Additive and Cancellative Interacting Particle Systems by David Griffeath

By David Griffeath

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

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14) ; we get Z Z+ 0 Z+ s ~t ' ~ n /~)->--~-- P(O¢ Z+ N t (0) > 0 S i n c e p a t h s c a n n o t jump o v e r o n e a n o t h e r , 0 i m p l y N t (0) > 0 . 1) The present result is of interest constant. this section one-dimensional lastthree theorems hold forthe for any + k > k. , We remark that for P(O ~ I t ) _ s 2 • . 0 + Nt (Z) > 0 together Z P ( 0 ~ [+~) > i _ lira i n f as desired. Yt yields because [] with a brief survey of further results contact • +,A} i(%k,t) systems. First, Intact, for the basic we note that none of the if A ~ S O then ~UAPt--6)~ k : in the nonergodie case the set of infected sites wanders off to the right if it does not die out.

O n e relevant quantity for systems which cluster is the asymptotic m e a n cluster size. Let C(A) , A ~ S , C(A) = lie n--~ A or entirely in are the connected components of A or be given by (Zn) d l{clusters of A in bn(0)} I provided the limit exists (undefined otherwise) • For the one-dimensional basic ~8 Z8 ' the asymptotic growth of C(~ t ) can be derived voter model starting from explicitly. First w e need a general result which states that mixing is preserved by local additive systems at any time to the limit as t~ t < ~o .

K In addition, let ~Z = z + ~Z (~B) in terms of ~I and Introduce m a k e a copy of ~L = m i n { t : d( ([%B U (z+C)) let ~l be the ~tz + C in terms of '~t ' by letting the flow A which starts from B use Thus, /~Z @i while the flow starting from z + C uses @Z until T L A and @i thereafter. ~[~(~B)~AZ(~t Z TL > t . ii) P Since is mixing, and the second term does not have influence from ~ [] A Theorem. Let lim t~ ~ Proof. s. 10)) D o ~ (0,1) n--~ vo C(l t 1 . if A(x) / A(x+l) . Birkhoff's theorem yields 1 h a s an e d g e at ~- ) Since V0 in [-n, n] } pt has a positive , Zn = C([t I {edges of It~0 ) = [P([t Z, I {clusters of A in [-n,n]} I by at most lira It f o l l o w s t h a t xe gO [{edges of It in [-n,n] }[ VO Zn = P(~t density of edges for 0 e (0, i) , in P-probability.

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