Solve the following:

Exercise 2." Show that a transitive graph G has a sequence Gn of subgmphs converging to it in the BenjaminiSchramm (local weak) sense iff it is amenable. Exercise 3. We considered the following simple model for a random dregular bipartite (multi)graph: take d independent uniform random permutations 7n : {1, . . . ,n} > {1, . . . ,n}, then take all the edges {(j,n+7r(j)) :j e {1,...,n}, i6 {1,...,d}}. (a) \" Show that for any k 2 2, the number of k-cycles is tight in n. (We saw this for k = 2.) (b)' Conclude from part (a) that this random graph converges to the dregular tree Td in the local weak sense. (Here the randomness for the measure ,um. comes from two sources: we take a random root pn in the random graph Gn, and want to show convergence in this joint probability space.)