A model for the bus system in Cuernavaca (Mexico)

Baik, Jinho; Borodin, Alexei; Deift, Percy; Suidan, Toufic

doi:10.1088/0305-4470/39/28/s11

Cited by 51 publications

(51 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Viewing them as random patterns of polymers or phase boundaries on a plane, the present system has been used as a model of polymer networks [26,33], or a model showing wetting (or melting) transitions [36] in statistical physics; see also [50,92]. Recently many authors have reported that notion of noncolliding BM and its discrete counterpart called vicious walk [36] is very useful to analyze the polynuclear growth models [78,46,82,44], time-dependent correlations of quantum spin chains [13], traffic problems [7], and the Chern-Simons theory [27,6].…”

Section: Introductionmentioning

confidence: 99%

Noncolliding Brownian Motion and Determinantal Processes

2007

View full text Add to dashboard Cite

A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson's BM model, which is a process of eigenvalues of hermitian matrixvalued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin-McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin-McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed.

show abstract

Section: Introductionmentioning

confidence: 99%

Noncolliding Brownian Motion and Determinantal Processes

2007

View full text Add to dashboard Cite

show abstract

“…This prediction is verified for the Cuernavaca bus system in [1]. A physicallymotivated model for the bus system was presented in [10] for which (1) and (3) hold.In this letter, we observe that (1) and (3) hold on a subset of the MTA system. We also find Poisson statistics within the MTA (which are also found in Puebla, Mexico [3]).…”

mentioning

confidence: 61%

“…This prediction is verified for the Cuernavaca bus system in [1]. A physicallymotivated model for the bus system was presented in [10] for which (1) and (3) hold.…”

mentioning

confidence: 63%

Random matrices and the New York City subway system

Jagannath

Trogdon

2017

Phys. Rev. E

View full text Add to dashboard Cite

We analyze subway arrival times in the New York City subway system. We find regimes where the gaps between trains exhibit both (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.The bus system in Cuernavaca, Mexico in the late 1990s has become a canonical physical system that is well-modeled by random matrix theory (RMT) [1][2][3][4]. This bus system has a built-in, yet naturally arising, mechanism to prevent buses from arriving in rapid succession. If a driver arrives at a stop just after another bus on the same route, there will be few fares to collect so the self-employed drivers introduced a scheme, using a cadre of observers along each route, to space themselves apart so as to maximize the number of fares they collect. Without this interaction, and mutual competition, one should expect that bus arrivals would be Poissonian [5]. While the New York City subway (MTA) system has a different, globally controlled, mechanism to space trains to eliminate collisions, much of the MTA system remains under manual control [6]. In this letter, we compare the predictions and results from Cuernavaca, Mexico with the MTA system.In particular, the authors in [1] noted that if one stood at bus stop in Cuernavaca, Mexico, near the city center, and recorded the set T of times between successive buses then for τ = T / Twhere · represents the sample mean and the function ρ(s) is known as the (β = 2) Wigner surmise (WS) [7]. This is the approximation of Eugene Wigner for the asymptotic (N → ∞) gap distribution for successive eigenvalues in the bulk of an N × N GUE (Gaussian Unitary Ensemble) matrix [8]. This is computed by considering the 2 × 2 case. This approximation of Wigner agrees surprisingly well with the true limiting distribution as N → ∞ [9]. The authors in [1] consider another statistic called the number variance. Fix a time T 0 and consider the time interval, [T 0 , T ], for T 0 ≤ T ≤ T 1 . Let n(T ) be the number of buses (or subway trains) that arrive in this time interval. Once one has made many statistically independent observations of n(T ), the number variance is computed byThis normalization is made so that n(T ) ≈ t. The aysmptotic prediction from RMT iswhere γ is the Euler constant [7]. This prediction is verified for the Cuernavaca bus system in [1]. A physicallymotivated model for the bus system was presented in [10] for which (1) and (3) hold.In this letter, we observe that (1) and (3) hold on a subset of the MTA system. We also find Poisson statistics within the MTA (which are also found in Puebla, Mexico [3]). For example, the southbound #1 train in northern Manhattan exhibits RMT statistics but the northbound #6 train exhibits Poisson statistics in the middle of its route. We also show that the train gap statistics tend to deviate more from RMT statistics as more stops are made. To quantitatively determin...

show abstract

“…We have thus derived our finitized bead process as the continuum limit of a tiling model. In fact a semi-continuous lattice paths model (see Figure 3) equivalent to our finitized bead process was proposed in [BBDS06] to model the bus transportation system in Cuernavaca (Mexico). In that context, for p ≤ t ≤ q the particle configuration {x…”

Section: Continuous Modelmentioning

confidence: 99%

A finitization of the bead process

Fleming

Forrester

Nordenstam

2010

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Abstract. The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after approriate identification of the anisotropy parameter therein.

show abstract

A model for the bus system in Cuernavaca (Mexico)

Cited by 51 publications

References 23 publications

Noncolliding Brownian Motion and Determinantal Processes

Noncolliding Brownian Motion and Determinantal Processes

Random matrices and the New York City subway system

A finitization of the bead process

Contact Info

Product

Resources

About