Regarding Ricci curvature of Markov chains In Ricci curvature of Markov chains on metric spaces Yann Ollivier, defines a coarse Ricci curvature for a Markov chain with transition kernels $\{m_x\}$ defined on a metric space $(X,d)$ as follows: The curvature along $(xy)$ is 
$$
\kappa(x,y) := 1 - \frac{W_1(m_x,m_y)}{d(x,y)},
$$
where $W_1$ is the $\ell_1$ transportation distance between $m_x$ and $m_y$ with transportation cost $d$. The space is positively curved if $\kappa(x,y) \ge \kappa > 0$. Then, it follows that the Markov chain shows contraction (in $W_1$ metric) to a unique invariant distribution, at a geometric rate (at least like $(1-\kappa)^n$). This is Corollary 21 in the paper. 
My question is what happens if we only have $\kappa(x,y) \ge \kappa > 0$ for $d(x,y) \ge M$ for $M$ large enoguh. That is, the space is positively curved if $x$ and $y$ are far apart (and maybe negatively curved if they are close.) Can we still show a contraction from any (or a subset) of starting distributions? If not what else is needed to get such a result.
 A: Just an example to show that you will certainly need something to get the kind of contraction you're after.
Let the space be $\mathbb N\times \{0,1\}$, where the distance from $(n,i)$ to $(m,j)$ is just $|n-m|+|i-j|$. Imagine a kernel which sends everything towards the origin, but leaving you in the same copy of the Markov chain e.g. $(n,i)$ transitions to 
$(\lfloor n/2\rfloor + Q,i)$, where $Q$ is a Poisson random variable. This satisfies the large scale geometric contraction you're looking for, but clearly two distributions on the two parts of the space will never approach each other.
A: I believe that you can establish a geometric bound if your contraction constraint is the other way around, i.e. if $\kappa(x,y) \geq \kappa > 0$ when $d(x,y) \leq M$ for some suitable $M$.  This kind of approach seems to me to generalise the small set coupling constructions for general state space Markov chains to more Wasserstein distances than the classical total variation metric.  It's explained in this paper (and other places most likely):
http://link.springer.com/article/10.1007/s11222-014-9511-z
