Borodin Alexei, Duits Maurice
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 USA.
Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25, 10044 Stockholm, Sweden.
Probab Theory Relat Fields. 2023;187(1-2):259-315. doi: 10.1007/s00440-023-01195-8. Epub 2023 Feb 14.
We study random domino tilings of the Aztec diamond with a biased periodic weight function 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.
我们研究了具有偏置周期权重函数的阿兹特克钻石的随机多米诺骨牌平铺，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核的双积分公式，其中被积函数是根据此流来表示的。对于参数的特殊选择，流是周期性的，这使我们能够对相关核进行鞍点分析。在这些情况下，我们计算了光滑无序（或气态）区域中的局部相关性。我们更详细地研究了流具有周期六的特殊示例，并表明在这种情况下，粗糙无序区域的边界是一条八次代数曲线。
