Additive and cancellative interacting particle systems - download pdf or read online

By David Griffeath

ISBN-10: 354009508X

ISBN-13: 9783540095088

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

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Extra info for Additive and cancellative interacting particle systems

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Etc. )+ x + ~ a(---) 1 ~(" ") + T 7 - ~ x__ i +zk ~( . . ) ' 2X ~(" - ") + % - - 7 - F (~ is strongly subadditive: ~( .... ) " 37 o(A I U A Z) < O(Al) + (~(AZ) - o(A I A AZ) In f a c t ; A I , A Z • S0 vI v1 (~(A I U A Z) - o ( A I) = ~o ( A I ) - g (A I U -A-z) by d u a l i t y , = VI({A : A n A I= ]~, A n ( A ~ n A 2 ) / @ } ) -- vI({A : A n (AI~]Az) = 9, A n ( A I A A Z) / @ } ) _ v = _ v1 = ~ (A I D A Z) I(A2) o(A2) o(A I [3 AZ) . I n particular, (~(. . ) + (~(. ) and Substituting into (C) and (D) w e get the n e w inequalities (C') g ( .

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.

Hence ' PX is increasing in k . If 31 k < k¢ , then invariant, Pk-- V k , l so are the ( {0 is infected}) = 0 . 6Z P ~ , t ~ T, is infected}) > 0 , this case. i) . v I / 69 . W i t h p k , k ¢ , PX and k~ Proposition. 2k - 2 pk--< 2 k _ 1 = whence k¢ -> 1 , 0 Thus k+ > 2 Proof. 1) , X-Z, =0 whence @ k >I, +_< k - 3 Pk k 1 X-

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Additive and cancellative interacting particle systems by David Griffeath

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