  Measuring distances on a lattice in noncommutative geometry involves square,
symmetric and real ``three-diagonal'' matrices, with the sum of their elements
obeying a supremum condition, together with a constraint forcing the absolute
value of the maximal eigenvalue to be equal to 1. In even dimensions, these
matrices are unipotent of order two, while in odd dimensions only their squares
are Markovian. We suggest that these bi-graded Markovian matrices (i.e.
consisting in the square roots of Markovian matrices) can be thought of as
non-local Dirac operators. The eigenvectors of these matrices are spinors.
Treating these matrices as determining the stochastic time evolution of states
might explain why one observes only left handed neutrinos. Some other physical
interpretations are suggested. We end by presenting a mathematical conjecture
applying to q-graded Markovian matrices.
