We prove the following type of discrete entropy monotonicity for sums of isotropic, log-concave, independent and identically distributed random vectors \(X_1,\dots ,X_{n+1}\) on \(\mathbb {Z}^d\) : \( H(X_1+\cdots +X_{n+1}) \ge H(X_1+\cdots +X_{n}) + \frac{d}{2}\log {\Bigl (\frac{n+1}{n}\Bigr )} +o(1), \) where o(1) vanishes as \(H(X_1) \rightarrow \infty \) . 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. In fact, we show this result under more general assumptions than log-concavity, which are preserved up to constants under convolution. Namely, we consider families of random variables for which, as the determinant of the covariance matrix increases, the probability mass function i) is bounded above in terms of the determinant of the covariance matrix, ii) has subexponential tails, iii) has (discrete) bounded variation. In order to show that log-concave distributions satisfy our assumptions in dimension \(d\ge 2\) , 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, barycenter and covariance matrix are close to their discrete counterparts. Moreover, in the log-concave case, we weaken the isotropicity assumption to what we call almost isotropicity. One of our technical tools is a discrete analogue to the upper bound on the isotropic constant of a log-concave function, which extends to dimensions \(d\ge 1\) a result of Bobkov, Marsiglietti and Melbourne (2022) in dimension one and may be of independent interest.
