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Register a new userThe statistical properties of the city transport in Cuernavaca (Mexico) and Random matrix ensembles
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We analyze statistical properties of the city bus transport in Cuernavaca (Mexico) and show that the bus arrivals display probability distributions conforming those given by the Unitary Ensemble of random matrices.
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The statistics of random-matrix spectra can be very sensitive to the unfolding procedure that separates global from local properties. In order to avoid the introduction of possible artifacts, recently it has been applied to ergodic ensembles of Random Matrix Theory (RMT) the singular value decomposition (SVD) method, based on normal mode analysis, which characterizes the long-range correlations of the spectral fluctuations in a direct way without performing any unfolding. However, in the case of more general ensembles, the ergodicity property is often broken leading to ambiguities between spectrum-unfolded and ensemble-unfolded fluctuation statistics. Here, we apply SVD to a disordered random-matrix ensemble with tunable nonergodicity, as a mathematical framework to characterize the nonergodicity. We show that ensemble-averaged and individual-spectrum averaged statistics are calculated consistently using the same normal mode basis, and the nonergodicity is explained as a breakdown of this common basis.
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matrix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.
There has been a long-standing and at times fractious debate whether complex and large systems can be stable. In ecology, the so-called `diversity-stability debate arose because mathematical analyses of ecosystem stability were either specific to a particular model (leading to results that were not general), or chosen for mathematical convenience, yielding results unlikely to be meaningful for any interesting realistic system. Mays work, and its subsequent elaborations, relied upon results from random matrix theory, particularly the circular law and its extensions, which only apply when the strengths of interactions between entities in the system are assumed to be independent and identically distributed (i.i.d.). Other studies have optimistically generalised from the analysis of very specific systems, in a way that does not hold up to closer scrutiny. We show here that this debate can be put to rest, once these two contrasting views have been reconciled --- which is possible in the statistical framework developed here. Here we use a range of illustrative examples of dynamical systems to demonstrate that (i) stability probability cannot be summarily deduced from any single property of the system (e.g. its diversity), and (ii) our assessment of stability depends on adequately capturing the details of the systems analysed. Failing to condition on the structure of dynamical systems will skew our analysis and can, even for very small systems, result in an unnecessarily pessimistic diagnosis of their stability.
We introduce a special class of random matrices (DUE) whose spectral statistics corresponds to statistics of microscopical quantities detected in vehicular flows. Comparing the level spacing distribution (for ordered eigenvalues in unfolded spectra of DUE matrices) with the time-clearance distribution extracted from various areas of the flux-density diagram (evaluated from original traffic data measured on Czech expressways with high occupancies) we demonstrate that the set of classical systems showing an universality associated with Random Matrix Ensembles can be extended by traffic systems.
We study probabilistic and combinatorial aspects of natural volume-and-trace weighted plane partitions and their continuous analogues. We prove asymptotic limit laws for the largest parts of these ensembles in terms of new and known hard- and soft-edge distributions of random matrix theory. As a corollary we obtain an asymptotic transition between Gumbel and Tracy--Widom GUE fluctuations for the largest part of such plane partitions, with the continuous Bessel kernel providing the interpolation. We interpret our results in terms of two natural models of directed last passage percolation (LPP): a discrete $(max, +)$ infinite-geometry model with rapidly decaying geometric weights, and a continuous $(min, cdot)$ model with power weights.
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