The Liouville theorem for discrete symmetric averaging operators

We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic, $\infty$-harmonic and the so-called game $p$-harmonic functions. Our approach provides an elementary alternative proof of the Liouville Theorem for positive $p$-harmonic functions on $\mathbb{Z}^d$.