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By M. Aizenman (Chief Editor)

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1) is verified for t = 0. 36 E. Caglioti, C. Marchioro, M. 20) for a suitable constant A to be fixed later and satisfying A > 2(Q(X)1/2 + 1). 16)). 24) and hence: R 4/6 |t ∗ −t1 | α therefore, for a suitable choice of α ∈ [1/2, 1], Furthermore 1 1 2 ∗ vi (t ) − vi2 (t1 ) = 2 2 = t∗ t1 vi · Fi,j j t∗ t1 ds H ds (vi − vj ) · Fi,j + j ∗ =− t1 +h h=1 t1 +(h−1) ∗ φ(xi (t ) − xj (t )) + j ds vj · Fi,j j φ(xi (t1 ) − xj (t1 )) j H + is integer. t1 +h h=1 t1 +(h−1) ds vj · Fi,j . 19): φ(xi (t ∗ ) − xj (t ∗ )) ≤ φ ∞ N (X n (t ∗ ); xi (t ∗ ), r)) ≤ C16 R(t)3/2 .

3) i∈Ik sup |xin (s) − xi (0)|. 5) where C2 = C1 T . Therefore the maximal number of particles that can be in the interaction sphere of a given particle xi (t), cannot be larger than the number of particles that, at time zero, were in a sphere of radius r + C2 ϕ(n). 1 and we have used that |xin (t)| ≤ C2 ϕ(n) + n. Writing Eq. 8) j where ∗j means the sum restricted to all the particles which can fall in the interaction sphere of xin (s) or xin+1 (s), for s ≤ t and C4 = ∇F ∞ . Notice that, since k + r + dn (s) + dn+1 (s) < k + r + 2C2 ϕ(n + 1) << n (provided that n is sufficiently large), the particle i cannot interact with the particles initially in B(0, n + 1)/B(0, n).

Introduction Static properties of random spin systems – spin glasses – have been the object of many recent papers by both physicists and mathematicians. Although many attempts from physicists have also been made to describe the dynamics of such models, rigourous results are few. ) Still the claims of physicists suggest a great variety of interesting features: very large relaxation times, aging properties . . (see [2]). The aim of the present paper is to discuss the convergence to equilibrium for quite general dynamical spin models.

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