Uniqueness, Spatial Mixing, and Approximation for Ferromagnetic 2-Spin Systems

Heng Guo, Pinyan Lu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

For anti-ferromagnetic 2-spin systems, a beautiful connection has been established, namely that the following three notions align perfectly: the uniqueness of Gibbs measures in infinite regular trees, the decay of correlations (also known as spatial mixing), and the approximability of the partition function. The uniqueness condition implies spatial mixing, and an FPTAS for the partition function exists based on spatial mixing. On the other hand, non-uniqueness implies some long range correlation, based on which NP-hardness reductions are built. These connections for ferromagnetic 2-spin systems are much less clear, despite their similarities to anti-ferromagnetic systems. The celebrated Jerrum-Sinclair Markov chain [JS93] works even if spatial mixing fails. Also, for a fixed degree the uniqueness condition is non-monotone with respect to the external field, which seems to have no meaningful interpretation in terms of computational complexity. However, it is still intriguing whether there are some relationship underneath the apparent disparities among them. We provide some answers to this question. Let ; be the (0; 0) and (1; 1) edge interactions respectively ( > 1), and the external field for spin “0”. For graphs with degree bound Δ Δc + 1 where Δc = p p +1 -1 , regardless of the field (even inconsistent fields are allowed), correlation decay always holds and FPTAS exists. If all fields satisfy < c (assuming ), where c = ( =) Δc+1 2 , then a weaker version of spatial mixing holds in all trees. Moreover, if 1, then < c is sufficient to guarantee strong spatial mixing and FPTAS. This improves the best previous algorithm, a Markov chain based FPRAS for = [LLZ14]. The bound c is almost optimal and can be viewed as a variant of the uniqueness condition with the degree d relaxed to be a real number instead of an integer. When 1, uniqueness holds in all infinite regular trees, if and only if int c , where int c = ( =) ⌈Δc⌉+1 2 . If we allow fields > int c ′, where int c ′ = ( =) ⌊Δc⌋+2 2 , then approximating the partition function is #BIS-hard. Interestingly, unless Δc is an integer, neither c nor int c is the tight bound in each own respect. We provide examples where correlation decay continues to hold in a small interval beyond c, and irregular trees in which spatial mixing fails for some < int c .
Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2016, September 7-9, 2016, Paris, France
Place of PublicationParis, France
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Pages31:1-31:26
Number of pages26
Volume60
ISBN (Electronic)978-3-95977-018-7
DOIs
Publication statusE-pub ahead of print - 9 Sep 2016
Event19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems / 20th International Workshop on Randomization and Computation - Paris, France
Duration: 7 Sep 20169 Sep 2016
http://cui.unige.ch/tcs/random-approx/2016/

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum fuer Informatik
Volume60
ISSN (Electronic)1868-8969

Conference

Conference19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems / 20th International Workshop on Randomization and Computation
Abbreviated titleAPPROX / RANDOM 2016
Country/TerritoryFrance
CityParis
Period7/09/169/09/16
Internet address

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