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The aim of this paper is to investigate the Gibbs sampling that's used for computing the mean of observables with respect to some function $f$ depending on a very small number of variables. For this type of observable, by using the $d_{l_{2}}$-metric one obtains the sharp concentration estimate for the empirical mean, which in particular yields the correct speed in the concentration for $f$ depending on a single observable.

Concentration inequalities for Gibbs sampling under $d_{l_{2}}$-metric