Percolation–like scaling exponents for minimal paths and trees in the stochastic mean field model

Abstract

In the mean field (or random link) model there arenpoints and inter-point distances are independent random variables. For 0 <ℓ< ∞ and in then→ ∞ limit, letδ(ℓ) = 1/ntimes the maximum number of steps in a path whose average step-length is ≤ℓ. The functionδ(ℓ) is analogous to thepercolation functionin percolation theory: there is a critical valueℓ*= e−1at whichδ(·) becomes non-zero, and (presumably) a scaling exponentβin the senseδ(ℓ) ≈ (ℓ−ℓ*)β. Recently developed probabilistic methodology (in some sense a rephrasing of the cavity method developed in the 1980s by Mézard and 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 thatβ= 3. A parallel study withtreesandconnected edge-setsin place of paths gives scaling exponent 2, while the analogue for classical percolation has scaling exponent 1. The new exponents coincide with those recently found in a different context (comparing optimal and near-optimal solutions of the mean-field travelling salesman problem (TSP) and the minimum spanning tree (MST) problem), and reinforce the suggestion that scaling exponents determine universality classes for optimization problems on random points.