# Large Degree Asymptotics and the Reconstruction Threshold of the Asymmetric Binary Channels

## TL;DR

This paper investigates the reconstruction problem on noisy tree networks with asymmetric binary channels, providing precise thresholds and asymptotic behavior for large degrees through refined analytical methods.

## Contribution

It extends previous work by rigorously determining the conditions under which the reconstruction threshold is tight for asymmetric binary channels on large-degree trees.

## Key findings

- Established the exact reconstruction threshold for asymmetric binary channels.
- Derived asymptotic behavior of the threshold as the degree grows large.
- Provided refined analysis techniques for moment recursion and concentration.

## Abstract

In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels), the exact threshold for the reconstruction problem for a binary asymmetric channel on the d-ary tree is establish, provided that the asymmetry is sufficiently small, which is the first exact reconstruction threshold obtained in roughly a decade. In this paper, by means of refined analyses of moment recursion on a weighted version of the magnetization, and concentration investigations, we rigorously give a complete answer to the question of how small it needs to be to establish the tightness of the reconstruction threshold and further determine its asymptotics of large degrees.