
exploit the well-known connections between $\mathcal{I}^{\Lambda,\tau}_{G}$ and
the topological construction called tree of self-avoiding walks. Our approach
is novel and gives new insights to the problem of establishing spectral
independence for Gibbs distributions. More importantly, it allows us to derive
new -- improved -- rapid mixing bounds for Glauber dynamics on distributions
such as the Hard-core model and the Ising model for graphs that the spectral
radius is smaller than the maximum degree.
