THE ARISING · Science & Frontiers
A Physics Law for Trade Costs Looks Elegant, but Road Data Already Point the Other Way
Our research desk proposed that the scatter in travel and trade costs grows with distance to the one-third power, a signature from the physics of random landscapes. A 2017 study of real road networks found much larger exponents.
Two cities the same distance apart can have very different costs to ship goods between them. Our research desk asked whether physics can predict how that unpredictability grows with distance. The idea is elegant. The best existing evidence suggests real geography does not cooperate.
Economists Treb Allen and Costas Arkolakis built an influential model of transport costs. In their 2022 paper in the Review of Economic Studies, the cost between two places is computed by adding up every possible route, each weighted by the product of its link costs. Shippers mostly take cheap routes, but random factors spread traffic across many.
The desk noticed that this sum has the same shape as a central object in statistical physics: the partition function of a path wandering through a random landscape. Physicists have studied such paths for decades under the name of first-passage percolation and the KPZ universality class, after Kardar, Parisi and Zhang's 1986 paper.
In two dimensions, that physics predicts a specific law. Across many pairs at distance L, the spread in cost grows as L to the one-third power. The best path wanders sideways as L to the two-thirds. The fluctuations follow a skewed distribution rather than a bell curve.
The desk's prediction: travel times and trade costs between real places should show that one-third exponent. It put the odds at 35% and proposed routing 100,000 random point pairs on OpenStreetMap using the OSRM routing engine.
The desk wrote that nobody has checked this against transport data. That is not correct. In 2017, physicists Alexandre Solon, Guy Bunin, Sherry Chu and Mehran Kardar posted Optimal paths on the road network as directed polymers. Kardar is the K in KPZ.
They used the same tools the desk proposed: OSRM on OpenStreetMap, with millions of shortest and fastest paths across three regions, including Europe and the United States. Long optimal paths did behave like directed polymers. But the fluctuation exponents they measured ranged from about 0.58 to 0.9, far above one-third. Sideways wandering scaled at about 0.69 to 0.72, slightly above two-thirds.
The authors attributed the gap to long-range correlations in geography and road design. That is exactly the caveat the desk raised itself. Under the desk's own scoring rule, a slope well above 0.5 means strongly correlated disorder, outside the simple one-third class.
Two more cautions. The Allen and Arkolakis sum runs over all routes, including ones that double back. A directed polymer moves forward only. The mapping is close but not exact. And the one-third law is proven only for certain exactly solvable models, such as those Kurt Johansson analyzed in 2000. For general first-passage percolation it is a strong conjecture, not a theorem.
The same math describes how far patients live from care. Travel time to a stroke center or trauma unit determines outcomes. If the variability in those times grows faster with distance than simple models assume, rural patients face more unpredictable access than planners estimate.
There is also a lesson in reading claims. A beautiful mapping between two fields can be exactly right on paper and still miss the real world. The first step is always to search for whether someone already ran the test.
• For travel times: the 2017 road data already lean against a one-third law. A replication with the desk's exact binning from 50 to 2,000 kilometers would settle it. • For trade costs: this remains untested. Freight rates or price-gap estimates between market pairs, binned by distance, could show whether economic costs smooth out the road-level correlations. • For the skewed shape: even if the exponent is wrong, the distribution of residuals may still be skewed toward unexpectedly cheap pairs. That is checkable with the same data.
• Whether the desk revises its 35% estimate in light of the 2017 road-network results. • Any test using freight rates rather than travel times, which is still open. • Use of the same methods to model variability in travel time to emergency care.
Claim What We Found Tier
A Physics Law for Trade Costs Looks Elegant, but Road Data Already Point the Other Way Could Meteorite Amino Acids Work as a Thermometer for Early Asteroids? An Idea Borrowed From Stars