Birds on a Wire

TL;DR

Birds on a Wire introduces the pushy-bird dynamics, where sequential landings trigger departures of nearby resting birds, producing a non-equilibrium steady state. The authors solve the 1D lattice version exactly, obtaining a steady-state density $\rho=\tfrac{1}{3}$ and a detailed void-length distribution $V_k$, along with the pair-correlation structure $C_j$. They also analyze a 1D continuum variant, showing $\rho=\tfrac{1}{2}$ and a super-exponentially decaying void distribution $V(x)$, with explicit avalanche probabilities $q_0$, $q_1$, $q_2$. Extending to higher dimensions, they conjecture densities $\rho=1/(2d+1)$ and develop a mean-field theory giving $\rho=1/(2d)$, with asymptotically small avalanches $q_n \to e^{-1}/n!$ as $d\to\infty$, elucidating the static spatial organization under arrival-departure dynamics and linking to RSA and forest-fire frameworks.

Abstract

We investigate the occupancy statistics of birds on a wire and on higher-dimensional substrates. In one dimension, birds land one by one on a wire and rest where they land. Whenever a newly arriving bird lands within a fixed distance of already resting birds, these resting birds immediately fly away. We determine the steady-state occupancy of the wire, the distribution of gaps between neighboring birds, and other basic statistical features of this process. We discuss conjectures for corresponding observables in higher dimensions.

Figures (5)

• Figure 1: Birds on wires.
• Figure 2: Processes that contribute to changes in the void densities in the PB model of Eq. \ref{['evol']}. The vertical arrows indicate the possible locations for a bird to land. In (d) only one of the two possible landing spots that creates a void of length $k$ is shown.
• Figure 3: Processes that contribute to changes in the void densities in Eq. \ref{['Vx12']}. (a) An $x$-void disappears if a bird lands anywhere inside the void (blue arrow) or within a unit distance of either bird outside the void (green arrow), (b) An $x$-void is created when a new bird lands a distance $x$ from an existing bird. Another bird may be anywhere in the range $[1,\infty]$ for $1<x<2$ or in the range $[x-1,\infty]$ for $x>2$.
• Figure 4: The void density $V(x)$ for $0<x<4$, showing the jump at $x=1$ and singularities in the first derivative at $x=2$ and $x=3$.
• Figure 5: The probability $q_2$ as a function of $b$ for $b\leq 1000$.