Intersecting random half spaces with a cube (Q1600173)

Consider independent standard normal random variables \((g_{i,k})_{i\leq N, k\leq M}\). Consider a number \(\tau>0\). The following subset \(U_k=\{x = (x_i)_{i\leq N}\in \mathbb R^N: \sum_{i\leq N} g_{i,k} x_i/\sqrt{N}\geq\tau\}\) is a random half-space of \(\mathbb R^N\). Its boundary is nearly at distance \(\tau\) from the origin. Consider the discrete cube \(\Sigma_N=\{-1, 1\}^N\). The problem is to find the ``typical'' cardinality of the set \(\Sigma_N \cap (\bigcap_{k\leq M} U_k)\). The typical cardinality is less than the average cardinality, that, of course, is easier to compute. The present work is directly motivated by the recent results by \textit{M. Shcherbina} and \textit{B. Tirozzi} [see the paper reviewed above and ``Rigorous solution to the Gardner problem'' (to appear)] who prove a similar result when the cube \(\Sigma_N\) is replaced by the sphere \(S_N =\{\sigma: \sum_{i\leq N} \sigma_i^2 =N\}\). It should be also mentioned \textit{M. Talagrand} [Ann. Probab. 28, 725-758 (2000)].