Non-Gaussian asymptotic minimizers in entropic uncertainty principles and the dimensional effect

In this paper we revisit the Bialynicki-Birula & Mycielski uncertainty principle (I. Bialynicki-Birula and J. Mycielski, 1975) and the associated cases of equality. This Shannon entropic version of the well-known Heisenberg inequality can be used when dealing with variables that admit no variance. In this paper, we extend this uncertainty principle to Renyi entropies. We recall that in both cases, equality occurs only for Gaussian random variables. However, we show that in the particular n-dimensional Laplace case, the bound is asymptotically attained as n grows. We also show numerically that this effect exists for Cauchy variables whatever the Renyi entropy considered, extending the results of S. Abe and A.K. Rajagopal (2001), These two cases are interesting since they show that this asymptotic behavior cannot be considered as a "Gaussianization" of the variable when the dimension increases, so that the effect is rather "dimensional"