How a random path winds round a point

The essays on loops have so far classified loops one at a time. A loop that cannot be pulled tight found the plane with a point removed to have loops that cannot be shrunk, and the count of how many times a loop goes round the point to be the whole of its class: the fundamental group of the punctured plane is the integers. The same loop, unrolled lifted loops to the helix that covers the punctured plane, where the winding becomes a height, and what homology forgets about a loop took two holes and found that the fundamental group is no longer commutative, so that a loop’s class is a word and some words record nothing that winding numbers can see.

This essay asks what class a random loop has. The question is natural in physics, where polymers wind round obstacles and charged particles round flux lines, and it has a surprising answer with a topological moral. A random path’s winding grows, its spread has a limiting law, and which law depends on whether the path can come arbitrarily close to the point. Brownian motion can, and its winding has a law so heavy-tailed that it has no mean. A walk on a grid cannot, and its winding has a law with exponential tails. Every heavy-tailed winding is made near the puncture.

The winding angle is a height on the helix #

A path in the plane that avoids the origin has a continuous angle $\theta(t)$, defined by following the direction from the origin to the path and never letting it jump by a whole turn. Lifted to the helix above the punctured plane — the universal cover, in which a point at angle $\theta$ and the same point at angle $\theta + 2\pi$ are different points one storey apart — the path becomes a path whose height is $\theta$. For a closed loop, $\theta$ at the end minus $\theta$ at the start is $2\pi$ times a whole number, and that number is the loop’s class. For an open path, $\theta$ is a real number, and it records the class of the path up to the last fraction of a turn.

The hero figure follows one walk on the square grid around a puncture placed at the centre of a square, so that the walk, which lives on the corners of squares, can never land on it. The walk’s angle about the puncture moves in bursts. When the path is close to the puncture, a single step can change the angle by a quarter of a turn or more, and the path can go round several times in a few dozen steps; when the path is far away, a step changes the angle by very little, and the winding barely moves for hundreds of steps. This walk ends nearly five turns clockwise from where it began. A typical walk winds less; this one was picked for staying near its start. What is typical is described by the laws below.

Spitzer’s law for Brownian motion #

Brownian motion in the plane, started away from the origin, never hits it, and so its winding angle $\theta(t)$ is defined for all time. Frank Spitzer proved in 1958 that it grows like the logarithm of time, and more precisely that

$$\frac{2\,\theta(t)}{\log t} \longrightarrow \text{Cauchy}, \qquad \text{density } \frac{1}{\pi(1 + x^2)}.$$

The Cauchy law is the archetype of a distribution without a mean: its tails fall like $1/x^2$, so slowly that the average of many independent Cauchy samples is itself Cauchy, no more concentrated than a single sample. Spitzer’s law says the winding of planar Brownian motion is that wild.

The reason it grows like $\log t$ is conformal invariance, the property the walk that becomes a curve gave Brownian motion in the plane. In polar coordinates, $\log r$ and $\theta$ are two independent one-dimensional Brownian motions, running not in ordinary time but in a clock that advances at rate $1/r^2$. Far from the origin the clock runs slowly and the angle hardly changes; near the origin it races, and the angle spins. By time $t$ the path has typically reached distance about $\sqrt{t}$, so $\log r$ has travelled about $\tfrac12 \log t$, and the angle, running on the same clock, has travelled a comparable amount — hence $2\theta/\log t$. The heavy tail comes from the rare paths that pass very close to the origin, where the clock races without limit and the angle can wind an unbounded number of times in very little real time.

That decomposition is also how the figures simulate Brownian motion exactly. The path is advanced in the clock, not in real time, with $\log r$ and $\theta$ each taking independent Gaussian steps and real time accumulating at rate $r^2$. A path that dives towards the origin takes many small clock steps that cost almost no real time, and spins as much as the mathematics says it should; a naive simulation in real time would step straight past the origin and miss exactly those windings.

A walk on the grid follows a different law #

A random walk on the square grid looks like Brownian motion from far away, and it is natural to expect its winding round the centre of a square to follow Spitzer’s law too. It does not.

Two thousand walks of ten thousand steps give a histogram of $2\theta/\log n$ that is far narrower than the Cauchy density, with a taller peak and thin tails. It is close to a different law, the hyperbolic secant, with density $\tfrac12 \operatorname{sech}(\pi x / 2)$, whose tails fall exponentially. The physicists Joseph Rudnick and Yuxing Hu found this law for lattice walks in 1987, and Christian Bélisle proved in 1989 that it is the limit for a broad class of random walks. The histogram is not yet at the limit — the middle half of the walks spread over $1.45$ units of the scaled angle, against the hyperbolic secant’s $1.12$ — because the scaling is by $\log n$, and a logarithm moves slowly. But it is nowhere near the Cauchy law’s $2$.

The reason is geometric. A lattice walk cannot come closer to the puncture than half the diagonal of a square, about $0.71$ of a step. It can go round the puncture by circling the four corners of its square, but every turn costs at least four steps, and it cannot spin arbitrarily fast. The windings that make Spitzer’s tail — many turns in a short time, very close to the point — are simply unavailable.

Close passes make the heavy tail #

The two laws differ in their tails, so that is where the comparison is sharpest.

On a logarithmic scale the difference is stark. The Brownian paths follow the Cauchy tail all the way out: 23.5% of them have a scaled winding beyond 3 in size, against the Cauchy law’s 20.5%, and 8.7% exceed 8. The lattice walks fall away along a nearly straight line — an exponential tail — and none of the walks of $10^5$ steps exceeds 4. They are still above the hyperbolic secant’s tail, by a margin that shrinks as the walks lengthen, which is the same slow logarithmic approach the histogram showed. The two kinds of path differ in nothing but their behaviour near the puncture: away from it a walk of many steps is a good approximation to Brownian motion, as the walk that becomes a curve showed, and the approximation fails exactly where the angle is most sensitive to position.

Big windings and small windings #

Jim Pitman and Marc Yor explained the two laws together in 1986, by splitting Brownian motion’s winding into two parts: the angle accumulated while the path is outside a fixed circle round the point, the big windings, and the angle accumulated inside it, the small windings.

Splitting the simulated paths at the unit circle reproduces their theorem. The big windings, made far from the point, follow the hyperbolic secant law closely: 27.2% beyond 1 against the law’s 26.1%, and 1.1% beyond 3 against 1.15%. The small windings carry all of the heavy tail: 21.0% of paths have small windings beyond 3 in size, and more than one in ten beyond 6. So the hyperbolic secant law is not a special property of lattices. It is the law of winding at a distance, which any path that looks like Brownian motion on large scales obeys, and a lattice walk obeys only that law because it has no small windings at all. Spitzer’s Cauchy law is the sum of the two, and the sum is dominated by the heavier tail.

There is a clean reason the big windings have exponential tails. In the clock in which $\log r$ is a Brownian motion, the time spent outside the unit circle up to the moment $\log r$ reaches a given level has a distribution whose Laplace transform is $1/\cosh$, and the winding accumulated in that time is a Gaussian with that random variance; averaging the Gaussian over the variance gives exactly the hyperbolic secant. The time spent inside the circle has no such bound — $\log r$ can make long excursions towards $-\infty$ — and averaging over it gives the heavy tail.

The class of a random loop #

So far the paths have been open. A closed path — a loop that returns to its start — has a whole number of windings, and that number is its class in the fundamental group of the punctured plane. Random loops on the grid can be sampled exactly: a closed walk on the square grid is the same thing as two independent closed walks on a line, one for $x + y$ and one for $x - y$, and each of those is a random arrangement of equally many steps up and down.

The distribution of the class is a narrow spike that widens slowly. Of loops of a hundred steps, 59% wind zero times round the puncture beside their start and can be shrunk to a point; of loops of ten thousand steps, 37% can. At every length about a fifth of the loops wind exactly once clockwise and about a fifth exactly once anticlockwise. The rest are spread over positive and negative winding numbers symmetrically, and the spread grows slowly with the length of the loop — as a logarithmic law of winding would lead one to expect, since a loop a hundred times longer is, in the clock that matters, only a fixed number of units longer. The loops started next to the puncture; a walk that always comes home showed that a walk in the plane returns to its start, and these loops are the walks that have, and the class records what they did on the way.

Why physicists counted windings #

The winding of a random path round a point entered physics before Spitzer’s theorem was widely known there. Sam Edwards asked in 1967 how a long flexible polymer, modelled as a random path, is entangled with a straight rod passing through it, and the question reduced to the distribution of the path’s winding round the point where the rod meets the plane. Electrons in a magnetic field confined to a thin tube give the same mathematics from a different direction: in the Aharonov–Bohm effect, a charged particle circling a tube of magnetic flux has its wave function multiplied by a factor fixed by its winding number, even though it never touches the field, and so the sum over all of its possible paths becomes a sum over paths weighted by a factor that depends only on their class in the fundamental group of the punctured plane. The distribution of winding numbers of random closed paths is, in that sense, the Fourier transform of a physical quantity, and it can be measured.

Both settings care about the difference the figures above have drawn. A real polymer has a thickness and a real flux tube a radius, so the path cannot approach the puncture arbitrarily closely, and by Pitman and Yor’s decomposition the windings that survive are the big ones: for long paths the relevant law is the hyperbolic secant rather than Cauchy’s, and the Cauchy law shows only over the range of scales between the obstacle’s size and the path’s finest wiggles, which for a polymer of short segments round a thin rod can be wide. The puncture of the mathematician is a limit that the physical systems approach and never reach, and the heavy tail lives entirely in that limit.

Two punctures: loops that homology cannot see #

With two punctures the fundamental group is free on two generators: a loop’s class is a word in $a$, for once round the first puncture, and $b$, for once round the second, with the inverses for the other direction. Cutting a space to find its group computed this group by van Kampen’s theorem, and what homology forgets about a loop measured, for loops in the figure eight, how much of the class is lost when only the winding numbers are kept. The same question can be asked of random loops.

Each loop’s word is read by placing a cut from each puncture straight upwards and recording a letter each time the loop crosses a cut — $a$ or $a^{-1}$ for the first, $b$ or $b^{-1}$ for the second, by direction — and cancelling any letter followed immediately by its inverse. The exponent sums of the word are the two winding numbers, the loop’s class in homology. The word itself is its class in the fundamental group.

Most loops that wind zero times round each puncture really are trivial: their words cancel completely. But some do not. Among loops of ten thousand steps, 18.8% have zero winding round both punctures, and 0.6% of all loops — about one in thirty of those — have a word that does not cancel. The shortest found is $b\,a\,b^{-1}a^{-1}$, the commutator: round the second puncture, round the first, back round the second, back round the first. It winds zero times round each puncture, so homology calls it trivial, and it cannot be shrunk without passing through one of them. The share of such hidden classes grows with the length of the loop, as does the average length of the reduced word, from 0.73 letters for loops of a hundred steps to 2.25 for loops of ten thousand. A longer random loop has more time to braid itself round the two punctures in ways that the winding numbers cannot record.

Still open: winding in other settings #

The planar laws are complete: Spitzer’s for Brownian motion, the hyperbolic secant for paths that cannot approach the point, and Pitman and Yor’s decomposition between them. What is less settled is winding in settings where the path interacts with itself or with its surroundings. A self-avoiding walk, which a walk that may not step where it has been followed, winds round a point with a Gaussian law at the scale of $\sqrt{\log n}$, according to a prediction of Bertrand Duplantier and Hubert Saleur from conformal field theory; it has been confirmed numerically and, through its connection with the Schramm–Loewner evolution, for some related curves rigorously, but not for the self-avoiding walk itself, whose scaling limit is still unproved.

For loops with several punctures, the law of the full word — not just its length, but the distribution of which words occur — is known for Brownian motion through work on the windings of planar Brownian motion around several points, which turn out to be governed by a joint Cauchy-type law with dependent components. For random walks on the grid, and for the hidden classes the last figure counted, the corresponding statements, and the rate at which the reduced word grows, are understood only partially. The free group is where the topology of the punctured plane becomes non-commutative, and it is also where the probability of random loops becomes hardest to compute.