Uniqueness of Gibbs measure for rotator model in one dimension I am trying to solve a problem in a course of Y. Velenik (models with continuous symmetry, exercice 8.18: http://www.unige.ch/math/folks/velenik/smbook/index.html): 

Show that in dimension $d=1$ there is a unique gibbs measure for the $O(N)$ model ($ N \geq 2$), and show that
  $$ |\langle S_i \cdot S_j \rangle | \leq e^{-c|i-j|}, $$
  for a constant $c >0$. 

For the uniqueness, one can apply the Dobrushin criterion, but is there a simpler way to do it? For the exponential decay, I have no idea.
 A: Concerning uniqueness, you can indeed use some uniqueness criterion, such as (the continuous spin version of) Theorem 6.58 in the book. You can also notice that nearest-neighbor one-dimensional models are really just (here, continuous-state) discrete-time Markov chains in disguise and use the standard argument establishing uniqueness of the invariant measure (the infinite-volume Gibbs measure if then simply the invariant path measure); note that exponential relaxation to the invariant measure then implies exponential decay of correlations.
For the second question (exponential decay of correlations), one simple way of doing that is to try to compute instead $\langle e^{\mathrm{i}(\theta_i-\theta_j)}\rangle_N^\varnothing$ (that is, in the box $\{-N,\ldots,N\}$ with free boundary condition) and observe that, changing variables to $\Psi_k = \theta_{k+1}-\theta_k$ factorizes everything, which allows you to compute explicitly this expectation (well, explicitly in terms of some Bessel function).
Alternatively, it should be possible to implement a variant of the high-temperature/random-cluster representations, which are described in the chapter on the Ising model. The point is that, in nearest-neighbor one-dimensional model the length of the "clusters" of interacting spins that these representations yield has an exponentially decaying tail and that spins in distinct "clusters" are independent. This should yield both uniqueness and exponential decay of correlations. I have not yet worked out the details for this approach (we'll work on solving the exercises we propose when the rest of the book is ready ;) . We'll then provide additional hints as to the best way to approach these problems, as well as solutions to the most difficult ones).
[EDIT: Yet another approach]
Thinking about it, the simplest way of proving exponential decay of correlation might be the following. First, since
$$
| \langle S_i \cdot S_j \rangle | \leq 
| \langle S_i(1) S_j(1) \rangle | +
| \langle S_i(2) S_j(2) \rangle | ,
$$
it is enough to prove it for, say, the first term in the rhs. This can be done as follows. Let us work in a finite box $\Lambda_N=\{-N,\ldots,N\}$ with arbitrary boundary condition. Now, condition on the values of the second component of all the spins in $\Lambda_N$. Note that, under the conditional measure, the absolute value of the first components of the spins become deterministic (the Euclidean length of the spins being $1$). Therefore, the only remaining degrees of freedom are given by the signs $\sigma_k = S_k(1)/|S_k(1)|$. Observe that $(\sigma_i)_{i\in\Lambda_N}$ is distributed exactly as a one-dimensional Ising model with inhomogeneous (but deterministic) coupling constants taking values between $0$ and $\beta$ (the coupling constant between $\sigma_k$ and $\sigma_{k+1}$ is given by $\beta |S_k(1)||S_{k+1}(1)|$). The desired conclusion thus follows from the exponential decay of the 2-point function in the one-dimensional Ising model, which can easily be proved in many ways (transfer matrix, high-temperature representation, random-cluster representation, etc.). (Of course, this does not give you the correct rate of decay, contrary to the approach sketched above. However, it should not be difficult to use the current approach to also prove uniqueness of the infinite-volume Gibbs measure, since the argument works for any boundary condition and the proven exponential decay should yield an efficient coupling between measures with different boundary conditions).
