  We provide the first rate of convergence analysis for RBM as the dimension
grows under natural uniformity conditions. In particular, if the underlying
routing matrix is uniformly contractive, uniform stability of the drift vector
holds, and the variances of the underlying Brownian Motion (BM) are bounded,
then we show that the RBM converges exponentially fast to stationarity with a
relaxation time of order $O(d^4\log(d)^2)$ as $d\to\infty$.
