Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model

In the mean field (or random link) model there are $n$ points and inter-point distances are independent random variables. For $0 < \ell < \infty$ and in the $n \to \infty$ limit, let $\delta(\ell) = 1/n \times$ (maximum number of steps in a path whose average step-length is $\leq \ell$). The function $\delta(\ell)$ is analogous to the percolation function in percolation theory: there is a critical value $\ell_* = e^{-1}$ at which $\delta(\cdot)$ becomes non-zero, and (presumably) a scaling exponent $\beta$ in the sense $\delta(\ell) \asymp (\ell - \ell_*)^\beta$. Recently developed probabilistic methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi) provides a simple albeit non-rigorous way of writing down such functions in terms of solutions of fixed-point equations for probability distributions. Solving numerically gives convincing evidence that $\beta = 3$. A parallel study with trees instead of paths gives scaling exponent $\beta = 2$. The new exponents coincide with those found in a different context (comparing optimal and near-optimal solutions of mean-field TSP and MST) and reinforce the suggestion that these scaling exponents determine universality classes for optimization problems on random points.