Randomly Coloring Constant Degree Graphs
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Optimal phylogenetic reconstruction
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The Kesten-Stigum Reconstruction Bound Is Tight for Roughly Symmetric Binary Channels
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Fast mixing for independent sets, colorings, and other models on trees
Random Structures & Algorithms
Reconstruction for Models on Random Graphs
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Reconstruction threshold for the hardcore model
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
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The reconstruction problem on the tree plays a key role in several important computational problems. Deep conjectures in statistical physics link the reconstruction problem to properties of random constraint satisfaction problems including random k-SAT and random colourings of random graphs. At this precise threshold the space of solutions is conjectured to undergo a phase transition from a single collected mass to exponentially many small clusters at which point local search algorithm must fail. In computational biology the reconstruction problem is central in phylogenetics. It has been shown [Mossel 04] that solvability of the reconstruction problem is equivalent to phylogenetic reconstruction with short sequences for the binary symmetric model. Rigorous reconstruction thresholds, however, have only been established in a small number of models. We confirm conjectures made by Mezard and Montanari for the Potts models proving the first exact reconstruction threshold in a non-binary model establishing the so-called Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the $q$-state Potts model when q ≥ 5. Moreover, we determine asymptotics for these reconstruction thresholds.