Quenched local limit theorem for random conductance models with long-range jumps

We establish the quenched local limit theorem for reversible random walk on ᵈ (with d≥ 2) among stationary ergodic random conductances that permit jumps of arbitrary length. The proof is based on the weak parabolic Harnack inequalities and on-diagonal heat-kernel estimates for long-range random walks on general ergodic environments. In particular, this partly solves [Open Problem 2.7]{BCKW}, where the quenched invariance principle was obtained. As a byproduct, we prove the maximal inequality with an extra tail term for long-range reversible random walks, which in turn yields the everywhere sublinear property for the associated corrector.