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Two open problems about concentation

Talagrand has shown that there exists universal constants \( {c,C>0} \) such that for any independent and identically distributed random variables \( {X_1,\ldots,X_n} \) in the unit disk \( {D} \) of \( {\mathbb{C}} \), any convex and Lipschitz function \( {f:D^n\rightarrow\mathbb{R}} \), and any real number \( {r>0} \),

\[ \mathbb{P}(|f(X_1,\ldots,X_n)-\mathbb{E}f(X_1,\ldots,X_n)|\geq r) \leq C\exp(-cr^2). \]

An accessible proof can be found in Ledoux’s monograph on the concentration phenomenon (chapter 4). This inequality is useful for instance in order to control the distance of a random vector to a sub-vector space of controlled dimension. In many situations, one would like a similar concentration result, beyond these restrictive assumptions. Let us consider for instance a \( {n\times n} \) random matrix \( {M} \) with i.i.d. entries (standard Gaussian or symmetric Bernoulli \( {\pm 1} \)). Here are two open questions about concentration of measure for which the Talagrand inequality is not enough as is due to the lack of one of the assumptions:

  • concentration for the function \( {\log|\det(M)|=\log\det\sqrt{MM^*}=\sum_{i=1}^n\log(s_i(M))} \)
  • concentration for the least singular value \( {s_n(M)=\min_{\left\Vert x\right\Vert=1}\left\Vert Mx\right\Vert_2} \)

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