Cutoff in random graphs with two communities

Random graph with 2 communities with parameters N and p is a graph which can be described
as follows. Initially Nvertices and their degrees are given and each one of them belongs to one of the
two communities. The graph is obtained by taking uniformly at random a graph from all random
graphs on these N vertices for which the number of edges going across the two communities is p.
For a simple random walk on a graph the epsilon-mixing time t mix(epsilon) is the smallest time such that the
distribution of the position of the walk after tmix(epsilon) steps is just epsilon away in a total variation distance
metric to an invariant distribution of the random walk on the graph (it is known that the random
walk will converges to this invariant distribution). Intuitively this means that around some times tn the walk will change
fast from being very far from the invariant distribution to being extremely close to it.

In the case of a non-backtracking random walk, which means that the walk can not go back to the vertex it has just came from, it
was proved that the cutoff happens in the case that alpha >> 1/ log(N) . The proof in this case relies on the
locally tree structure of a random graph on which non-backtracking walk can only move forward,
and so the issues which appear in the case of a walk which can backtrack are complex. The idea to
solve them is to use some of the techniques used to analyse the cutoff for a random walk on a quasi
random graph and to solve additional issues using techniques like the spectral prole and evolving
sets.
An important question in probability is to find some conditions which would imply that a certain
Markov chain collection exhibits a cutoff. Therefore, proving such a result could be beneficial for
further research on this topic. After completing the proof in the case of two communities random
graphs we will look into extending it to other models as well, for example a model with more
communities.