研究了在对称测度下具有有限时域$ [0,β] $的格点$ \Z^d $上的$ N $随机过程的大偏差原理，其中所有的始点和终点都由一个随机置换一致给定.也就是说，给定$ N $元素的置换$ σ$和$ N $初始点的向量$ (x_1,...,x_N) $，我们让随机过程终止于点$ (x_{σ(1)},...,x_{σ(N)}) $，然后对所有可能的置换和初始点求和，用初始分布加权。有一个两级随机机制，我们证明了两级大偏差原则的平均经验路径措施，平均路径和平均占领当地时间在此对称措施。对称测度不能写成任何单个随机过程分布的乘积。我们展示了这些结果在量子统计力学中的几个重要应用，使用Feynman-Kac公式表示某些迹类算子的迹。特别地，我们证明了具有玻色-爱因斯坦统计和平均场相互作用的量子自旋系统的非对易Varadhan引理。对于$ N $简单随机游动的局部时的平均值，我们的大偏差原理的一个特例是对于有限时间$ β$，但对于有限时间$ N\to\infty $，Donsker-Varadhan速率函数作为速率函数.对这一令人惊讶的结果给出了量子统计力学的解释。
英文摘要：
We study large deviations principles for $ N $ random processes on the lattice $ \Z^d $ with finite time horizon $ [0,β] $ under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation $ σ$ of $ N $ elements and a vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes terminate in the points $ (x_{σ(1)},...,x_{σ(N)}) $ and then sum over all possible permutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviations principles for the mean of empirical path measures, for the mean of paths and for the mean of occupation local times under this symmetrised measure. The symmetrised measure cannot be written as any product of single random process distributions. We show a couple of important applications of these results in quantum statistical mechanics using the Feynman-Kac formulae representing traces of certain trace class operators. In particular we prove a non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation local times of $ N $ simple random walks has the Donsker-Varadhan rate function as the rate function for the limit $ N\to\infty $ but for finite time $ β$. We give an interpretation in quantum statistical mechanics for this surprising result.
Star-exponential for Fermi systems and the Feynman-Kac formula
Facultad de Ciencias, Universidad Autónoma de San Luis Potosí（圣路易斯波托西自治大学理学院）；Departamento de Física, Centro de Investigación y de Estudios Avanzados del IPN（墨西哥国立理工学院高级研究中心物理系）
