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 early 1990s Winkler introduced a fascinating class of dependent percolation problems, the so-called co-ordinate percolation problems, where the vertices are open or closed depending on variables on co-ordinate axes. Long-range dependence makes these problems not amenable to the tools of Bernoulli percolation. It turns out that several natural questions about embedding one random sequence into another following certain rules can be reformulated as problems in this class (see e.g. @cite_7 @cite_11 @cite_12 @cite_22 ). In particular, Grimmett, Liggett and Richthammer @cite_2 asked whether there exists a Lipschitz embedding of one Bernoulli sequence (indexed by @math ) into an independent copy. This question was recently answered in @cite_16 (see also @cite_17 ). The problem we investigate in this paper is a natural generalisation of the above question to higher dimensions.
@cite_22 @cite_7 @cite_17 @cite_2 @cite_16 @cite_12 @cite_11
1971206237 2000931246 2034477758 2037520821
We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing (M )-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough (M ). A closely related problem is to show that two independent Poisson processes on ( R ) are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 In this paper, we provide general conditions on a one parameter family of random infinite subsets of Zd to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab. 17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation... 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.