We prove the following type of discrete entropy monotonicity for isotropic log-concave sums of independent identically distributed random vectors $X_1,\dots,X_{n+1}$ on $\mathbb{Z}^d$: $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$ where $o(1)$ vanishes as $H(X_1) \to \infty$. Moreover, for the $o(1)$-term we obtain a rate of convergence $ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$, where the implied constants depend on $d$ and $n$. This generalizes to $\mathbb{Z}^d$ the one-dimensional result of the second named author (2023). As in dimension one, our strategy is to establish that the discrete entropy $H(X_1+\cdots+X_{n})$ is close to the differential (continuous) entropy $h(X_1+U_1+\cdots+X_{n}+U_{n})$, where $U_1,\dots, U_n$ are independent and identically distributed uniform random vectors on $[0,1]^d$ and to apply the theorem of Artstein, Ball, Barthe and Naor (2004) on the monotonicity of differential entropy. However, in dimension $d\ge2$, more involved tools from convex geometry are needed because a suitable position is required. We show that for a log-concave function on $\mathbb{R}^d$ in isotropic position, its integral, its barycenter and its covariance matrix are close to their discrete counterparts. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which generalises a result of Bobkov, Marsiglietti and Melbourne (2022) and may be of independent interest.

翻译：我们证明了以下关于$\mathbb{Z}^d$上独立同分布各向同性对数凹随机向量和$X_1,\dots,X_{n+1}$的离散熵单调性：$$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{d}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} +o(1), $$其中$o(1)$项在$H(X_1) \to \infty$时趋于零。此外，对于$o(1)$项，我们得到了收敛速度$ O\Bigl({H(X_1)}{e^{-\frac{1}{d}H(X_1)}}\Bigr)$，其中隐含常数依赖于$d$和$n$。这推广了第二作者（2023年）在一维情形下的结果至$\mathbb{Z}^d$。与一维情况类似，我们的策略是证明离散熵$H(X_1+\cdots+X_{n})$接近于微分（连续）熵$h(X_1+U_1+\cdots+X_{n}+U_{n})$，其中$U_1,\dots, U_n$是$[0,1]^d$上独立同分布的均匀随机向量，并应用Artstein、Ball、Barthe和Naor（2004年）关于微分熵单调性的定理。然而，在维数$d\ge2$时，需要用到凸几何中更复杂的工具，因为需要恰当的位置。我们证明了在$\mathbb{R}^d$上处于各向同性位置的对数凹函数，其积分、重心和协方差矩阵与其离散对应量相近。我们的一个技术工具是对数凹函数各向同性常数上界的离散类似物，这推广了Bobkov、Marsiglietti和Melbourne（2022年）的结果，并可能具有独立意义。