The spectral cap for the Kawasaki dynamics at low temperature

In this paper we analyze the convergence to equilibrium of Kawasaki dynamic s for the Ising model in the phase coexistence region. First we show, in st rict analogy. with the nonconservative case, that in any lattice dimension, for any boundary condition and any positive temperature and particle densi ty, the spectral gap in a box of side L does not shrink faster than a negat ive exponential of the surface Ld-1. Then we prove that, in two dimensions and for free boundary condition, the spectral gap in a box of side L is sma ller than a negative exponential of L provided that the temperature is belo w the critical one and the particle density rho satisfies p is an element o f (p(-)*, p(+)*), where p(+/-)* represents the particle density of the plus and minus phase, respectively.