  These are self-contained lecture notes for spectral independence. For an
$n$-vertex graph, the spectral independence condition is a bound on the maximum
eigenvalue of the $n\times n$ influence matrix whose entries capture the
influence between pairs of vertices, it is closely related to the covariance
matrix. We will present recent results showing that spectral independence
implies the mixing time of the Glauber dynamics is polynomial (where the degree
of the polynomial depends on certain parameters). The proof utilizes
local-to-global theorems which we will detail in these notes. Finally, we will
present more recent results showing that spectral independence implies an
optimal bound on the relaxation time (inverse spectral gap) and with some
additional conditions implies an optimal mixing time bound of $O(n\log{n})$ for
the Glauber dynamics.
  We also present the results of Anari, Liu, Oveis Gharan, and Vinzant (2019)
for generating a random basis of a matroid. The analysis of the associated
bases-exchange walk utilizes the local-to-global theorems used for spectral
independence with the Trickle-Down Theorem of Oppenheim (2018) to analyze the
local walks. Our focus in these notes is on the analysis of the spectral gap of
the associated Markov chains from a functional analysis perspective, and we
present proofs of the associated local-to-global theorems from this same Markov
chain perspective.
