Abstract
We consider the problem of embedding one i.i.d. collection of Bernoulli random variables indexed by \({\mathbb {Z}}^d\) into an independent copy in an injective M-Lipschitz manner. For the case \(d=1\), it was shown in Basu and Sly (Probab Theory Relat Fields 159:721–775, 2014) to be possible almost surely for sufficiently large M. In this paper we provide a multi-scale argument extending this result to higher dimensions.
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Notes
Throughout, by saying a function f is bi-Lipschitz with Lipschitz constant M, we shall mean that for all x, y.
$$\begin{aligned} \frac{1}{M}|x-y|\le |f(x)-f(y)|\le M|x-y|. \end{aligned}$$Although we define \(\alpha \)- canonical maps from a union of domains to another union of domains of the same shape while matching up certain of their respective sub-blocks, for the definition we only need to know the domains, and not the configurations \({\mathbb {X}}\) and \({\mathbb {Y}}\). Often we shall refer to \(\alpha \)-canonical maps between potential domains of multiblocks (i.e., union of potential domains that are compatible) \(\hat{U}_{X}\), \(\hat{U}_{Y}\) with respect to lattice animals \({\mathcal {T}}=\{{T}_1,{T}_2,\ldots ,{T}_k\}\) and \({\mathcal {T}}'=\{{T'}_1,\ldots ,{T'}_{k'}\}\), where each of these lattice animals will be assumed to be equipped with one of the corresponding potential domains.
Observe that we can get a crude estimate of such lattice animals as follows. Starting with u, enumerate the vertices of the lattice animal in the following depth first way. For any current vertex, explore any unexplored neighbour of the current vertex first. If none exists, move to the previous vertex and so on. This procedure terminated in v steps and at each step there are at most 8 choices for the next unexplored vertex, giving a crude upper bound of \(8^v\) of lattice animals of size v containing u.
To see this bound observe the following. We are choosing \(k'\) many sites from at most \(2vL_{j+1}\) many ones, so a crude upper bound is \((2v L_{j+1})^{(k')}/(k')!\), the given bound now follows from upper bounding \(v^{k'}/(k')!\) by \(16^{v}\).
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Acknowledgements
This work was completed when R. B. was a graduate student at the Department of Statistics at UC Berkeley and the result in this paper appeared in Chapter 4 of his Ph.D. dissertation at UC Berkeley: Lipschitz Embeddings of Random Objects and Related Topics, 2015. R. B. gratefully acknowledges the support of UC Berkeley graduate fellowship. V. S. was supported by CNPq grant Bolsa de Produtividade. A. S. was supported by NSF grant DMS-1352013, and a Simons Investigator grant. We also thank an anonymous referee for many useful comments and suggestions that helped improve both the technical and editorial quality of the paper.
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Basu, R., Sidoravicius, V. & Sly, A. Lipschitz embeddings of random fields. Probab. Theory Relat. Fields 172, 1121–1179 (2018). https://doi.org/10.1007/s00440-017-0826-5
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DOI: https://doi.org/10.1007/s00440-017-0826-5