I can provide you concentration inequality around the mean, in particular the decay is exponential et reflects the correct size of the variance. It is important to notice that Classical concentration inequality for Lipschitz functions can not be applied.
As a reference see :
http://perso.math.univ-toulouse.fr/ktanguy/files/2012/04/Article-3-brouillon.pdfhttps://perso.math.univ-toulouse.fr/ktanguy/files/2012/04/Article-3-brouillon.pdf
Kevin Tanguy
