An elementary approach to free entropy theory for convex potentials
An elementary approach to free entropy theory for convex potentials
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凸势自由熵理论的基本方法
DOI：
10.2140/apde.2020.13.2289
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
David Jekel
中科院分区：
文献类型：
--
作者：
David Jekel
我们提出了另一种方法的理论自由吉布斯状态凸势在几篇论文中的Guionnet，Shlyakhtenko，和Dabrowski。而不是解决的问题，我们结合联合收割机PDE技术与跟踪多项式的渐近逼近的一系列功能$M_N（\mathbb{C}）_{sa}^m$的概念，以证明以下。设$\mu_N$是M_N（\mathbb{C}）_{sa}^m$上由一致凸势和半凹势V_N$给出的一个概率测度，且序列DV_N$可由迹多项式渐近逼近.然后$\mu_N$的矩收敛到一个非交换律$\lambda$。此外，自由熵$\chi（\lambda）$、$\underline{\chi}（\lambda）$和$\chi^*（\lambda）$一致并等于标准化经典熵$\mu_N$的极限。一个关键的步骤是显示跟踪多项式的渐近逼近的属性被保存在几个操作下，包括限制，组合，高斯卷积，并最终在某些抛物偏微分方程的演变。这使我们能够证明收敛的时刻$\mu_N$和Fisher信息的高斯扰动$\mu_N$。
We present an alternative approach to the theory of free Gibbs states with convex potentials developed in several papers of Guionnet, Shlyakhtenko, and Dabrowski. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by uniformly convex and semi-concave potentials $V_N$, and suppose that the sequence $DV_N$ is asymptotically approximable by trace polynomials. Then the moments of $\mu_N$ converge to a non-commutative law $\lambda$. Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$, and $\chi^*(\lambda)$ agree and equal the limit of the normalized classical entropies of $\mu_N$. A key step is to show that the property of asymptotic approximation by trace polynomials is preserved under several operations, including limits, composition, Gaussian convolution, and ultimately evolution under certain parabolic PDE. This allows us to prove convergence of the moments of $\mu_N$ and of the Fisher information of Gaussian perturbations of $\mu_N$.