论文标题
通过空间混合的相对熵的块分解
Block factorization of the relative entropy via spatial mixing
论文作者
论文摘要
我们考虑在满足所谓的强空间混合条件的$ d $维晶格中的旋转系统。我们表明，相应的Gibbs测量的相对熵功能满足了一个不平等的家族，该家族控制给定区域$ v \ subset z^d $在块上的熵的加权总和$ a \ subset v $时，每个$ a $ a $均为任意非负性权重$α_A$。这些不平等概括了Glauber动力学的众所周知的对数Sobolev不等式。此外，它们提供了Shannon熵满足的经典剪力机不平等的自然扩展。最后，他们暗示了一个经过改进的对数Sobolev不平等的家族，这些家族对热浴类型的任意加权块动力学的均衡进行了定量控制。
We consider spin systems in the $d$-dimensional lattice $Z^d$ satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region $V\subset Z^d$ in terms of a weighted sum of the entropies on blocks $A\subset V$ when each $A$ is given an arbitrary nonnegative weight $α_A$. These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.
