标题： 高维酉群上的傅里叶变换及其在随机铺砌中的应用 显示英文标题

标题： Fourier transform on high-dimensional unitary groups with applications to random tilings

摘要： 单位群上的直接和逆傅里叶变换的组合 $U(N)$将归一化特征标与$N$-元组的测度联系起来。 我们通过匹配特征标对数在单位元处的部分导数的渐进行为与相应随机$N$-元组全局行为的大数定律和中心极限定理，来发展这种对应关系的$N\to\infty$版本。 作为一项应用，我们研究了随机多米诺和菱形镶嵌在一个丰富类域中的高度函数的波动。 在这方面，我们证明了 Kenyon-Okounkov 的猜想（该猜想预测渐近高斯性和协方差的精确形式）对于一类非单连通多边形。 另一个应用是带有随机初始数据的$U(N)$量子随机游走的中心极限定理。 显示英文摘要

摘要： A combination of direct and inverse Fourier transforms on the unitary group $U(N)$ identifies normalized characters with probability measures on $N$-tuples of integers. We develop the $N\to\infty$ version of this correspondence by matching the asymptotics of partial derivatives at the identity of logarithm of characters with Law of Large Numbers and Central Limit Theorem for global behavior of corresponding random $N$-tuples. As one application we study fluctuations of the height function of random domino and lozenge tilings of a rich class of domains. In this direction we prove the Kenyon-Okounkov's conjecture (which predicts asymptotic Gaussianity and exact form of the covariance) for a family of non-simply connected polygons. Another application is a central limit theorem for the $U(N)$ quantum random walk with random initial data.