I have a distribution on $\mathbb{Z}^n$:
$$p_X(x;a)=\begin{cases} C \prod\limits_{i=1}^n a^{|x_i|} & \sum\limits_{i=1}^n i x_i = 0 \text{ and } \sum\limits_{i=1}^n x_i=0\\ 0 & \text{otherwise} \end{cases}$$
where $a \in (0,1)$ is a parameter and $C$ is a normalization constant. I am looking to compute the expected value of a certain bounded function $f$ depending on $(X_1,X_2)$ only; beyond that I don't think the identity of $f$ is relevant. I actually just want to compute $\lim_{n \to \infty} E[f(X)]$, so estimation is OK provided it becomes exact as $n \to \infty$.
It is clear that without the constraints, the $X_i$ would be independent and identically distributed. My intuition suggests that the sum constraints become "delocalized" as $n \to \infty$, so that the joint distribution of any fixed subset of the $X_i$ should not "see" them. Thus I expect to see $P(X_1=x_1,X_2=x_2) \approx C a^{|x_1|} a^{|x_2|}$ (for a new $C$ of course). In this case the expectation is straightforward to calculate for the particular $f$ I have in mind. However, I am not quite certain of how to justify this approximation.
I am aware of this paper: https://projecteuclid.org/download/pdf_1/euclid.cmp/1104178139 In section 2 of this paper, the authors consider a very similar problem, but instead it has $\sum\limits_{i=1}^n i x_i = V$ where $V$ is proportional to $n^2$. As best I can tell this form of the constraint is essentially "baked into" their technique, but I may be missing some way to adapt it.
Any suggestions would be very helpful.
I notice now that this little $2 \times n$ linear system can be explicitly solved; you have $x_1=\sum_{i=3}^n (i-2) x_i$ and $x_2=-\sum_{i=3}^n (i-1) x_i$, with the other $x_i$ being free variables. Thus the expected value with the constraints looks like
$$C \sum_{(x_3,\dots,x_n) \in \mathbb{Z}^{n-2}} \left ( \prod_{j=3}^n a^{|x_j|} \right ) f \left ( \sum_{i=3}^n (i-2) x_i,-\sum_{i=3}^n (i-1) x_i \right ) a^{\left | \sum_{i=3}^n (i-2) x_i \right |} a^{\left | \sum_{i=3}^n (i-1) x_i \right |}.$$
Thus my approach to the problem seems to boil down to showing that $g(y,z)=\sum_{x \in S_n(y,z)} \prod_{i=1}^{n-2} a^{|x_i|}$ where $S_n(y,z)=\left \{ x \in \mathbb{Z}^{n-2} : \sum_{i=1}^{n-2} i x_i = y,\sum_{i=1}^{n-2} x_i=z \right \}$, is in some sense "asymptotically constant"...perhaps this approach is oversimplified, then?