A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds
A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds
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DOI：
10.1214/18-aihp922
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发表时间：
2017-03
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通讯作者：
David Garc'ia-Zelada
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作者：
David Garc'ia-Zelada
我们证明了波兰空间上由吉布斯概率测度定义的点过程序列的一个大偏差原理。这是由非归一化吉布斯测度的更一般的拉普拉斯原理得到的。我们考虑了紧化空间上的条件吉布斯测度、紧化黎曼流形上的库仑气体以及欧几里德空间中的通常吉布斯测度的三种主要应用。最后，我们研究了Fekete点的推广，并证明了拉普拉斯原理的一个确定性版本，称为$\Gamma$-收敛。这种方法的部分灵感来自于Dupuis及其合作者的作品。与吉布斯测度的常用策略相比，它是非常自然和普遍的。
We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $\Gamma$-convergence. The approach is partly inspired by the works of Dupuis and co-authors. It is remarkably natural and general compared to the usual strategies for singular Gibbs measures.