论文标题
有偏见$ 2 \ times 2 $定期阿兹台克钻石和椭圆曲线
Biased $2 \times 2$ periodic Aztec diamond and an elliptic curve
论文作者
论文摘要
我们研究了一个有偏见的$ 2 \ times 2 $定期的阿兹台克钻石的随机多米诺骨牌，并将椭圆曲线上的线性流与该模型相关联。我们的主要结果是相关内核的双重积分公式，其中积分以该流量表示。对于参数的特殊选择，流是周期性的，这使我们能够对相关内核执行鞍点分析。在这些情况下，我们计算平滑无序（或气态）区域中的局部相关性。流量具有六个周期的特殊示例更详细地研究了，我们表明，在这种情况下，粗糙无序区域的边界是八度的代数曲线。
We study a biased $2\times 2$ periodic random domino tilings of the Aztec diamond and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.
