Reconstruction thresholds on regular trees

We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_{11}=0$. The latter case is closely related to the "hard-core model" from statistical physics; a corollary of our results is that, for the hard-core model on the (k+1)-regular tree with activity $\lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k.Comment: 12 page

Reconstruction on trees and spin glass transition

Consider an information source generating a symbol at the root of a tree network whose links correspond to noisy communication channels, and broadcasting it through the network. We study the problem of reconstructing the transmitted symbol from the information received at the leaves. In the large system limit, reconstruction is possible when the channel noise is smaller than a threshold. We show that this threshold coincides with the dynamical (replica symmetry breaking) glass transition for an associated statistical physics problem. Motivated by this correspondence, we derive a variational principle which implies new rigorous bounds on the reconstruction threshold. Finally, we apply a standard numerical procedure used in statistical physics, to predict the reconstruction thresholds in various channels. In particular, we prove a bound on the reconstruction problem for the antiferromagnetic ``Potts'' channels, which implies, in the noiseless limit, new results on random proper colorings of infinite regular trees. This relation to the reconstruction problem also offers interesting perspective for putting on a clean mathematical basis the theory of glasses on random graphs.Comment: 34 pages, 16 eps figure

Reconstruction Threshold for the Hardcore Model

In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.Comment: 14 pages, 2 figure

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.Comment: Added references, updated notatio

Quiet Planting in the Locked Constraint Satisfaction Problems

We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles. The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to be crucial. Our main result is the location of the hard region in the planted ensemble. In a part of that hard region instances have with high probability a single satisfying assignment.Comment: 21 pages, revised versio

Non-linear Log-Sobolev inequalities for the Potts semigroup and applications to reconstruction problems

Consider a Markov process with state space $[k]$, which jumps continuously to a new state chosen uniformly at random and regardless of the previous state. The collection of transition kernels (indexed by time $t\ge 0$) is the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $\alpha_2$ in the $2$-log-Sobolev inequality ($2$-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., $p$-NLSI, $p\ge1$). As an example, we show $\alpha_1 = 1+\frac{1+o(1)}{\log k}$. The more precise NLSIs have been shown by Polyanskiy and Samorodnitsky to imply various geometric and Fourier-analytic results. Beyond the Potts semigroup, we also analyze Potts channels -- Markov transition matrices $[k]\times [k]$ constant on and off diagonal. (Potts semigroup corresponds to a (ferromagnetic) subset of matrices with positive second eigenvalue). By integrating the $1$-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a $k$-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least $$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$ This extends previous results (of Sly and Bhatnagar et al.) to general trees, and avoids the need for any specialized arguments. Similarly, we improve the state-of-the-art on reconstruction threshold for the stochastic block model with $k$ balanced groups, for all $k\ge 3$. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics

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 statistical physics and cs. In this work, we consider the colouring model. A basic question here is whether the root's assignment affects the distribution of the colourings at the vertices at distance h from the root. This is the so-called "reconstruction problem". For a d-ary tree it is well known that d/ln (d) is the reconstruction threshold. That is, for k=(1+eps)d/ln(d) we have non-reconstruction while for k=(1-eps)d/ln(d) we have. Here, we consider the largely unstudied case where the underlying tree is chosen according to a predefined distribution. In particular, our focus is on the well-known Galton-Watson trees. This model arises naturally in many contexts, e.g. the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The aforementioned study 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)