Random matrices and the New York City subway system

We analyze subway arrival times in the New York City subway system. We find regimes where the gaps between trains are well modeled by (unitarily invariant) random matrix statistics and Poisson statistics. The departure from random matrix statistics is captured by the value of the Coulomb potential along the subway route. This departure becomes more pronounced as trains make more stops.

Figure 1

The KS test for the No. 1 train (a) $[u=0]$ and the No. 6 train (b) $[u=1]$. Circles and triangles represent southbound and northbound trains, respectively. The dashed lines in order of decreasing height represent the significance levels $\alpha =0.01,0.05,0.1$. Stations that lie below a line pass the associated KS test.

Figure 2

The KS fit for the No. 1 train (a) $[u=0]$ and the No. 6 train (b) $[u=1]$. Circles and triangles represent southbound and northbound trains, respectively. The dashed lines give the $u^{*}=0.43$ and the $u^{*}=0.94$ thresholds. Values of $u^{*}$ above 0.94 indicate Poisson statistics and values of $u^{*}$ below 0.43 indicate RMT-like statistics.

Figure 3

The normalized train gap histograms for the northbound (bottom) and southbound (middle) No. 1 trains at station 112. The solid curves give the exponential and WS density. The triangles represent the best-fit density $p(s;u^{*})$. The southbound train exhibits (highly significant) RMT statistics and our ansatz that determined $p(s;u^{*})$ is not sufficient to capture the behavior of northbound trains.

Figure 4

The normalized train gap histograms for the northbound No. 6 trains at station 619. The solid curves give the exponential and WS density. The triangles represent the best-fit density $p(s;u^{*})$. This station exhibits (highly significant) Poissonian statistics.

Figure 5

The empirical number variance for southbound No. 1 trains at stations 116 (dots), 117 (triangles), and 112 (crosses) plotted against the theoretical curve (3). Agreement appears particularly good for station 117 for small values of $t$. Agreement is not as good for station 112.

Figure 6

Trajectories of the shifted southbound No. 1 trains ${\mu }_j\left(\ell \right), j=1,2,...,10$. The horizontal axis represents the distance the train has traveled (measured from stop 103). Theses shifted trajectories are qualitatively similar to that of nonintersecting Dyson Brownian motion, at least for short distances.

Figure 7

The averaged Coulomb potential (4) for southbound No. 1 trains plotted as a function of distance from station 103. The scaled KS distance from WS is also plotted to show that when the Coulomb potential increases, so does the scaled KS statistic, indicating increased deviation from RMT statistics.