Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
复制标题
多边形的均匀随机菱形平铺的渐近性。
DOI：
10.1214/12-aop823
复制
发表时间：
2012
期刊：
影响因子：
--
通讯作者：
L. Petrov
中科院分区：
文献类型：
--
作者：
L. Petrov
我们研究了三角形格子上多边形均匀随机菱形镶嵌对应的随机阶梯曲面的大尺度高度涨落。对于一类多边形(允许任意多个边)，我们证明了这些涨落是由无质量(无质量)高斯场渐近控制的。这与在Kenyon[Comm.数学课。太棒了。281(2008)675-709]关于极限形状没有冻结面的区域的平铺。在我们的渐近分析中，我们使用了以前在Petrov[通过Gelfand-Tsetlin方案(2012)预印的随机菱形切片的渐近性]中得到的模型的行列式相关核的显式双轮廓积分公式。
We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large number of sides), we show that these fluctuations are asymptotically governed by a Gaussian free (massless) field. This complements the similar result obtained in Kenyon [Comm. Math. Phys. 281 (2008) 675-709] about tilings of regions without frozen facets of the limit shape. In our asymptotic analysis we use the explicit double contour integral formula for the determinantal correlation kernel of the model obtained previously in Petrov [Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes (2012) Preprint].