Determinantal point processes and fermions on complex manifolds: Bulk universality
Determinantal point processes and fermions on complex manifolds: Bulk universality
复制标题
复流形上的行列式点过程和费米子：整体普适性
DOI：
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发表时间：
2008
期刊：
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通讯作者：
R. Berman
中科院分区：
文献类型：
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作者：
R. Berman
在多粒子极限下研究了紧致复流形X上的行列式点过程。过程的相关核是X上Hermitian全纯线丛L的高次幂的Bergman核。它示出的过程中，描述粒子的位置，在X上的定义随机措施，收敛概率朝着一个pluripotential平衡措施，表示为蒙格-安培措施。它的平滑波动在散装被证明是渐近正常的和明确计算的极限方差。的相关函数的缩放限制示出是普遍的，并表示在条款（高维模拟）的Ginibre合奏。这种设置特别适用于正常的随机矩阵集合和多元正交多项式。还探讨了强磁场中基态费米子的相变、直接像束和隧穿（即指数小的Dolbeault Laplacian本征值）的关系。
Determinantal point processes on a compact complex manifold X are considered in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line bundle L over X. It is shown that the defining random measure on X of the process, describing the particle locations, converges in probability towards a pluripotential equilibrium measure, expressed as a Monge-Ampere measure. Its smooth fluctuations in the bulk are shown to be asymptotically normal and the limiting variance is explicitly computed. A scaling limit of the correlation functions is shown to be universal and expressed in terms of (the higher dimensional analog of) the Ginibre ensemble. This setting applies in particular to normal random matrix ensembles and multivariate orthogonal polynomials. Relation to phase transitions, direct image bundles and tunneling of ground state fermions in strong magnetic fields (i.e. exponentially small eigenvalues of the Dolbeault Laplacian) are also explored.