Random matrices can help things run more smoothly jentzphoto /Alamy Stock Photo

With antiquated trains, rusty rails and straphangers who keep the doors from closing, the New York City subway could hardly be described as efficient. And yet, some trains arrive with a certain regularity, following a neat statistical model similar to that seen in quantum systems.

Aukosh Jagannath at the University of Toronto, Canada, and Tom Trogdon at the University of California, Irvine, used the subway system’s real-time data feeds to analyse gaps between arrival times on two lines. They found that the 6 line that runs up the east side of Manhattan is inefficient. Its trains follow the Poisson distribution, a statistical model that describes particles that arrive more or less randomly.

“If you were waiting at a stop for 5 minutes, waiting for the next 5 minutes does you no good,” says Trogdon. In a more functional transit system, you’d expect that after waiting for a while, the probability of a train arriving soon would be quite high. The Poisson distribution does not guarantee this.


In contrast, the southbound 1 line that runs down the west side of Manhattan show random matrix patterns, which are “a sign of greater efficiency”, says Jagannath, now at Harvard University. These trains run at more regular intervals.

“I think the data is confirming people’s intuition about the two lines,” says Trogdon. Indeed, the 1 line is one of the three local subway lines serving the west side of Manhattan, so it’s far less crowded than the 6, which at the time of the study was the only local line on the east side.

Inspired by buses

The efficiency analysis hinges on a landmark 1990 study in Cuernavaca, Mexico. Despite operating with no central control, buses there run without much clustering, thanks to the drivers’ effort to maximise their profit. In that study, buses also conformed to random matrix patterns.

The parallel isn’t exact for the New York City subway system, however. The random matrix patterns break down at the last 10 stations of the southbound 1 line. Moreover, the northbound 1 line does not follow those patterns.

“The analysis of the New York system is less clear [than the Cuernavaca bus system],” says Ariel Amir at Harvard University.

Still, Amir says this kind of analysis is the first step towards optimising the subway system. For straphangers in New York, that’s always going to be a plus.

Journal reference: Physical Review E, DOI: 10.1103/PhysRevE.96.030101