Spin Waves in the Dilute Heisenberg Ferromagnet

The energy of long wave length spin waves in dilute Heisenberg ferromagnets is calculated. It is shown that spin waves can exist only within an infinite cluster of magnetic spins and that such spin waves are mechanically stable. Heuristic arguments based on dimensionality of the infinite cluster imply that there is no thermal instability of the type discussed by Mermin and Wagner. Thus, the critical concentration for ferromagnetism is essentially equal to percolation concentration.