PaperlayerView exact revision on arXiv

Source-linked AI summary

Limit Theorems and Optimal Policies for Flock Flight through Poisson Forests

Lav R. Varshney

arXiv:2610.39378v1eess.SYcs.ROcs.SYmath.PR

TL;DR

A single-bird speed limit does not answer how many birds can cross a cluttered forest per unit time. This paper models many-bird transport through a Poisson forest with a directed corridor network and common waypoint headway; its capacity and large-scale current follow from cut and asymptotic results.

  • Problem

    The single-bird question of infinite collision-free flight does not determine the throughput of many birds crossing clutter, where finite body size and separation constrain route reuse.

    Source: §I · p. 1

  • Method

    The paper models flight on an explicit corridor network with common waypoint headway and uses vertex-disjoint path flows to schedule repeated traversals.

    Source: §I · p. 1

  • Results

    The schedule attains vertex-cut capacity and maximizes deliveries at every finite horizon; joint-growth current matches the width-first, depth-second limit for diverging depth no greater than order width.

    Source: §IX · p. 13

  • Takeaways & Limitations

    Under the stated arrival-moment and dispatch assumptions, an upstream admission queue is stable at every rate below finite-network capacity.

    Source: B. Stochastic Arrivals · p. 6

    The graph exactly models the specified protocol, not all continuous multi-agent trajectories.

    Source: §IX · p. 13

Abstract

from arXiv · show

Inspired by birds flying through cluttered environments, Karaman and Frazzoli showed that high-speed flight through an ergodic random forest exhibits an ultimate speed limit: above a critical speed, almost surely no infinite collision-free trajectory exists for a single bird. This paper formulates a corresponding bulk-transport problem for many birds. We construct an explicit directed corridor network in a Poisson disk forest and impose a common temporal headway at its waypoints. The maximum current equals a minimum vertex-cut capacity and is achieved by repeated flight along vertex-disjoint paths. The same schedule maximizes completed traversals at every finite horizon, with constant forward speed and no internal waiting. For a forest belt of fixed depth, exact transverse superadditivity yields an almost-sure deterministic current-density limit. A uniform shape theorem for an asymmetric planar dual identifies the current when depth and width grow together. For diverging depth bounded by a constant times width, this current equals the width-first, depth-second limit and is independent of the positive aspect ratio. When diagonal corridors are available, bulk current is positive at sufficiently low obstacle intensity and zero at sufficiently high intensity. For stochastic demand, an upstream admission queue is stable at every rate below the finite-network capacity.

I. INTRODUCTION

Prior work establishes a speed limit for a single bird, but does not determine how many birds can cross clutter per unit time. This paper formulates that bulk-transport question for a Poisson disk forest and introduces a corridor protocol whose finite-network capacity and asymptotic currents can be analyzed.

I. INTRODUCTION

  • The many-bird problem is throughput through a random medium: finite body size and separation constrain route reuse, while parallel corridors may support extensive current.This transport question is distinct from whether one bird has an infinite feasible trajectory.Source: §I · p. 1
  • The prescribed corridor-and-headway protocol makes current a vertex-cut capacity achieved by repeated flight on vertex-disjoint paths, with no internal stopping.Equal route transit times also make this schedule maximize deliveries at every finite horizon.Source: §I · p. 1
  • For fixed-depth belts, adjacent transverse bands combine by exact superadditivity, yielding a deterministic current-density limit; upstream admission is stable below finite-network capacity under stated assumptions.Kingman’s ergodic theorem gives the limit, while the queueing result requires the stated arrival-moment and dispatch assumptions.Source: §I · p. 1
  • When depth and width grow together, a uniform asymmetric-dual shape theorem identifies the current with the width-first, depth-second limit, independent of positive aspect ratio.With diagonal corridors, current is positive at sufficiently low obstacle intensity and zero at sufficiently high intensity.Source: §I · p. 2

B. Explicit Forward Corridor Library

  • The corridor library uses noncrossing forward segments in a Poisson disk forest; straight arcs advance by 2d, while diagonal arcs advance by d and require lateral speed νb/d.An arc is open when every tree center lies farther than R from its centerline, so its swept tube avoids the forest.Source: B. Explicit Forward Corridor Library · p. 2

C. Admissible Traffic

Under a prescribed corridor-and-headway rule, finite-network current is determined by a vertex cut. Repeated launches on disjoint paths attain the bound and maximize deliveries at every finite horizon without internal waiting.

C. Admissible Traffic

  • Birds traverse open graph paths at constant forward speed, with at least H between successive passages at every waypoint; waiting for admission occurs outside the forest.The optimum is for the prescribed corridor-and-headway structure, not asserted to be extremal over all continuous collision-free trajectories.Source: C. Admissible Traffic · p. 3

A. Vertex Cuts and Integral Flows

  • The finite-network current capacity is K/H, where K is both the maximum number of vertex-disjoint inlet-to-outlet paths and the size of a minimum waypoint cut.A waypoint cut meets every inlet-to-outlet path; the minimum cut need not be a straight transverse cross-section.Source: A. Vertex Cuts and Integral Flows · p. 3

B. Geometric Separation

  • Vertex-disjoint paths have centerlines more than 2q apart under the stated parameter condition, separating their embedded flights.The separation follows from the minimum distance between candidate arcs without a common endpoint.Source: B. Geometric Separation · p. 3

C. A Matching Converse and Achievability Theorem

  • A maximum family of K vertex-disjoint paths, launched at H-spaced times, attains the delivery bounds at every horizon without internal waiting.Equal crossing times and the common waypoint headway make the schedule feasible under the stated collision constraints.Source: C. A Matching Converse and… · p. 4

IV. TRANSVERSE CURRENT-DENSITY LIMIT

Adjacent transverse bands yield an exact superadditive path-count bound, while the interface cost qualifies its use across growing dimensions. Ergodicity then gives a deterministic current-density limit and a finite-width approximation.

A. A Pathwise Band Inequality

  • Combining optimal path families in adjacent bands gives exact superadditivity, while the reverse bound pays an interface cost proportional to depth that need not vanish after width normalization when depth and width grow together.The exact lower inequality needs neither a boundary correction nor independence.Source: A. A Pathwise Band Inequality · p. 4

B. Existence and Finite-Width Approximation

  • Kingman’s theorem yields an almost-sure deterministic limit for normalized path counts, and the normalized expectations converge to their supremum.The deterministic bound also gives L1 convergence.Source: B. Existence and Finite-Width… · p. 4
  • The argument relies on stationarity and ergodicity of the forest, not independence of neighboring corridor states; the finite-width bound concerns the ensemble mean.Source: B. Existence and Finite-Width… · p. 5

C. Concentration About the Mean

The section establishes concentration at fixed depth, positive current through every finite belt, and conditions under which current vanishes as depth and width grow. It relates the joint-growth limit to the width-first, depth-second limit.

C. Concentration About the Mean

  • At fixed depth, normalized-current fluctuations are controlled at scale n^-1/2, while the mean bias is at most N/n.The concentration bound alone does not establish a joint-growth law when N is proportional to n.Source: C. Concentration About the Mean · p. 5

A. A Positive Current at Every Finite Depth

  • Positive current occurs at every fixed finite depth; in the straight-only regime, the fraction of rows crossing a belt decreases exponentially with depth.Clear horizontal paths in distinct inlet rows provide the lower-bound mechanism.Source: A. A Positive Current at Every Finite… · p. 5
  • Positive finite-belt current does not imply an infinite collision-free trajectory: in the straight-only regime, each fixed row is eventually blocked along an infinite ray.A positive fraction of rows can still cross any specified finite depth.Source: A. A Positive Current at Every Finite… · p. 5

B. Increasing the Forest Depth

  • At fixed width, increasing depth cannot increase the crossing family, yielding the width-first, depth-second limit.Every path to a deeper belt visits the shorter belt's final layer, so truncation preserves a vertex-disjoint crossing family.Source: B. Increasing the Forest Depth · p. 5
  • If the probability of an infinite open forward path is zero, normalized current converges to zero almost surely and in L1 along any diverging depth and width sequences.The inlet count bounds crossings by the fraction of inlets that can reach any fixed depth.Source: B. Increasing the Forest Depth · pp. 5–6

C. Separating Service Rate from Maneuverability

The section separates service rate from maneuverability, describes a fixed-path schedule attaining capacity, and gives an upstream queue policy for stochastic arrivals. The stochastic guarantee depends on dispatch assumptions and strict slack below capacity.

C. Separating Service Rate from Maneuverability

  • Service-normalized current is nonincreasing with speed, although current itself need not be; straight-only finite-depth current grows proportionally to speed while infinite-depth current is zero.The corridor library has only one lateral-maneuver threshold.Source: C. Separating Service Rate from… · p. 6

A. Offline Route Construction

  • A maximum-flow path family launched at common headway attains capacity and maximizes completed traversals at every finite horizon without intermediate forest waiting.Each selected path carries at most one bird per launch epoch; all transit takes L/ν.Source: A. Offline Route Construction · p. 6

B. Stochastic Arrivals

  • For stochastic arrivals, the upstream admission policy strongly stabilizes the queue below capacity, and admitted birds traverse without waiting inside the forest.Long-run mean departures equal the arrival rate under the stated assumptions.Source: B. Stochastic Arrivals · p. 6
  • Aggregate arrival rate alone does not guarantee stability when fixed arrivals at separate inlets cannot be dispatched to all selected waypoints.The guarantee assumes a staging facility that can dispatch to all selected inlet waypoints.Source: B. Stochastic Arrivals · p. 6

C. Bulk Load Condition

The section gives a bulk-load condition for stable admission and develops a planar-dual route to identify current under joint spatial scaling. The dual represents finite-network capacity as a passage cost across the forest belt.

C. Bulk Load Condition

  • At loads below the limiting current density, sufficiently wide realized belts almost surely admit a stable upstream policy under the stated moment and dispatch assumptions.Above the finite-network cut capacity, un rejected demand grows upstream; exact critical loading does not meet the stochastic theorem's strict-slack condition.Source: C. Bulk Load Condition · p. 7

A. Another Scaling for the Flight Problem

  • When depth and width grow proportionally, the fixed-depth interface estimate no longer vanishes after width normalization, so the paper uses a different probabilistic route.The route proves an asymmetric dual shape theorem for dependent corridor capacities.Source: A. Another Scaling for the Flight… · p. 7

B. A Bounded Planar Representation of Waypoint Capacity

  • A bounded planar realization preserves the finite-band maximum flow while making the capacity field stationary and finite-range dependent, despite dependence between neighboring corridors.The realization uses waypoint splitting and deterministic zero-capacity arcs to complete bounded cells.Source: B. A Bounded Planar Representation of… · p. 7

C. Asymmetric Duality and the Slab Boundary

  • Finite-band capacity equals the minimum passage cost of a dual path between upper and lower ports, with outgoing cut arcs charged and incoming arcs uncharged.The end correction involves two arcs; terminal chains exclude an artificial exterior bypass.Source: C. Asymmetric Duality and the Slab… · pp. 7–8

D. A Uniform Asymmetric Shape Theorem

The theorem gives a deterministic uniform shape for passage times in the asymmetric dual. Its limiting function is finite, nonnegative, Lipschitz, convex, and positively homogeneous, but may be asymmetric or vanish in some directions.

D. A Uniform Asymmetric Shape Theorem

  • Uniformly over compact regions, rescaled dual passage times converge almost surely to a deterministic shape function that may be asymmetric or vanish in some directions.The function is finite, nonnegative, Lipschitz continuous, convex, and positively homogeneous.Source: D. A Uniform Asymmetric Shape Theorem · pp. 8–9
  • Kingman's theorem applied to stationary subadditive passage-time arrays yields deterministic directional limits, which extend to a Lipschitz, positively homogeneous function.The deterministic routing bound supplies integrability and endpoint control without requiring a positive time constant.Source: D. A Uniform Asymmetric Shape Theorem · pp. 8–9
  • A density argument for good origins and deterministic endpoint estimates extends directional convergence to moving endpoints uniformly over compact regions.The spatial ergodic theorem controls the density of good origins without requiring independence of their defining events.Source: D. A Uniform Asymmetric Shape Theorem · p. 9

E. Reflection Symmetry and the Vertical Direction

The corridor library treats the two lateral directions alike, yielding horizontal evenness in the dual time constant rather than full symmetry.

E. Reflection Symmetry and the Vertical Direction

  • The symmetry is only horizontal evenness because forward and backward motion remain distinct in the corridor library.Source: E. Reflection Symmetry and the… · p. 9
  • Reflection across the horizontal axis preserves the corridor library, Poisson law, and split-vertex construction, implying horizontal evenness of the dual time constant.The reflection reverses vertical completion arcs, but both dual crossing directions of those zero-capacity arcs have zero length.Source: E. Reflection Symmetry and the… · p. 9
  • Reflecting and reversing a dual path preserves its passage cost in the reflected forest and maps displacement (x, y) to (−x, y).Compatible dual representatives and bounded adjustments leave the time constant unchanged.Source: E. Reflection Symmetry and the… · p. 9

F. Confined Vertical Blocks and the Iterated Limit

The section identifies the iterated current by confining near-optimal dual paths to vertical blocks, then taking the width limit before the depth limit.

F. Confined Vertical Blocks and the Iterated Limit

  • Confinement is needed because an unrestricted near-optimal dual path may leave the finite slab.Only vertical blocks are needed because of horizontal evenness.Source: F. Confined Vertical Blocks and the… · p. 10
  • The iterated current is identified with the vertical dual time constant by first fixing depth and then taking width to infinity.The depth used in the confinement argument is fixed before the width limit; the proof does not interchange the limits.Source: F. Confined Vertical Blocks and the… · p. 10
  • The proof stacks restricted-cost blocks inside the slab and joins them with bounded-cost end connections to control the passage cost.The block depth is selected for the prescribed accuracy and remains fixed as slab width grows.Source: F. Confined Vertical Blocks and the… · p. 10

G. The Joint-Growth Current Law

For diverging depth at most proportional to width, the paper identifies an almost-sure deterministic bulk current that is independent of positive aspect ratio. It also shows that fixed-width depth growth eventually blocks transport, so the order and rate of spatial growth matter.

G. The Joint-Growth Current Law

  • For N_n→∞ with N_n=O(n), normalized current converges almost surely and in L1 to the bulk limit, which is independent of any positive finite aspect ratio.The repeated-path no-stopping policy attains the finite-slab currents; the limit also equals the width-first, depth-second current.Source: G. The Joint-Growth Current Law · p. 10
  • For arrival rates below the bulk current, the upstream admission queue is stable under the stated arrival-moment and dispatch assumptions, while rates above it overload a cut.The claim concerns long-run operation in each fixed finite slab; it gives no delay bound uniform in slab size.Source: G. The Joint-Growth Current Law · p. 11

H. Fixed Width and Reversed Limits

VIII. OBSTACLE INTENSITY AND POSITIVE BULK CURRENT

  • At sufficiently low obstacle intensity, the common bulk current is positive when diagonal corridors are available, while sufficiently high intensity gives zero current.The positive-current proof controls disjoint crossings directly; the extinction bound also gives zero current for jointly diverging dimensions.Source: §VIII · p. 11

A. An Explicit Extinction Bound

The paper bounds current using forward-path extinction at high obstacle intensity and a sparse-forest dual-cut argument for positive current at low intensity.

A. An Explicit Extinction Bound

  • At sufficiently high obstacle intensity, normalized current vanishes for arbitrary jointly diverging depths and widths.The explicit extinction bound counts at most 5N candidate paths from an inlet and applies a Poisson void bound to each forbidden region.Source: A. An Explicit Extinction Bound · p. 11

B. A Sparse-Forest Cut Bound

  • The dual cut bound is converted to waypoint capacity by discarding straight arcs and reducing diagonal edge flow by a factor of two.Source: B. A Sparse-Forest Cut Bound · pp. 11–12
  • The sparse-forest argument bounds minimum-cut cost by counting cheap dual paths, which must cross a substantial fraction of closed diagonals.A finite coloring separates clearance regions to handle dependence; the proof does not assume independent edges.Source: B. A Sparse-Forest Cut Bound · p. 12
  • With diagonal corridors, the common current is positive at sufficiently low intensity and approaches its upper bound as intensity decreases to zero.The sparse-forest proof establishes positivity directly, without invoking the shape theorem.Source: B. A Sparse-Forest Cut Bound · p. 12

C. A Nontrivial Critical Intensity

With diagonal corridors, bulk current is positive below a critical obstacle intensity and zero above it, though the estimates do not determine the threshold exactly. The conclusion also identifies current limits and scope boundaries for the prescribed corridor protocol.

C. A Nontrivial Critical Intensity

  • Bulk current is positive for 0 < ζ < ζc and zero for ζ > ζc; the estimates do not determine the exact threshold or its critical behavior.In the straight-only regime, current is zero at every positive intensity.Source: C. A Nontrivial Critical Intensity · p. 12

IX. CONCLUSION

  • For the prescribed corridor protocol, maximum current equals a vertex-cut capacity and is achieved by repeated flight along disjoint paths at constant speed.The schedule also maximizes deliveries at every finite horizon; the graph models the specified protocol, not all continuous multi-agent trajectories.Source: §IX · p. 13
  • For diverging depth no greater than a constant times width, joint-growth current equals the width-first, depth-second limit and is independent of positive aspect ratio.At fixed finite width, a sufficiently deep forest is eventually blocked almost surely.Source: §IX · p. 13
  • A sufficiently sparse forest supports positive extensive current with diagonal corridors, while sufficiently high intensity extinguishes it.As obstacle intensity vanishes, current approaches the inlet-capacity limit; the critical value and behavior at criticality remain open questions.Source: §IX · p. 13
  • Congestion depends on bottlenecks in the feasible-corridor network, and traffic above the cut limit overloads every policy respecting the protocol.The conclusion identifies information, tracking, and coordination costs as needed to quantify how closely less structured controllers approach this benchmark.Source: §IX · p. 13
Loading 2610.39378v1…