Large deviations for configurations generated by Gibbs distributions with energy functionals consisting of singular interaction and weakly confining potentials
Large deviations for configurations generated by Gibbs distributions with energy functionals consisting of singular interaction and weakly confining potentials
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DOI：
10.1214/20-ejp449
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发表时间：
2015-11
影响因子：
1.4
通讯作者：
P. Dupuis;Vaios Laschos;K. Ramanan
中科院分区：
文献类型：
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作者：
P. Dupuis;Vaios Laschos;K. Ramanan
我们建立大偏差原则（LDPs）的经验措施与吉布斯分布的序列上$n$粒子配置，其中每一个被定义在一个逆温度$% \beta_n$和能量泛函组成的（可能是奇异的）相互作用势和（可能是弱的）限制势。在相当一般的假设下的潜力，我们使用一个共同的框架来建立LDPs的速度$\beta_n/n \rightarrow \infty$，在这种情况下，速率函数表示的功能涉及的潜力，和速度$\beta_n =n$，当速率函数包含一个额外的熵项。这种LDPs的动机随机矩阵理论，采样，模拟退火和渐近凸几何中出现的问题。我们的方法，它使用Dupuis和Ellis开发的弱收敛方法，建立LDPs相对于更强的Wasserstein型拓扑结构。我们的研究结果解决了几个有趣的例子没有涵盖以前的作品，包括弱限制的潜力，这使得率函数的极小值，没有紧凑的支持，从而解决了几个开放的问题中提出的工作Chafa\"{\i}等人的情况。
We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on $n$-particle configurations, each of which is defined in terms of an inverse temperature $% \beta_n$ and an energy functional consisting of a (possibly singular) interaction potential and a (possibly weakly) confining potential. Under fairly general assumptions on the potentials, we use a common framework to establish LDPs both with speeds $\beta_n/n \rightarrow \infty$, in which case the rate function is expressed in terms of a functional involving the potentials, and with speed $\beta_n =n$, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling, simulated annealing and asymptotic convex geometry. Our approach, which uses the weak convergence method developed by Dupuis and Ellis, establishes LDPs with respect to stronger Wasserstein-type topologies. Our results address several interesting examples not covered by previous works, including the case of a weakly confining potential, which allows for rate functions with minimizers that do not have compact support, thus resolving several open questions raised in a work of Chafa\"{\i} et al.