Random sub-neighborhood discrepancy on the Boolean cube

Definitions and mathematical statement

For $n\ge1$, let $Q_n$ be the graph with vertex set $V_n=\{0,1\}^n$, where two vertices are adjacent when they differ in exactly one coordinate. For each $v\in V_n$, choose a subset $S_v$ of its $n$ neighbours uniformly from all $2^n$ subsets. Make these choices independently for different vertices. Equivalently, each ordered neighbour pair $(v,u)$ independently has probability $1/2$ of placing $u$ in $S_v$. In particular, membership of $u$ in $S_v$ need not agree with membership of $v$ in $S_u$.

For the resulting random set system $\mathcal S_n=(S_v:v\in V_n)$, its discrepancy is the random variable \[ D_n=\min_{\varepsilon:V_n\to\{-1,1\}} \max_{v\in V_n}\left|\sum_{u\in S_v}\varepsilon(u)\right|. \] The signing is chosen after the entire set system has been sampled and is shared by all its sets. Empty sets contribute zero to the maximum. The problem is to determine the typical asymptotic size of $D_n$ as $n\to\infty$, ideally by matching upper and lower bounds holding with probability tending to one. For example, an order-of-magnitude answer would specify a deterministic function $f(n)$ and constants $c,C>0$ for which $\Pr(c f(n)\le D_n\le C f(n))\to1$. The independence and uniform-subset conventions make the workshop's phrase "random subset" precise.

Short English statement

How well can one balance independently chosen random subsets of the neighbourhoods of a high-dimensional cube?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20000913 (AIM-COMBINATORICS-0038), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Hereditary discrepancy and factorization norms, item 1.25. Original workshop question underlying AIM-COMBINATORICS-0038. http://aimpl.org/hereddiscrep/1/.