a unique unbounded component. We now move on to the proofs.

Uniqueness monotonicity
=======================

In this section we first present a short proof for Theorem \[uniqmon1\] in the symmetric case, and then a proof which only needs the assumption that the space is homogeneous.

First we present an essential ingredient to the first proof, the mass transport principle which is due to Benjamini and Schramm [@itai1]. We denote the group of isometries on the symmetric space $M$ by Isom($M$).

\[diagonalinv\] A measure $\nu$ on $M\times M$ is said to be *diagonally invariant* if for all measurable $A,\,B\subset M$ and $g\in$*Isom(M)* $$\nu(g A\times g B)=\nu(A\times
B).$$

([Mass Transport Principle on M]{}) \[masstransport\] If $\nu$ is a positive diagonally invariant measure on $ M\times M$ such that $\nu(A\times M)<\infty$ for some open $A\subset M$, then $$\nu(B\times M)=\nu(M \times B)$$ for all measurable $B\subset
M$.

Actually the mass transport principle is proved in [@itai1] for the case when $M=\hyp$, but as is remarked there, it holds for any symmetric space.

[*Proof of Theorem \[uniqmon1\] in the symmetric case:*]{} Suppose $\lambda_c<\lambda_1<\lambda_2$. We couple $C^{\lambda_1}$ and $C^{\lambda_2}$ using the monotone coupling. We are done if we can show that any unbounded component of $C^{\lambda_2}$ contains 