We develop transportation-entropy inequalities which are saturated by measures such that their log-density with respect to the background measure is an affine function, in the setting of the uniform measure on the discrete hypercube and the exponential measure. In this sense, this extends the well-known result of Talagrand in the Gaussian case. By duality, these transportation-entropy inequalities imply a strong integrability inequality for Bernoulli and exponential processes. As a result, we obtain on the discrete hypercube a dimension-free mean-field approximation of the free energy of a Gibbs measure and a nonlinear large deviation bound with only a logarithmic dependence on the dimension. Applied to the Ising model, we deduce that the mean-field approximation is within \(O(\sqrt{n} ||J||_2)\) of the free energy, where n is the number of spins and \(||J||_2\) is the Hilbert–Schmidt norm of the interaction matrix. Finally, we obtain a reverse log-Sobolev inequality on the discrete hypercube similar to the one proved recently in the Gaussian case by Eldan and Ledoux.

在离散超立方体和指数测度的统一测度的设置中，我们发展了运输熵不等式，这些不等式被测度饱和，使得它们的相对于背景测度的对数密度是仿射函数。从这个意义上讲，这扩展了塔拉格朗在高斯案例中的著名结果。通过对偶性，这些输运熵不等式暗示了伯努利和指数过程的强大可积性不等式。结果，我们在离散超立方体上获得了吉布斯测度的自由能和非线性大偏差的无量纲均值近似值，该偏差仅对量纲有对数依赖性。应用于伊辛模型，我们推论平均场近似值在\（O（\ sqrt {n} || J || _2）\）内的自由能，其中n是自旋数，\（|| J || _2 \）是相互作用矩阵的Hilbert-Schmidt范数。最后，我们在离散超立方体上获得了一个对数Sobolev不等式，类似于Eldan和Ledoux最近在高斯案例中证明的一个。