This question is inspired by Joseph O'Rourke's beautiful answer to [my previous question][1].

Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum to the zero vector.  I want to approximate the uniform distribution on this set by running a diffusion process.  Specifically, consider the following random transition: Given a member of $\mathbb{S}^{d\times n}$, select some columns at random (say, each column is independently active with probability $1/2$), and then apply a random rotation to these columns that fixes their sum.  This transition can be expressed in terms of a conditional density on $\mathbb{S}^{d\times n}$, namely $f(x^{(i+1)}|x^{(i)})$.  Next, define the operator $A:L^1(\mathbb{S}^{d\times n})\rightarrow L^1(\mathbb{S}^{d\times n})$ that uses this transition rule to update the distribution on $\mathbb{S}^{d\times n}$:
$$
g^{(i+1)}(x)
=(Ag^{(i)})(x)
:=\int_{\mathbb{S}^{d\times n}}f(x|y)g^{(i)}(y)dy.
$$
My questions concern this operator: 

(1) As expected, $A$ sends the uniform distribution to itself.  But is this stationary distribution also the limiting distribution?

(2) Assuming this is the limiting distribution, how fast is the convergence?  I assume there is an analog to [expander walk sampling][2] (i.e., the rate of convergence should be expressed in terms of the spectrum of $A$), but I would like a reference for the  continuous-state case.  Yoav Kallus commented on Joseph O'Rourke's answer to [my previous question][3] that polymer people might use the phrase "Rouse relaxation time" to describe this, but these keywords haven't helped me find the theorem I want.

(3) Assuming the rate of convergence is completely expressible in terms of the spectrum of $A$, how do I actually calculate the spectrum?  Do the symmetries in the transition rule naturally lead to a Fourier-type eigenbasis?


  [1]: http://mathoverflow.net/questions/131156/how-can-i-randomly-draw-an-ensemble-of-unit-vectors-that-sum-to-zero
  [2]: http://en.wikipedia.org/wiki/Expander_walk_sampling
  [3]: http://mathoverflow.net/questions/131156/how-can-i-randomly-draw-an-ensemble-of-unit-vectors-that-sum-to-zero