By David Griffeath

Griffeath D. Additive and Cancellative Interacting Particle platforms (LNM0724, Springer, 1979)(ISBN 354009508X)(1s)_Mln_

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O n e can s h o w that the growth is of order at most open for d -> Z . 4) Notes. Z) but the exact order remains This concludes our discussion of contact processes. 5). td , is due to Harris (1974) It improves a result of Dobrushln is from Holley and Liggett (1978). in Griffeath (1975). (1971). 3) and Holley and The upper bound in is based on a computation The rest of the results mentioned in this section m a y be found in Harris (1974, 1976, 1978), except for s o m e refinements in Griffeath (1978).

5 J 2 X~ 3 4 5 I 6 I ! 7 X figure vi. 36 To s u m m a r i z e , we h a v e seen that pk= lim P(O ~ ~Z is increasing in k , 0 x ' t ) = p(~ ' t ~ t--oo equals 0 for k -< 1 , Vt) and is strictly positive for k >-Z . In fact, PX is s a n d w i c h e d b e t w e e n the two curves s h o w n in figure v . While we have drawn PX to be continuous at X = k~ , there is no k n o w n rigorous basis for this. The analogous graphs for the one-sided s y s t e m s are s h o w n in figure vi. 4). We illustrate this with our next result.

For the remainder of the discussion notation. ~ = ~k so w e discuss the k will be fixed, and often suppressed from the By self-duality of the basic contact process (cf. 9) t -< 0 = Unfortunately, however, VA ~ SO . 9) a renewal m e a s u r e w h i c h works provided k for all A . There i__ss, The is large enough. oo renewal m e a s u r e ~f ~ ~ is determined by a probability density f = (fk)k= 1 co such that m = ~ kf k < co . k=l abilities given by ~f is translation invariant, ~f({A : A(x) = A(X+Yl) .