Critical inverse-temperature conjecture for macroscopic cycles and loops

Consider the cycle or loop model on the cubic lattice $\Lambda_n=\{1,\dots,n\}^d$ in $\mathbb{Z}^d$. Let $\lambda^{(i)}$ be the length of the $i$th-longest cycle, and let $\eta_{\mathrm{macro}}(\beta)$ be the fraction of sites lying in macroscopic cycles. Critical-temperature conjecture. Suppose $d\geq 3$. There exists $\beta_c>0$ such that, for $\beta>\beta_c$, the fractions of sites in infinite and macroscopic cycles (or loops) approach the same typical value

$$\eta:=\eta_{\infty}(\beta)=\eta_{\mathrm{macro}}(\beta)>0,$$

and the vector of ordered normalized cycle lengths

$$\left(\frac{\lambda^{(1)}}{\eta n^d},\frac{\lambda^{(2)}}{\eta n^d},\ldots\right)$$

converges weakly to a random variable $\xi$ in $\Delta_1$ as $n\to\infty$. This conjecture predicts the emergence of macroscopic cycles above a positive critical inverse temperature; the subsequent Poisson–Dirichlet claim specifies the expected law of $\xi$.