Large deviations for empirical measures generated by Gibbs measures with singular energy functionals
Large deviations for empirical measures generated by Gibbs measures with singular energy functionals
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发表时间：
2015-11
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通讯作者：
P. Dupuis;Vaios Laschos;K. Ramanan
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作者：
P. Dupuis;Vaios Laschos;K. Ramanan
我们建立了大偏差原理（LDPs），用于与$n$ -粒子构型上的吉布斯分布序列相关的经验测量，每个Gibbs分布都是根据逆温度$\beta_n$和能量泛函数定义的，该泛函数是（可能是单一的）相互作用和限制势的总和。在对势的相当一般的假设下，我们建立了具有速度$\beta_n/n \rightarrow \infty$的LDPs，在这种情况下，速率函数用包含势的函数表示，当速率函数包含额外的熵项时，我们建立了具有速度$\beta_n =n$的LDPs。这种LDPs是由随机矩阵理论、抽样和模拟退火中出现的问题所驱动的。我们的方法使用了“大偏差理论的弱收敛方法”中开发的弱收敛方法，建立了相对于更强的wasserstein型拓扑的大偏差原理，从而解决了“Calogero-Sutherland气体的一阶全局渐近性”中的一个开放问题。它还为分析所有速度的ldp提供了一个通用框架，并包括由于先前工作中技术原因而未涵盖的情况。
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 that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we 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 the 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 and simulated annealing. Our approach, which uses the weak convergence methods developed in "A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in "First order global asymptotics for Calogero-Sutherland gases". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.