Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees

The broadcasting models on trees arise in many contexts such as discrete mathematics, biology, information theory, statistical physics and computer science. In this work, we consider the k-colouring model. A basic question here is whether the assignment at the root affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called reconstruction problem. For the case where the underlying tree is d-ary it is well known that d/ ln d is the reconstruction threshold. That is, for k = (1 + )d/ ln d we have non-reconstruction while for k = (1− )d/ ln d we have reconstruction. Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, we consider the well-known Galton-Watson trees. The corresponding model arises naturally in many contexts such as the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The study on rCSP focuses on Galton-Watson trees with offspring distribution B(n, d/n), i.e. the binomial with parameters n and d/n, where d is fixed. Here we consider a broader version of the problem, as we assume general offspring distribution which includes B(n, d/n) as a special case. Our approach relates the corresponding bounds for (non)reconstruction to certain concentration properties of the offspring distribution. This allows to derive reconstruction thresholds for a very wide family of offspring distributions, which includes B(n, d/n). A very interesting corollary is that for distributions with expected offspring d, we get reconstruction threshold d/ ln d under weaker concentration conditions than what we have in B(n, d/n). Furthermore, our reconstruction threshold for the random colorings of Galton-Watson with offspring B(n, d/n), implies the reconstruction threshold for the random colourings of G(n, d/n). 1998 ACM Subject Classification G.2 Discrete Mathematics, G.3 Probability and Statistics, E.4 Coding and Information Theory