See e.g. Section 2.1 "Talagrand's transport inequalities and Gaussian dimension-free concentration" of [Gozlan's survey][1]. 

Theorem 2.3 there is Talagrand's result that the standard Gaussian measure on $\mathbb R^d$ satisfies Talagrand’s transport inequality $\mathbf T_2(2)$. On the other hand, Theorem 2.4 in that survey, with a very short proof, states that, for any real $C>0$, the $\mathbf T_2(C)$ property of a probability measure implies a concentration property for that measure. 

A corollary of this result is


  [1]: https://www.esaim-proc.org/articles/proc/pdf/2015/04/proc145101.pdf