发表机构
St. Petersburg State University; Beijing Institute of Mathematical Sciences and Applications (BIMSA)（圣彼得堡国立大学; 北京数学与应用科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明 Christoffel 变换下投影算子仅差秩一算子，并应用于斜 Sp Howe 对偶性，得到随机 Young 图局部涨落的四种渐近区域，包括临界区域的离散 Hermite 核。
AI 中文摘要
对于有限离散格点上的对称权重 $w(x)$ 及其 Christoffel 变换 $x^{2}w(x),x^{2}(x^{2}w(x)),\dots$，我们证明，在每一步 Christoffel 变换中，与变换后的 Christoffel--Darboux 核相关联的共轭投影与原始正交投影在原点处消失的函数子空间上相差一个秩一算子。这为当格点大小趋于无穷时，将局部渐近结果从正交多项式系综传递到其 Christoffel 变换后的对应系综提供了一种一般机制。作为主要应用，我们研究了由斜 $(\mathrm{Sp}_{2n},\mathrm{Sp}_{2k})$ Howe 对偶性产生的随机 Young 图的局部涨落。相应的粒子系综是通过 Christoffel 变换从二次格点上的 Krawtchouk 正交多项式系综获得的。在 $n,k\to\infty$ 且 $n/k\to c\in(0,\infty)$ 的极限下，我们确定了四种局部涨落的渐近区域。除了由离散正弦核控制的普适体涨落和极限形状右边缘的普适 Airy 涨落外，我们在临界区域 $(k-n)/\sqrt{n+k}\longrightarrow r\in\mathbb{R}$ 中得到了离散 Hermite 核，并在左角处得到了离散硬壁正弦核。
英文摘要
For a symmetric weight $w(x)$ on a finite discrete lattice and its Christoffel transforms $x^{2}w(x),x^{2}(x^{2}w(x)),\dots$, we prove that, at each Christoffel step, the conjugated projection associated with the transformed Christoffel--Darboux kernel differs from the original orthogonal projection by a rank-one operator on the subspace of functions vanishing at the origin. This provides a general mechanism for transferring local asymptotic results from an orthogonal polynomial ensemble to its Christoffel-transformed counterpart as the lattice size tends to infinity. As a main application, we study local fluctuations of random Young diagrams arising from skew $(\mathrm{Sp}_{2n},\mathrm{Sp}_{2k})$ Howe duality. The corresponding particle ensemble is obtained from the Krawtchouk orthogonal polynomial ensemble on a quadratic lattice by a Christoffel transform. We identify four asymptotic regimes of local fluctuations in the limit $n,k\to\infty$ with $n/k\to c\in(0,\infty)$. Besides the universal bulk fluctuations governed by the discrete sine kernel and universal Airy fluctuations at the right edge of the limit shape, we obtain the discrete Hermite kernel in the critical regime $(k-n)/\sqrt{n+k}\longrightarrow r\in\mathbb{R}$, and the discrete hard-wall sine kernel at the left corner.
Comments31 pages, 1 figure
