Lipschitz Embeddings of Random Fields

TL;DR

This paper extends the known results on Lipschitz embeddings of i.i.d. Bernoulli random fields from one dimension to higher dimensions using a multi-scale approach.

Contribution

It introduces a multi-scale method to achieve Lipschitz embeddings of i.i.d. Bernoulli fields in higher dimensions, generalizing previous one-dimensional results.

Findings

Embedding is possible in higher dimensions with large enough Lipschitz constant

Multi-scale argument successfully extends 1D results

Provides a framework for Lipschitz embeddings of random fields

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 by Basu and Sly (PTRF, 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.