Does there exist any uniform distribution on the space of (right) stochastic matrices (a real square matrix, with each row summing to 1)? If so, is it unique?
Besides the uniform distribution, does there exist any other probability distribution (computable) on stochastic matrices? I want to sample numerically some random stochastic matrix.
There is a uniform distribution on the space of stochastic matrices, because it is a bounded polytope. Sampling from this distribution is also easy. The first observation is that we can sample each row independently (because there are no constraints among elements from different rows). Each row is a probability distribution, therefore it suffices to sample uniformly from the unit simplex $\Delta^n=\{x: x_i\ge 0,\sum_i x_i=1\}\subset\mathbb{R}^n$. To do this, first sample $y_1,\dots, y_n$ independently from an exponential distribution, and set $x_i=y_i/\sum_i y_i$. then the vector $(x_1,\dots, x_n)$ is sampled uniformly from $\Delta^n$.