Rainbow percolation
Scatter points at random along a line and give each one a reach, a distance drawn from a heavy-tailed law. Most reaches are short and a few are very long. Join two points by an edge when each of them lies inside the reach of the other.
That is the model. The question is whether the resulting graph has an infinite connected component. The panels below simulate the model, its behaviour under rescaling, the two structures the lower bound in the paper uses, and the rainbows around the origin at every scale, with the ones that cut a component off.
1 The model and its connection rule
Points of a Poisson process on the line, each with a reach R that is Pareto distributed with scale β. The reaches are drawn below the axis, the longest at the bottom, and the edges as arcs above it, coloured by connected component. Each point keeps its mark as β changes, so raising β only adds edges.
Hover over a point, or tap it, to see the connection rule. Its reach is highlighted and every point inside it is marked: green where that point reaches back, so that the edge is present, and red where it does not. An edge of length d requires two reaches of at least d, and each reach is shared by all the edges at its point, so edges meeting at a vertex are not independent. The lower plot measures the same sample on a wider window of 104 points, since a window of a few hundred points is spanned by a single point of large reach at values of β below the critical one.
The precise statement
The vertex set is a Poisson process of intensity λ on ℝ × (0,1). A vertex (x,t) is joined to (y,s) exactly when (t ∨ s)·|x − y| ≤ β, that is, when |x − y| ≤ min(Rx, Ry) for the reach R = β/t. Since t is uniform on (0,1), the reach is Pareto with scale β and tail exponent exactly one: P(R > r) = β/r. Averaging over the marks, two points at distance d > β are joined with probability β2/d2, the critical decay of one-dimensional long-range percolation.
The paper proves that the model has a non-degenerate phase transition: all components are finite once λβ < 1, and an infinite one exists once λβ ≥ 31. At intensity one this reads 1 ≤ βc ≤ 31, and the simulations reported in the paper place βc near 2.
The plotted quantity is the paper's θL(β), the fraction of the points of the central half of the window that lie in the largest component. This is one sample at one window size; the numerical study in the paper measures how it moves with L. The movement is already visible at small sizes. Taking one sample at L = 103, 104, 105 gives θ(1) = 0.61, 0.41, 0.28 and θ(0.5) = 0.16, 0.06, 0.01, falling towards zero, while θ(2) = 0.94, 0.98, 0.97 and θ(3) = 1.00, 0.995, 0.998 are stable. Curves that decay with L below the threshold and are stable above it locate βc. The paper runs this to L = 109, where every finite-size effect decays like 1/log L rather than like a power of L, because the reach tail exponent is exactly one.
2 The same picture at every scale
The same model, viewed through a window whose width you can set. Only the points whose reach is at least a fixed fraction of the window are drawn, since the others are too small to resolve at that width. Widen the window by a factor of a million and compare the picture and the counter with what they were.
The number of visible points, the length of the visible reaches and the density of the arcs stay the same as the window width runs through the decades. This is a property of the model and not of the drawing: rescaled to the unit window, the visible picture has the same law at every width. Colours mark connected components, and two components whose spans overlap never share one. The components are traced once, at a resolution finer than the drawing, and remembered as the window moves, so two points joined only through reaches too short to draw keep a common colour and a colour never splits as you zoom. Two colours become one when a connection between them is found; the colour that stays is that of the point nearest the origin, and a component that the merged span now overlaps in that colour is given a new one. Panel 4 shows the same line at the same λβ, pinned to the origin: a new sample or a move of the slider here is one there as well. Drag inside the picture to pan, and scroll or pinch to zoom.
The rescaling
Fix a resolution ε and draw only the points whose reach exceeds εW, where W is the window width. Such a point has t ≤ β/(εW), so the visible points form a Poisson process of intensity λβ/(εW) per unit length. Rescale the picture to the unit window, u = x/W. The visible intensity becomes λβ/ε per unit u, and the rescaled reach ρ = R/W satisfies P(ρ > s) = ε/s for s ≥ ε. Neither depends on W. Since the connection rule |u − u'| ≤ min(ρ, ρ') involves no other scale, the rescaled picture has the same law at every window width, and the two parameters enter only through the product λβ.
Varying β or λ therefore moves only the constant and never the exponent, so the model cannot be compared to an off-critical regime, and no stochastic domination argument settles percolation. The paper also treats the other exponents: under the rule (t ∨ s)γ|x − y| ≤ β the reaches have tail exponent 1/γ, and the threshold is degenerate for every γ ≠ 1, infinite for γ < 1 and zero for γ > 1.
3 Rainbows and cut points
The lower bound in the paper is proved on a discrete skeleton of the model, with the points pinned to the integers. In the skeleton every component is finite for every admissible parameter, although almost surely infinitely many edges cross every site. The two tabs show the objects the proof uses.
A rainbow is a pair of edges {a,b} and {c,d} with a < c < d < b whose two diagonals {a,c} and {d,b} are absent and whose overhangs are balanced. Drag the four points along the axis, and drag the handles on the two inner reaches to change Rc and Rd. The checklist records which conditions of the definition hold. Extending an inner reach past the far outer endpoint produces a long diagonal, which the balance condition rules out.
The definition
Write ℓ = c − a, m = d − c, r = b − d. The outer arch forces Ra, Rb ≥ b − a, so the two absent diagonals constrain only the inner reaches: Rc ∈ [m, ℓ) and Rd ∈ [m, r), which already forces ℓ, r > m. Balance, |ℓ − r| ≤ m, then caps the inner reaches further, by Rc < m + r and Rd < m + ℓ, so the long diagonals are absent too.
A rainbow does not by itself separate the inner arch from the outer one, since a path through an overhang could still join them. The paper adds two things: the containment event that no interior vertex reaches an outer endpoint, and a cut point in each overhang. Containment has probability ∏k(1 − xm/Dk), where Dk = min(ℓ+k, r+m−k) is the distance from the interior vertex c+k to the nearer outer endpoint, and it is bounded below by (4/9)2xm uniformly in ℓ, r and m. The bound does not depend on the scale, so the argument can be run at infinitely many independent scales.
Each mark in the lower strip is a cut point, a position that no edge crosses, so that the graph is disconnected there. The full-height dark marks are the cut points certified by the criterion of the paper, whose probability factorises over the vertices; their number agrees with n1−xm/Γ(1−xm) already at these sizes. Raising xm towards one thins the cut points out, but the exponent 1 − xm stays positive for every xm < 1, so almost surely there are infinitely many of them on the line and every component lies between two. Raising the site count instead increases the peak crossing number, which grows like log n.
Crossing edges and cut points
Almost surely infinitely many edges cross every fixed site: the expected number of edges straddling a point is xm2∑dd·d−2, a harmonic sum. Components of arbitrarily large diameter therefore exist, and a finite window appears connected. All components are nevertheless finite for every xm < 1.
The certified cut points are counted through an event that factorises over the vertices, which gives E N𝒞 ≈ n1−xm, with a positive exponent exactly when xm < 1. For independent edges with the same marginals the corresponding count is of order n1−xm2, so the dependence through the shared reaches costs an exponent; both exponents are positive on (0,1).
The transition on the integers is degenerate. At xm = 1 every reach is at least one, so every nearest-neighbour edge is present and the graph is connected. The model jumps from total fragmentation to trivial connectivity, with no supercritical percolation phase in between. Pinning the vertices to ℤ caps the number of points per connection range at every scale, whereas in the continuum the parameter λβ pushes that density arbitrarily high.
4 Rainbows around the origin
The window of panel 2, pinned to the origin. The arches over the origin are coloured by the component the origin belongs to at that scale, so a change of colour is a component cut off. Where two nested arches satisfy the definition of panel 3 they are drawn as a rainbow, pale when both arches lie in one component and solid when the outer arch cuts the inner one off.
Arches of one colour lie in one connected component, and the first arch of a new colour is the outer arch of a separating rainbow: nothing under it connects to it, so the component below is trapped inside it and is finite. The solid rainbow joins that arch to the longest arch of the component it cuts off, and a tick on the axis marks its feet. Below the critical value the colours keep changing as the window widens. Above it they stop, once the origin belongs to the infinite component. The checkbox drops the balance condition of the definition; the cuts do not depend on it, only which pairs count as rainbows. The picture is drawn on the same line as panel 2 at the same λβ: a new sample or a move of the slider in either panel is one in both. Scroll to zoom, or press zoom out.
What the colours certify
If two edges {a,b} and {u,v} cross, with a < u < b < v, then b − u is shorter than both, so {u,b} is an edge as well. Take an arch {a,b} over the origin. Any point between a and b with an edge that leaves [a,b] crosses the arch, and so is joined to a or to b directly. Whether the component of a smaller arch reaches the outer arch is therefore decided by the points under the outer arch alone, and if it does not reach it, that component cannot leave (a,b) at all: it is finite. The same argument shows that the arches over the origin fall into components that are contiguous in length, which is what makes the colours a sequence.
The colours are computed once for each sample and each value of λβ and are never revisited, so nothing recolours as the window widens. The line is walked outward octave by octave. A cheap pass over the points with reach at least 1/1000 of the octave joins each new arch to the current component whenever it finds a connection, and a connection found on a subset of the points is a real one. Each candidate for a cut is then checked in its own window under the arch, with the points down to 1/8000 of their distance from the origin. Only that second verdict can be wrong, and only in one direction: it can miss a chain through points below that resolution, never invent one. A marked connection is always right; a marked cut can in rare cases be spurious, most plausibly near λβ ≈ 2, where every scale contributes to connectivity. A cut that could not be checked within the point budget, which happens only after a very long gap without any arch, is drawn dashed.
In the integer skeleton of panel 3, the points whose reach is at least a fixed fraction of a large window form, at that scale, the picture of the continuum with λβ = xm < 1, the regime in which every component is finite and the cuts never stop. In the continuum the parameter can be raised past the threshold, and the cuts stop once the origin sits in the infinite component.