TY - GEN
T1 - A log-sobolev inequality for the multislice, with applications
AU - Filmus, Yuval
AU - O’Donnell, Ryan
AU - Wu, Xinyu
N1 - Publisher Copyright:
© Yuval Filmus, Ryan O’Donnell, and Xinyu Wu.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - Let κ ϵ Nℓ+ satisfy κ1+···+κℓ = n, and let Uκ denote the multislice of all strings u ∈ [ℓ]n having exactly κi coordinates equal to i, for all i ϵ [ℓ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ρκ for the chain satisfies (Formula presented), which is sharp up to constants whenever ℓ is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal–Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan–Szegedy Theorem.
AB - Let κ ϵ Nℓ+ satisfy κ1+···+κℓ = n, and let Uκ denote the multislice of all strings u ∈ [ℓ]n having exactly κi coordinates equal to i, for all i ϵ [ℓ]. Consider the Markov chain on Uκ where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant ρκ for the chain satisfies (Formula presented), which is sharp up to constants whenever ℓ is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal–Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan–Szegedy Theorem.
KW - Combinatorics
KW - Conductance
KW - Fourier analysis
KW - Hypercontractivity
KW - Log-Sobolev inequality
KW - Markov chains
KW - Representation theory
KW - Small-set expansion
UR - http://www.scopus.com/inward/record.url?scp=85069451110&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.ITCS.2019.34
DO - 10.4230/LIPIcs.ITCS.2019.34
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AN - SCOPUS:85069451110
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 10th Innovations in Theoretical Computer Science, ITCS 2019
A2 - Blum, Avrim
T2 - 10th Innovations in Theoretical Computer Science, ITCS 2019
Y2 - 10 January 2019 through 12 January 2019
ER -