Asymptotic formula for the traveling salesperson tour in randomly embedded random graphs

Let $G_{{\mathcal X},p}$ be the randomly embedded random graph on $n$ points in $[0,1]^d$, with edge-probability $p$, and let $T(G_{{\mathcal X},p})$ denote the length of a minimum tour. Suppose that

$$p=\frac{1}{n^\alpha}$$

for a constant $0<\alpha<1$. Asymptotic tour-length conjecture. There exists a constant $\beta_\alpha^d$ such that, asymptotically almost surely,

$$T(G_{{\mathcal X},p})\sim \beta_\alpha\frac{n^{\frac{d-1}{d}}}{p^{1/d}}.$$

The preceding results motivate seeking an asymptotic formula for the minimum tour length, while the conjecture restricts attention to power-law edge probabilities rather than arbitrary decreasing functions $p$.