Borel Local Lemma: arbitrary random variables and limited exponential growth

TL;DR

This paper extends the Borel Local Lemma to handle continuous random variables and dependency graphs with limited exponential growth, broadening its applicability in probabilistic combinatorics.

Contribution

It provides an alternative proof of the Borel Local Lemma that works for continuous variables and graphs with limited exponential growth, unlike previous finite-valued restrictions.

Findings

Applicable to continuous random variables

Works with dependency graphs of limited exponential growth

Broadens the scope of the Borel Local Lemma

Abstract

The Lov\'asz Local Lemma (the LLL for short) is a powerful tool in probabilistic combinatorics that is used to verify the existence of combinatorial objects with desirable properties. Recent years saw the development of various "constructive" versions of the LLL. A major success of this research direction is the Borel version of the LLL due to Cs\'oka, Grabowski, M\'ath\'e, Pikhurko, and Tyros, which holds under a subexponential growth assumption. A drawback of their approach is that it only applies when the underlying random variables take values in a finite set. We present an alternative proof of a Borel version of the LLL that holds even if the underlying random variables are continuous and applies to dependency graphs of limited exponential growth.