  For each $n,r \geq 0$, let $KG(n,r)$ denote the Kneser Graph; that whose
vertices are labeled by $r$-element subsets of $n$, and whose edges indicate
that the corresponding subsets are disjoint. Fixing $r$ and allowing $n$ to
vary, one obtains a family of nested graphs, each equipped with a natural
action by a symmetric group $S_n$, such that these actions are compatible.
Collections of graphs of this type are common in algebraic combinatorics and
include families such as the Johnson Graphs, Crown Graphs and Rook Graphs. In
previous work, the authors systematically studied families of this type using
the language of representation stability and FI-modules. In that work, it is
shown that such families of graphs exhibit a large variety of asymptotic
regular behaviors.
  The present work applies the theory developed in that previous work, later
refined in work of the authors and Speyer, to study random walks on the graphs
of such families. We show that the moments of hitting times exhibit rational
function behavior asymptotically. By consequence we conclude similar facts
about the entries of the discrete Green's functions, as defined by Chung and
Yau. Finally, we illustrate how the algebro-combinatorial structure of the
graphs in these families give bounds on the mixing times of random walks on
those graphs. We suggest some possible directions for future study, including
of the appearance, or not, of the cut-off phenomenon, originally presented by
Diaconis.
