实 Gram 矩阵 Hafnian 与对称高斯 Hafnian 的平移反集中性
Shifted Anticoncentration for Real Gram Hafnians and Symmetric Gaussian Hafnians
- School of Statistics, University of Minnesota（明尼苏达大学统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明实高斯 Gram 矩阵 hafnian 的平移反集中定理，给出密度界与多项式增长系数，并推广至对称高斯情形。
AI中文摘要：
我们证明了实高斯 Gram 矩阵的 hafnian 在行维数的显式条件下满足一致的平移反集中定理。经均方根归一化后，其分布具有有界连续密度，在零处达到最大，且每个区间的概率被一个显式系数乘以区间半径所界定。在行维数的适当增长条件下，该系数随 hafnian 阶数（即 Gram 矩阵维数的一半）至多呈多项式增长。我们还计算了精确的二阶矩。证明利用了完美匹配结构，结合条件高斯表示、行悬挂和双线性插值来控制条件方差的逆矩。在固定的 hafnian 阶数下，随着行维数增长的重新标度极限，为具有对角线上方独立元素的实对称高斯矩阵的 hafnian 提供了相应的界。
英文摘要：
We prove a uniform shifted anticoncentration theorem for the hafnian of a real Gaussian Gram matrix under an explicit condition on the row dimension. After normalization by its root mean square, the law has a bounded continuous density, maximal at zero, and every interval has probability bounded by an explicit coefficient times its radius. Under suitable growth conditions on the row dimension, this coefficient grows at most polynomially in the hafnian order, meaning half the dimension of the Gram matrix. We also compute the exact second moment. The proof exploits the perfect matching structure, combining conditional Gaussian representations, row suspension, and bilinear interpolation to control an inverse moment of the conditional variance. At fixed hafnian order, a rescaled limit as the row dimension grows yields corresponding bounds for the hafnian of a real symmetric Gaussian matrix with independent entries above the diagonal.
