Random metrics on hierarchical graphs

Take a «diamond»-shaped graph. Replace each of its four edges by a copy of a «diamond» graph. Then each of sixteen edges again by a diamond, etc. The resulting object is called a hierarchical [diamond] graph. Moreover, a graph can be turned into a metric space by choosing lengths of its edges; if we agree to divide lengths by two on each step of this procedure, we have a [Gromov–Hausdorff] convergence of metric spaces, thus turning the limit object into a metric space. Now, instead of multiplying the lengths by deterministic constant (1/2), let us multiply them by random constants, chosen independently and identically distributed for all the replacements. Does the sequence of (random) metric graphs that we have defined converge (perhaps, after a suitable normalization)? If yes, what can be said about the limiting metric space? This is a «baby version» of a more (and very) complicated problem (originating, in particular, from physics) of giving a rigorous sense to a likewise–defined two–dimensional object. However, even this baby version turned out to be sufficiently difficult. I will speak on a joint work of Mikhail Khristoforov, Michele Triestino and myself, devoted to its study (and on some corollaries and nearby topics).