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Polynomial mixing of the edge-flip Markov chain for unbiased dyadic tilings [article]

Sarah Cannon, David Levin, Alexandre Stauffer
2016 arXiv   pre-print
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002  ...  Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n^4.09), which implies that the mixing time is at most O(n^5.09).  ...  Acknowledgments This work started during the 2016 AIM workshop Markov chain mixing times. We thank the organizers for the invitation and the stimulating atmosphere.  ... 
arXiv:1611.03636v1 fatcat:gssfcjs5lrhadobw7tjngdlqwm

Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings *

Sarah Cannon, David Levin, Alexandre Stauffer
unpublished
We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall, and Spencer in 2002  ...  Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n 4.09), which implies that the mixing time is at most O(n 5.09).  ...  This work started during the 2016 AIM workshop Markov chain  ... 
fatcat:dxiztbpuhnalpmi2hncxebdl44

Pivotal, cluster, and interface measures for critical planar percolation

Christophe Garban, Gábor Pete, Oded Schramm
2013 Journal of The American Mathematical Society  
and Chuck Newman for explaining that the results of [CN06] imply the uniqueness of the quad-crossing scaling limit, and to Asaf Nachmias for encouraging us to include the statement about the rotational  ...  Acknowledgments We are grateful to Artëm Sapozhnikov for explaining to us the simple proof of the Strong Separation Lemma (Proposition A.1) in emails, before it was written up in [DS11] , to Vincent Beffara  ...  for our Markov process.  ... 
doi:10.1090/s0894-0347-2013-00772-9 fatcat:7pdlroy3hnbsjpmu445q3caqs4

Dagstuhl Reports, Volume 7, Issue 8, August 2017, Complete Issue [article]

2018
The unique scientific atmosphere and the historic building provided not only all the room we needed for our program and the working groups, but also plenty of opportunities for continued discussions and  ...  On behalf of all participants the organizers want to express their deep gratitude to the entire Dagstuhl staff for their outstanding support and service accompanying this seminar.  ...  The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling.  ... 
doi:10.4230/dagrep.7.8 fatcat:gksmijgk5ff6reblxsqnt33aze

2D growth processes: SLE and Loewner chains

Michel Bauer, Denis Bernard
2006 Physics reports  
Some of the SLE remarkable properties are explained, as well as the tools for computing with SLE.  ...  This review has been written with the aim of filling the gap between the mathematical and the physical literatures on the subject.  ...  Acknowledgements: We benefited from discussions with many of our colleagues.  ... 
doi:10.1016/j.physrep.2006.06.002 fatcat:vahjyrl3z5haffla3gwhm7ut4a