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.

中文翻译：

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使用指数度量对有偏差的单调表面进行采样

跨越 ℤ 有限区域的单调曲面d出现在许多情况下，包括基于 DNA 的自组装、洗牌和菱形拼贴。一种用于均匀生成这些表面的方法是马尔可夫链，它在一个步骤中迭代地添加或移除表面下方的单个立方体。我们考虑了一个有偏差的链，我们更有可能添加一个立方体而不是删除它，从而有利于“更高”或在其下方有更多立方体的表面。我们证明对于 ℤ 中的任何均匀偏差，该链正在快速混合2对于偏差 λ >d在ℤd什么时候d> 2. 在ℤ2我们匹配了 Benjamini、Berger、Hoffman 和 Mossel 在有偏见的洗牌 [2] 的背景下实现的最佳混合时间，但使用了更简单的论点。证明使用几何距离函数和路径耦合的变体，以处理可能呈指数增长的距离。我们还提供了在以下情况下的第一个结果波动偏差，其中偏差可能会因图块的位置而异。我们表明，如果偏差接近均匀，则链会继续快速混合，但在一般情况下，链会以指数方式缓慢收敛。