The balanced-event probability conjecture for the hard-core model on the hypercube

Let $\mathbb{Z}_{2}^{d}$ be the $d$-dimensional hypercube, let $\Omega_{\mathrm{hc}}^{\mathbb{Z}_{2}^{d}}$ be its hard-core configuration space, and let $\mu^{\mathbb{Z}_{2}^{d}}$ be the hard-core measure with fugacity $\lambda$. Let $E_{\mathrm{Bal}}$ denote the balanced-event set introduced in the paper. Here $C,d_{0}>0$ are constants, and $\Theta$ denotes bounds tight up to universal multiplicative constants.

Balanced-event probability conjecture. There exist $C,d_{0}>0$ such that for all dimensions $d\ge d_{0}$,

$$\mu^{\mathbb{Z}_{2}^{d}}\left(E_{\mathrm{Bal}}\right)=\begin{cases} e^{-\Theta(\lambda^{2}2^{d})} & \frac{C}{d}\le\lambda\le\frac{1}{\sqrt{d}},\\ e^{-\Theta(\frac{\log(1+\lambda)}{\sqrt{d}}2^{d})} & \lambda\ge\frac{1}{\sqrt{d}}. \end{cases}$$

The conjecture predicts the leading exponential scale of the probability of balanced configurations in two fugacity regimes. The preceding discussion provides matching-motivated lower bounds in part of the range, while the corresponding full bounds, especially from $\lambda\ge C/d$, remain unproved.