An irreducible stochastic matrix may be constructed by partitioning a line of unit length into a finite number of intervals, shifting the line to the right $(\mod 1)$ by a small amount, and defining transition probabilities in terms of the overlaps among the intervals before and after the shift. It is proved that every $2 \times 2$ irreducible stochastic matrix arises from this construction. Does every $n \times n$ irreducible stochastic matrix arise this way?
"A Geometric Representation of a Stochastic Matrix: Theorem and Conjecture." Ann. Probab. 9 (5) 899 - 901, October, 1981. https://doi.org/10.1214/aop/1176994319