We consider the problem of embedding one i.i.d. collection of Bernoulli random variables indexed by @math into an independent copy in an injective @math -Lipschitz manner. For the case @math , it was shown by Basu and Sly (PTRF, 2014) to be possible almost surely for sufficiently large @math . In this paper we provide a multi-scale argument extending this result to higher dimensions.
In a related direction, a series of works by Grimmett, Holroyd and their collaborators @cite_25 @cite_12 @cite_8 @cite_23 @cite_13 investigated a number of problems including when one can embed @math into site percolation in @math and showed that this was possible almost surely for @math when @math and the the site percolation parameter was sufficiently large but almost surely impossible for any @math when @math . Recently Holroyd and Martin @cite_5 showed that a comb can be embedded in @math . Another series of work in this area involves embedding words into higher dimensional percolation clusters @cite_9 @cite_3 @cite_0 @cite_6 .
@cite_8 @cite_9 @cite_3 @cite_6 @cite_0 @cite_23 @cite_5 @cite_13 @cite_25 @cite_12
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We consider critical site percolation on the triangular lattice, that is, we choose @math or 1 with probability 1 2 each, independently for all vertices @math of the triangular lattice. We say that a word @math is seen in the percolation configuration if there exists a selfavoiding path @math on the triangular lattice with @math . We prove that with probability 1 "almost all" words, as well as all periodic words, except the two words @math and @math , are seen. "Almost all" words here means almost all with respect to the measure @math under which the @math are i.i.d. with @math (for an arbitrary @math ). Does there exist a Lipschitz injection of ℤd into the open set of a site percolation process on ℤD, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D and also when d ≥ 2 if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d < D and M = 2. As a result, the above question is answered for all d, D and M. Our proof in the case d = D uses Tucker’s lemma from topological combinatorics, together with the aforementioned result for d < D. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions. There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^ d-1 joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on Z, and first-passage percolation on general graphs. For a natural number k, define an oriented site percolation on ℤ2 as follows. Let x i , y j be independent random variables with values uniformly distributed in 1, …, k . Declare a site (i, j) ∈ℤ2closed if x i = y j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i0, j0)(i1, j1) · · · such that 0 = i0≤i1≤· · ·, 0 = j0≤j1≤· · ·, and each site (i n , j n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x i ) and (y j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods.