Parking and the visual perception of space

Abstract

Using measured data we demonstrate that there is an amazing correspondence among the statistical properties of spacings between parked cars and the distances between birds perching on a power line. We show that this observation is easily explained by the fact that birds and human use the same mechanism of distance estimation. We give a simple mathematical model of this phenomenon and prove its validity using measured data.

Rayleigh's Distribution, Wigner's Surmise and Equation of the Diffusion

“…eba compared distribution of the bird-to-bird distances with the bumper-to-bumper distances between parking autos and found that the both distributions are in good agreement with RWD [20].…”

Asymmetry in repeated isotropic rotations

“…Random matrix theory asserts that spectral and statistical properties of complex physical systems are well described by those of random operators given (statistically) the same symmetry. Applications started with Wigner explaining the spacing statistics of energy levels in atomic nuclei [1] and today cover fields as diverse as quantum chaos [3], traffic dynamics [8], economics [10], and neurophysics [4,12,13] as well as generic complex systems [5][6][7].…”

“…Random operations ubiquitously appear in complex systems models where they often reflect a statistical or approximate symmetry of the real system [1][2][3][4][5][6][7][8][9][10]. Such operations play a special role in physics and are the basic objects of random matrix theory [9,11].…”

Gaussian determinantal processes: A new model for directionality in data

“…whereψ(x, y) = x − y, u x − y, v 1{ x − y ≤ r}1{ x ≤ R − r} . (Xi, Xj)ψ(X k , X l )] − E i =jψ(Xi, Xj) ψ(x, y)ψ(z, w)ρ4(x, y, z, w)dxdydzdw + ψ(x, y)ψ(y, w)ρ3(x, y, w)dxdydw + ψ(x, y)ψ(x, w)ρ3(x, y, w)dxdydw+ ψ(x, y)ψ(z, y)ρ3(x, y, z)dxdydz + ψ(x, y)ψ(z, x)ρ3(x, y, z)dxdydz + ψ(x, y) 2 ρ2(x, y)dxdy − ψ(x,y)ρ2(x, y)dxdyBy symmetry of K (and thus of ρ3), all triple integrals are the same, equal to ψ(x, y)ψ(x, z)ρ3(x, y, z)dxdydz .Moreover, since 4-point correlation ρ4 is given byρ4(x1, x2, x3, x4) = det    K(x1, x1) K(x1, x2) K(x1, x3) K(x1, x4) K(x2, x1) K(x2, x2) K(x2, x3) K(x2, x4) K(x3, x1) K(x3, x2) K(x3, x3) K(x3, x4) K(x4, x1) K(x4, x2) K(x4, x3) K(x4, x4) the symmetry of K and Fischer's inequality thatρ4(x1, x2, x3, x4) ≤ det K(x1, x1) K(x1, x2) K(x2, x1) K(x2, x2) • det K(x3, x3) K(x3, x4) K(x4, x3) K(x4, x4) = ρ2(x1, x2)ρ2(x3, x4) [12]. Therefore, ψ(x, y)ψ(z, w)ρ4(x, y, z, w)dxdydzdw − ψ(x, y)ρ2(x, y)dxdy 2…”