Percolation beyond ℤ d : the contributions of Oded Schramm*

Abstract

Oded Schramm (1961Schramm ( -2008 influenced greatly the development of percolation theory beyond the usual 7!.,d setting, in particular the case of nonamenable lattices. Here we review some of his work in this field. Introd uctionOded Schramm was born in 1961 and died in a hiking accident in 2008, in what otherwise seemed to be the middle of an extraordinary mathematical career. Although he made seminal contributions to many areas of mathematics in general and probability in particular, I will here restrict attention to his work on percolation processes taking place on graph structures more exotic than the usual 7J.,d setting. The title I've chosen alludes to the short but highly influential paper Percolation beyond 7J.,d, many questions and a few answers from 1996 by Itai Benjamini and Oded Schramm [13].I need to point out, however, that there are at least two respects in which I will fail to deliver on what my chosen title suggests. Firstly, I will not come anywhere near an exhaustive exposition of Oded's contributions to the field. All I can offer is a personal and highly subjective selection of highlights. Secondly, Oded was a very collaborative mathematician, and I will make no attempt (if it even makes sense) at identifying his individual contributions as opposed to his coauthors'. Suffice it to say that everyone who worked with him knew him as a very generous person and as ' someone who would not put his name on a paper unless he had contributed at least his fair share. I'll just quote one recollection from Oded's long-time collaborator and friend Russ Lyons:To me, Oded's most distinctive mathematical talent was his extraordinary clarity of thought, which led to dazzling proofs and results. Technical difficulties did not obscure his vision. Indeed, they often melted away under his gaze. At one point when the four of us [Oded, Russ, Itai Benjamini and Yuval Peres] were working on uniform spanning forests, Oded came up with a brilliant new application of the Mass-Transport Principle. We weren't sure it was kosher, and I still recall Yuval asking me if I believed it, saying that it seemed to be "smoke and mirrors". However, when Oded explained it again, the smoke vanished. [59] Following the spirit of the aformentioned paper [13], I will take "percolation beyond Zd" to mean percolation process that are not naturally thought of as embedded in d How it beganIt will be assumed throughout that G = (V, E) is an infinite but locally finite connected graph. In i.i.d. site percolation on G with retention parameter P E [0, 1], each vertex v E V is declared open (retained, value 1) with probability p and closed (deleted, value 0) with the remaining probability 1-p, and this is done independently for different vertices. Alternatively, one may consider LLd. bond percolation, which is similar except that it is the edges rather than the vertices that are declared open or closed. Write lP' p,site and lP' p,bond for the resulting probability measures on {O, I} v and {O,l}E, respectively. The choice whether t...

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