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Order out of noise — page 2

Random processes that reliably produce the same shape, and the reason that is less mysterious than it looks.
Where a long orbit of the logistic map spends its time. A histogram of 60000 iterates in 32 bins, with the density the map preserves drawn over it as the exact share each bin should hold. Dynamics

The histogram an orbit leaves

When no single step of an orbit is worth reporting, what is left is where it spends its time. That distribution is not uniform, it does not depend on where the orbit started, and it can be computed in closed form.

 Sampling the orders, and how fast the answer arrives. The largest error in the estimated shares against the number of orderings sampled, both on logarithmic axes, with the square-root rate drawn through the first point. Applied

Too many orders to list

The rule is an average over every order the players could have arrived in. At seven players that is five thousand orders and at twenty it is more than there are seconds in the age of the universe — so the average is sampled, and the error falls at a rate that can be measured.

 One disc, and two paths that stop being near each other. Two nearly identical billiard paths drawn on an empty square and on a square with a circular obstacle, with the separation between them plotted against distance travelled. Dynamics

The obstacle that makes a table chaotic

Put one round post in the middle of a square table and every trace of order goes. Two paths that start a hundred-thousandth of a degree apart end up on opposite sides of the table, and the reason is that a wall curving outwards multiplies a gap where a flat one only adds to it.

 Which regions each rule favours. Average seats above or below exact quota for the largest and the smallest region, under each of the five methods, over 400 generated instances. Applied

The rule with no favourites

Over four hundred instances, Jefferson's method gives the largest region a third of a seat more than its exact share and the smallest a third of a seat less. Adams reverses both. Webster's average is a hundredth of a seat, and that is not luck.

 Nineteen circles, one per vertex. A circle packing of a triangulation with seven interior vertices and twelve on the boundary: two circles touch exactly when their vertices are joined, and the radii were solved for rather than chosen. Geometry

Every flat graph is a pile of circles

A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.

 A walk on a weighted graph, and a cycle whose traffic goes one way. A weighted graph with the long-run share of each state read off its total weight, beside a three-state cycle whose shares are equal and whose traffic circulates. Probability

The chain that runs the same backwards

Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.

 The time a single walk spends in each state, against the share it should hold. Paired bars for each state, one the fraction of a long run's time spent there and one the computed stationary share, above a table of expected return times. Probability

The time spent and the share held

The first rung's shares were a limit of distributions — where the walk probably is after many steps. This one is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.

 A scatter walking towards its own centres. 4 panels of the same 24 sites: the initial clumpy scatter and the Voronoi diagram after 1, 3, 12 rounds of Lloyd's iteration, with the cost falling to 52% of the scatter's as the cells even out. Geometry

Every site in the middle of its own cell

Move each point to the centre of mass of its own Voronoi cell, then redraw the diagram, then do it again. The rule is two lines long, it never mentions hexagons, and what it settles into is a honeycomb.

 Downhill on average, and never on purpose. The logarithms of 4 Collatz orbits plotted against step number, each wandering upward and downward and each ending at one. A separate sample of four thousand starts gives an average fall of -0.15 per step. Dynamics

The heuristic that cannot be a proof

There is a two-line argument that the Collatz conjecture is true, it is convincing, and everybody who works on the problem believes it. It also cannot be turned into a proof, and understanding exactly where it fails is more instructive than the argument itself.

 The same modulus, four multipliers, four qualities. 4 linear generators at modulus 1021, drawn as scatters of consecutive pairs and ranked by the spacing of the lines their points fall on. The spacings differ by more than a factor of two. Computation

The test that ranks the generators

Every linear generator's output lies on a family of parallel planes. Which generator is better is decided by how far apart those planes are, and that distance is the length of the shortest whole-number vector the modulus annihilates — a quantity that can be computed exactly rather than estimated by testing.

 A scatter with no lines in it. 900 consecutive pairs from a generator that squares modulo a product of two primes. The points show no family of parallel lines, and an exhaustive search for a short relation between consecutive outputs finds none. Computation

Randomness that has to be earned

A generator that resists prediction cannot be built out of a rule anybody can fit. It has to be built out of a computation believed hard to undo, and the belief is the load-bearing part — which makes cryptographic randomness a conditional statement rather than a construction.

 Every pattern, exactly as often. A table over 4 window lengths of a shift register's output stream: how many bit patterns are possible, how many actually occur, and the difference between the most and least frequent, which is one in every row. Computation

Nineteen thousand bits of state

The generator most simulations actually use is not clever. It is a linear recurrence over the two-element field with an enormous state, and its virtues are a proved period, a proved equidistribution and speed — none of which is unpredictability, which it does not have and does not claim.

 How fast a chain forgets where it started. The total variation distance to the stationary distribution plotted logarithmically against the number of steps, for each of 3 starting states. The curves are straight lines of equal slope. Probability

How long until it forgets

The ladder's four rungs settle where a chain ends up and how much time it spends there, and none of them asks how long the settling takes. That question has an exact answer, it is a single number, and it is the only thing any practical use of a chain depends on.

 The moment a giant piece appears. The largest component's share of 900 points plotted against the average degree, with the measured values as dots and the predicted curve behind them. The curve is flat at zero below an average degree of one and rises steeply above it. Probability

The moment a giant appears

Raise the chance of an edge slowly and a random graph does nothing for a long time, then in a narrow window acquires a component holding a definite fraction of everything. The fraction is the root of an equation, and the equation says why the transition is where it is.

 One piece, and connected, are different thresholds. Two curves against the average degree for graphs of 400 points: the largest component's share, rising from an average degree of one, and the probability of connectivity, rising only near the logarithm of the point count. Probability

Two thresholds, not one

A random graph acquires a piece holding most of its points at average degree one, and is still not connected. Connectivity waits until the average degree reaches the logarithm of the size, and what holds it up is the very last isolated point.

 A triangle appears when the count says it should. The measured probability of containing a triangle against the edge chance, on graphs of 200 points, beside the expected number of triangles capped at one. Probability

Finding a threshold with two moments

Every monotone property of a random graph has a threshold, and locating one is nearly always the same two calculations — count what the property needs, and check the count does not concentrate on rare cases. The triangle is where the method is cleanest.

 At the threshold the largest part is neither of the two obvious sizes. A table with one row per graph size, giving the largest component at the critical edge probability and that value divided by three candidate scalings. Probability

The window where the giant is born

Below the threshold the largest piece is a few dozen points, above it a definite fraction of everything. At the threshold it is neither, and the size it does take — the two-thirds power — is an exponent with no elementary derivation that a measurement finds immediately.

 One threshold narrowing, one staying wide. Probability curves for connectivity and for containing a triangle, plotted against the edge probability as a multiple of each property's own threshold, at several graph sizes. Probability

Sharp, or merely a threshold

Every monotone property of a random graph has a threshold. Some of them turn on over a range that shrinks relative to the threshold as the graph grows, and some do not — and which kind a property is turns out to be decided by whether it is about a local structure or about the whole graph.

 11 points, equally spaced. 1/(1 + x²) and the polynomial of degree 10 through 11 of its points, spaced evenly across the interval. The worst error is 2.48e-1, at x = -2.350. Analysis

The points that ruin the fit

A polynomial through eleven points of a gentle curve should be a good approximation to it, and adding more points should make it better. On evenly spaced points it makes it worse, without limit, and the reason is not the polynomial but where the points were put.

 3 carries in base 2, and 2 divides it 3 times. The addition of 5 and 7 written in base 2, column by column, with the carries marked. There are 3, and 2 divides the binomial coefficient 792 exactly 3 times. Discrete

The carries decide the divisibility

How many times a prime divides a binomial coefficient is not a fact about the coefficient at all. It is a count of the carries that happen when two numbers are added in that prime's base, which is a question about column addition and has nothing to do with choosing anything.

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