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Sampling biased monotonic surfaces using exponential metrics

Published online by Cambridge University Press:  30 June 2020

Sam Greenberg ,
Dana Randall  and
Amanda Pascoe Streib
Sam Greenberg
Affiliation:
Department of Defense, Arlington, VA, USA
Dana Randall
Affiliation:
School of Computer Science, Georgia Institute of Technology, Atlanta, GA 30332, USA
Amanda Pascoe Streib*
Affiliation:
Center for Computing Sciences, Bowie, MD 20715, USA
*
*Corresponding author. Email: [email protected]
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Abstract

Monotonic surfaces spanning finite regions of ℤd arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that iteratively adds or removes a single cube below the surface during a step. We consider a biased version of the chain, where we are more likely to add a cube than to remove it, thereby favouring surfaces that are ‘higher’ or have more cubes below it. We prove that the chain is rapidly mixing for any uniform bias in ℤ2 and for bias λ > d in ℤd when d > 2. In ℤ2 we match the optimal mixing time achieved by Benjamini, Berger, Hoffman and Mossel in the context of biased card shuffling [2], but using much simpler arguments. The proofs use a geometric distance function and a variant of path coupling in order to handle distances that can be exponentially large. We also provide the first results in the case of fluctuating bias, where the bias can vary depending on the location of the tile. We show that the chain continues to be rapidly mixing if the biases are close to uniform, but that the chain can converge exponentially slowly in the general setting.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 5 , September 2020 , pp. 672 - 697
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

A preliminary version of this paper appeared in Proceedings of the 20th ACM–SIAM Symposium on Discrete Algorithms (2009), pp. 76–85.

‡

Supported in part by NSF grants CCF-1526900, CCF-1637031 and CCF-1733812.

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