广秩HCIZ积分渐近的高效计算
Efficient computation of the asymptotics of extensive-rank HCIZ integrals
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中文总结 AI 辅助
本文提出一种基于粒子离散化的高效数值方法，用于求解广秩HCIZ积分的高维渐近极限，并证明其收敛性，为高维模型数值探索开辟新途径。
中文摘要 AI 辅助
我们研究了广秩区域中Harish-Chandra-Itzykson-Zuber（HCIZ）积分的高维渐近行为。这些积分的极限由一个一维边值流体动力学问题所控制，该问题最初由Matytsin（1994）推导，并由Guionnet和Zeitouni（2002）严格证明。尽管其应用广泛，但该问题的显式解仅在少数特定情况下已知。在本工作中，我们引入了一种基于粒子离散化的高效数值方案，并证明了其对一般边界密度连续边值问题的收敛性。我们针对已知解析解验证了我们的方法，并将其应用于一般密度，揭示了有趣的动力学现象。HCIZ积分的高维极限出现在各种背景中，从随机矩阵谱的大偏差到无序系统的极限自由能、高维统计和机器学习。因此，我们的贡献为数值探索先前难以处理的各种高维模型开辟了道路。
英文摘要
We study the high-dimensional asymptotics of Harish-Chandra-Itzykson-Zuber (HCIZ) integrals in the extensive-rank regime. The limit of these integrals is governed by a one-dimensional boundary-value hydrodynamical problem originally derived by Matytsin (1994) and rigorously proved by Guionnet and Zeitouni (2002). Despite its wide-ranging applications, explicit solutions to this problem are known only in a few specific cases. In this work, we introduce an efficient numerical scheme based on a particle discretization and prove its convergence to the continuous boundary-value problem for generic boundary densities. We validate our approach against known analytical solutions and apply it to generic densities, uncovering interesting dynamical phenomena. The high-dimensional limit of HCIZ integrals appears in various contexts, from the large deviations of random matrix spectra to the limiting free energy of disordered systems, high-dimensional statistics, and machine learning. As such, our contribution opens the way towards the numerical exploration of a wide range of high-dimensional models that were previously intractable.
