Decoupling of clusters in independent sets in a percolated hypercube
Mriganka Basu Roy Chowdhury, Shirshendu Ganguly, Vilas Winstein
TL;DR
This work extends the understanding of independent sets in random subgraphs of the hypercube by developing a polymer-model framework that captures defect clustering around ground states on the Even/Odd halves. The authors prove that for p>0.465 the number of independent sets i(Q_{d,p}) concentrates around a two-term exponential form, i(Q_{d,p})/2^{2^{d-1}} ∼ e^{Ψ^Even}+e^{Ψ^Odd}, with Ψ^Even,Ψ^Odd obeying a joint central limit theorem to independent Gaussian limits after proper centering and scaling. They introduce a two-stage sampling scheme, ApproxSampler, whose failure probability vanishes and which can be coupled to a uniform independent set, enabling efficient approximate sampling in this disordered regime. The analysis reveals no barrier at p=1/2 for the qualitative behavior of independent sets, and identifies a transition window c.586<p<1 where dimers begin to influence clustering, while providing a robust framework to tackle even lower p where larger polymers emerge. Altogether, the paper delivers precise in-probability approximations, distributional limits, and a practical sampling algorithm, advancing the understanding of hard-core constraints on disordered high-dimensional graphs with potential applicability to related percolation and lattice-gas models.
Abstract
Independent sets in graphs are sets of vertices containing no neighbors, and they represent a canonical spin system with hardcore constraints. Of particular interest is the setting of the boolean hypercube, where counting independent sets was the original motivator for Sapozhenko's famous graph container method. A modern perspective on such problems is to consider the effect of disorder, and the study of independent sets in random subgraphs of the hypercube obtained via bond percolation with parameter $p$ was initiated by Kronenberg and Spinka. They employed tools from statistical mechanics to obtain detailed information about the moments of the number of independent sets (now a random variable), and posed many interesting questions. Previous work by the authors addressed many of these questions in the regime $p \geq \frac{2}{3}$, where the behavior is relatively simple and can be modeled well by a related family of independent particles. As $p$ decreases, though, typical independent sets become larger and feature more intricate clustering behavior. In the present article we overcome many of the challenges presented by this phenomenon and analyze the model for all $p> 0.465$. We obtain a sharp in-probability approximation for the number of independent sets in the percolated hypercube in terms of explicit random variables, as well as provide a sampling algorithm. Note that this shows, curiously, that $p = \frac{1}{2}$ is not a natural barrier for this problem unlike in many other problems where it appears as a point of a phase transition. A key contribution of this work is the introduction of a new probabilistic framework to handle the clustering behavior for these low values of $p$. Although our analysis is restricted to $p > 0.465$, our arguments are expected to be helpful for studying this model at even lower values of $p$, and possibly for other related problems.