Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-6f8dk Total loading time: 0.225 Render date: 2021-03-05T17:46:44.786Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Mixing properties of colourings of the ℤd lattice

Published online by Cambridge University Press:  19 October 2020

Noga Alon
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA, and Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 6997801, Israel
Raimundo Briceño
Affiliation:
Pontificia Universidad Católica de Chile, Santiago, Chile
Nishant Chandgotia
Affiliation:
School of Mathematical Sciences, Hebrew University of Jerusalem, Israel
Alexander Magazinov
Affiliation:
Higher School of Economics, National Research University, 6 Usacheva Street, Moscow 119048, Russia
Yinon Spinka
Affiliation:
University of British Columbia, Department of Mathematics, Vancouver, BC V6T 1Z2, Canada
Corresponding

Abstract

We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When $q\le d+1$ , there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when $q\ge d+2$ , any proper q-colouring of the boundary of a box of side length $n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When $q\geq 2d+1$ , the latter holds for any $n \ge 1$ . Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.

Type
Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.

References

Alon, N., Krivelevich, M. and Sudakov, B. (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 7382.CrossRefGoogle Scholar
Alon, N. and Tarsi, M. (1992) Colorings and orientations of graphs. Combinatorica 12 125134.CrossRefGoogle Scholar
Boyle, M., Pavlov, R. and Schraudner, M. (2010) Multidimensional sofic shifts without separation and their factors. Trans. Amer. Math. Soc. 362 46174653.CrossRefGoogle Scholar
Briceño, R. (2018) The topological strong spatial mixing property and new conditions for pressure approximation. Ergodic Theory Dynam. Systems 38 16581696.CrossRefGoogle Scholar
Briceño, R. and Bulatov, A.A. and Dalmau, V. and Larose, B. (2019) Dismantlability, Connectedness, and Mixing in Relational Structures, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019), Leibniz International Proceedings in Informatics (LIPIcs) 132 18688969.Google Scholar
Briceño, R., McGoff, K. and Pavlov, R. (2018) Factoring onto ℤd subshifts with the finite extension property. Proc. Amer. Math. Soc. 146 51295140.CrossRefGoogle Scholar
Briceño, R. and Pavlov, R. (2017) Strong spatial mixing in homomorphism spaces. SIAM J. Discrete Math. 31 21102137.CrossRefGoogle Scholar
Brightwell, G. R. and Winkler, P. (2000) Gibbs measures and dismantlable graphs. J. Combin. Theory Ser. B 78 141166.CrossRefGoogle Scholar
Brightwell, G. R. and Winkler, P. (2002) Random colorings of a Cayley tree. Contemporary Combinatorics 10 247276.Google Scholar
Burton, R. and Steif, J. E. (1994) Non-uniqueness of measures of maximal entropy for subshifts of finite type. Ergodic Theory Dynam. Systems 14 213235.Google Scholar
Chandgotia, N. (2018) A short note on the pivot property. http://math.huji.ac.il/~nishant/Research_files/Notes_files/Pivot.pdf Google Scholar
Chandgotia, N. and Marcus, B. (2018) Mixing properties for hom-shifts and the distance between walks on associated graphs. Pacific J. Math. 294 4169.CrossRefGoogle Scholar
Chandgotia, N. and Meyerovitch, T. (2016) Markov random fields, Markov cocycles and the 3-colored chessboard. Israel J. Math. 215 909964.CrossRefGoogle Scholar
Gamarnik, D., Katz, D. and Misra, S. (2015) Strong spatial mixing of list coloring of graphs. Random Struct. Algorithms 46 599613.CrossRefGoogle Scholar
Johansson, A. (1996) Asymptotic choice number for triangle free graphs. DIMACS technical report.Google Scholar
Jonasson, J. (2002) Uniqueness of uniform random colorings of regular trees. Statist. Probab. Lett. 57 243248.CrossRefGoogle Scholar
Marcus, B. and Pavlov, R. (2015) An integral representation for topological pressure in terms of conditional probabilities. Israel J. Math. 207 395433.CrossRefGoogle Scholar
Pak, I., Sheffer, A. and Tassy, M. (2016) Fast domino tileability. Discrete Comput. Geom. 56 377394.CrossRefGoogle Scholar
Peled, R. and Spinka, Y. (2018) Rigidity of proper colorings of ℤd. arXiv:1808.03597Google Scholar
Schmidt, K. (1995) The cohomology of higher-dimensional shifts of finite type. Pacific J. Math. 170 237269.CrossRefGoogle Scholar
Sheffield, S. (2005) Random Surfaces (Astérisque 304). Société mathématique de France.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 11 *
View data table for this chart

* Views captured on Cambridge Core between 19th October 2020 - 5th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Mixing properties of colourings of the ℤd lattice
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Mixing properties of colourings of the ℤd lattice
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Mixing properties of colourings of the ℤd lattice
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *