Basically : the goal is to bound from above the variance of the maximum of Gaussian familly Gaussiennes when the Poincaré inequality (satisfied by Gaussian measures) provides sub optimal bound. For example : consider X_i, for i=1 to n, a sequence of i.i.d. standard Gaussian random variables. Poincaré inequality implues that the variance of the maximum of the X_i’s is bounded by 1, in fact one can show that the variance is of order 1/log(n).
This simple example is a prototype of the so-called superconcentration phenomenon introduced by S.Chatterjee in his book Superconcentration and related topics. Such phenomenon appears naturally in a lot of differents mathematical area : percolation, spin glasses, Gaussian free field, random matrix ….
The subject of my thesis is to find new examples of superconcentration, new methods to obtain the sharpest bound for the variance of the maximum of Gaussian family and concentration inequality reflecting the variance bounds.
I am also interested in functional inequalities, concentration of measure phenomenon, boolean analysis.
e-mail : kevin.tanguyATuniv-angersDOTfr
LAREMA, Faculté des Sciences
2 Boulevard Lavoisier
49045 Angers cedex 01