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We present an improved coupling technique for analyzing the mixing time ofMarkov chains. Using our technique, we simplify and extend previous results forsampling colorings and independent sets. Our approach uses properties of thestationary distribution to avoid worst-case configurations which arise in thetraditional approach. As an application, we show that for $k/\Delta >1.764$,the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with maximumdegree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =\Omega(\logn)$ and that the graph is triangle-free. Previously, girth $\ge 5$ was needed.As a second application, we give a polynomial-time algorithm for samplingweighted independent sets from the Gibbs distribution of the hard-core latticegas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular graph $G$on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$. The bestknown algorithm for general graphs currently assumes $\lambda <2/(\Delta -2)$.

Coupling with the stationary distribution and improved sampling for colorings and independent sets