Random Walks

Brownian, self-avoiding and Lévy — the fat tail is visible.

patterns.walk · v1 · renderer canvas

Brownian | Levy | Saw · 22 walkers × 2,600 steps · α = 1.35 · centred start

Three walks from the same number of steps, each scaled to cover the same fraction of its panel. Brownian reaches √n, the self-avoiding walk n^(3/4), and the Lévy flight n^(1/α) — here n^0.74. Same step count, three different exponents, three shapes you would never mistake for one another.

[data-canvas-stage] { display: none; }

Drawn in the browser

This generator does not send you an image. The server sends a short description of one — the parameters below, plus a 32-byte key — and the drawing is done here, from the same byte stream the server would have used. That needs JavaScript, so with it switched off there is nothing to show.

The description itself is still public: /api/v1/g/patterns.walk returns it, and the maths is in resources/js.

Brownian | Levy | Saw · 22 walkers × 2,600 steps · α = 1.35 · centred start

Controls · from schema()

Width

200 2048

Height

200 2048

Walk

Walkers

1 160

Each one is an independent walk and is never reused.

Steps each

20 20000

Lévy tail α

1.05 3

The Pareto exponent. Below 2 the variance is infinite and the walk is dominated by its rare huge jumps; above 2 the central limit theorem takes back over and it becomes Brownian again.

Coverage

0.2 1.6

How much of the frame a typical walk should reach. Step length is solved backwards from this and each mode's own scaling exponent, so all three fill the frame the same amount however many steps you ask for.

Start from

Line alpha

0.01 0.9

Per-segment opacity. A walk that crosses its own path a thousand times is drawn out of the overlap, so this wants to stay low.

Line width

0.4 4

Additive blending

Ground

Colour

Palette size

2 12

Grain

0 0.08

Entropy source

Picking a source changes the story on the receipt, never the quality: anything below class A is mixed with the OS CSPRNG before it reaches you. This result actually came from Bitcoin Proof-of-Work.

seed token JCZNA28N07X05NZ1XDR4J7SQK8

Re-render keeps this token. Generate replaces it. That is the whole idea.

Burned into the corner before the file is written, so the image keeps its provenance wherever it ends up:

entropy in

16 bytes

stream used

36 bytes

rejections

duration

1,256 µs

palette

background

#010103

segments

171,600

exponents

brownian r∝n^(1/2), levy r∝n^(1/α), saw r∝n^(3/4)

levy variance

infinite (α < 2)

Seed receipt

Class B · Public beacon cached

Bitcoin block 973,812 — 19 leading zeros of proof-of-work, found about ten minutes ago.

Published openly — anyone can look this value up after the fact.

source: Bitcoin Proof-of-Work
observed: 2026-10-30T11:13:18+00:00
latency: 0ms
mixed with: csprng — always

https://mempool.space/block/00000000000000000001437feed9aae961c1ad1a4747d5b76de354fe6846c2cf ↗

Permalink

The token is the seed, so this link recomputes the result rather than looking it up. No database, and it works for as long as v1 does.

This page and /api/v1/g/patterns.walk run the same code. There is no private endpoint.

Random Walks

Result

Measurements

Seed receipt

Permalink