  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]. We mainly focus on the Hard-core model and the Ising
model. We obtain bounds for fast mixing with the parameters expressed in terms
of the spectral radius of the adjacency matrix, improving on the seminal work
in [Hayes: FOCS 2006]. Furthermore, we go beyond the adjacency matrix and
establish -- for the first time -- rapid mixing results for Glauber dynamics
expressed in terms of the spectral radius of the Hashimoto non-backtracking
matrix of the underlying graph $G$. Working with the non-backtracking spectrum
is extremely challenging, but also more desirable. Its eigenvalues are less
correlated with the high-degree vertices than those of the adjacency matrix and
express more accurately invariants of the graph such as the growth rate. Our
results require ``weak normality" from the Hashimoto matrix. This condition is
mild and allows us to obtain very interesting bound. We study the pairwise
influence matrix ${I}^{\Lambda,\tau}_{G}$ by exploiting the connection between
the matrix and the trees of self-avoiding walks, however, we go beyond the
standard treatment of the distributional recursions. The common framework that
underlies our techniques we call the topological method. Our approach is novel
and gives new insights into how to establish 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.
