Lattice fermions with solvable wide range interactions

Exactly solvable (spinless) lattice fermions with wide range interactions are constructed explicitly based on {\em exactly solvable stationary and reversible Markov chains} ${\mathcal{K}}^R$ reported a few years earlier by Odake and myself. The reversibility of ${\mathcal{K}}^R$ with the stationary distribution $\pi$ leads to a positive classical Hamiltonian ${\mathcal{H}}^R$. The exact solvability of ${\mathcal{H}}^R$ warrants that of a spinless lattice fermion $c_x$, $c_x^{†}$, ${\mathcal{H}}_f^R={\sum }_{x,y\in \mathcal{X}}{c}_x^{†}{\mathcal{H}}^R(x,y)c_y$ based on the principle advocated recently by myself. The reversible Markov chains ${\mathcal{K}}^R$ are constructed by convolutions of the orthogonality measures of the discrete orthogonal polynomials of Askey scheme. Several explicit examples of the fermion systems with wide range interactions are presented.