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The number of colorings of the middle layers of the Hamming cube
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The number of colorings of the middle layers of the Hamming cube
abstract
For an odd integer $n = 2d-1$, let $\mathcal B_d$ be the subgraph of the hypercube $Q_n$ induced by the two largest layers. In this paper, we describe the typical structure of proper $q$-colorings of $V(\mathcal B_d)$ and give asymptotics on the number of them. The proofs use various tools including information theory (entropy), Sapozhenko's graph container method and a recently developed method of M. Jenssen and W. Perkins that combines Sapozhenko's graph container lemma with the cluster expansion for polymer models from statistical physics.
Forward citations
Cited by 1 Pith paper
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Range of random $\mathbb Z$-homomorphisms on weak expanders
Random Z-homomorphisms on weak expanders are O(log log n)-flat with high probability, answering a question of Peled-Samotij-Yehudayoff, and at most 5-valued on Hamming-cube middle layers.
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