By Peter Ney (auth.), Kenneth S. Alexander, Joseph C. Watkins (eds.)

This quantity has been created in honor of the 70th birthday of Ted Harris, which used to be celebrated on January eleventh, 1989. The papers rep­ resent the wide variety of subfields of likelihood conception within which Ted has made profound and primary contributions. This breadth in Ted's study complicates the duty of placing jointly in his honor a e-book with a unified subject matter. One universal thread famous was once the spatial, or geometric, point of the phenomena Ted investigated. This quantity has been equipped round that subject matter, with papers overlaying 4 significant topic components of Ted's examine: branching procedures, percola­ tion, interacting particle structures, and stochastic flows. those 4 issues don't· exhaust his study pursuits; his significant paintings on Markov chains is venerated within the common expertise "Harris chain" and "Harris recurrent" . The editors want to take this chance to thank the audio system on the symposium and the members to this quantity. Their enthusi­ astic help is a tribute to Ted Harris. we want to precise our appreciation to Annette Mosley for her efforts in typing the manuscripts and to Arthur Ogawa for typesetting the quantity. ultimately, we gratefully recognize the nationwide technology starting place and the college of South­ ern California for his or her monetary support.

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D. BARSKY, G. GRIMMETT AND C. NEWMAN 48 In the subsequent proof of Prop. 2, the connections across the size L cube C are constructed by attaching together several building blocks like those from Lemma 1 of smaller size(s) K. The attachments require some mortar at the corners of the building blocks. The mortar is provided by Lemmas 2 and 3. Lemma 2 will be used to reach the north face of C and Lemma 3 (along with Lemma 2) to reach the west and east faces. In Lemma 2, we consider the "face" Fh = {-L,··· ,L}d-l x {h} which for h = L or L - 1 is approximately halfway between the north face Fn of C and the north face F of the adjoining cube C'.

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