Pathfinder · a study of self-avoiding random walks on the 4000 × 4000 lattice
compiled 2026-08-01 11:52
34m ago · refresh 300s
Some 45.5K walks have been judged
interesting enough to keep — the saved tail of a far larger population of blind,
self-trapping walks that begin at the origin and step to a random unvisited neighbour until
they wall themselves in. Together the kept walks trace 89.16M
steps — some 89,165 km of lattice
travel at one metre to the cell. What follows is their collective portrait: how far they
wander, how they turn, where they die, and how faithfully this blind process reproduces the
exact critical laws that theory demands of it.
Library
45.5K
kept paths
Marathoners
881
length ≥ 3,000 steps
Voyagers
44.8K
reached radius ≥ 300
Legends
133
both at once
Contributors
16
machines in the fleet
I
The anatomy of a walk
45,509 paths · median rule marked
Fig. 1. How many kept paths there are of each length, from the shortest
(1,069 steps) to the longest (4,117).
45,509 paths in 139 equal bins of
22 steps; the dashed rule is the median
(1,920), the mode sits at 1,975.
End-to-end distance R —
med 301.379 · p99 361.935Maximum radius —
med 306.811 · p99 380.291Radius of gyration —
med 93.945 · p99 121.482Tortuosity N/R —
med 6.331 · p99 15.839Fractal dimension —
med 1.297 · p99 1.399Asphericity —
med 0.889 · p99 0.973Convex-hull area —
med 17,661.098 · p99 37,501.18Orbits of home —
med 0.528 · p99 1.815Winding angle —
med -1.308 · p99 444.353Self-contacts —
med 2,823.356 · p99 4,000Choice entropy —
med 2.43 · p99 2.476
Table 1. Every measure broken out by family — the two species the collector
saves for, and the rarer Legends that are both at once.
Family
Kept
Mean length
Mean R
Mean Rmax
Mean Rg
Mean N/R
Marathoners · length ≥ 3,000
748
3,131
174.5
211.6
75.2
21.7
Voyagers · radius ≥ 300
44.6K
1,936
303
315
95.4
6.4
Legends · both
133
3,140
313.3
334.9
104.1
10.2
Pearson r across
45.5K paths ·
positive
negative · opacity = strength
II
Hall of fame
extremes of the library
Table 2. The record-holder for each superlative, with its trace. The
● marks the origin, the
● the death.
Var(winding) vs. ln N — the Gaussian-winding slope
Var ∼ const·ln N (Duplantier & Saleur 1988)
The shape of dying
mean free neighbours in the steps before the trap
Fig. 5. Every walk suffocates the same way — options
dwindle to zero as the trail seals it in.
First-passage scaling
median step to first reach each ring (log–log)
Fig. 6. Time to reach radius r grows as r1/ν = r4/3
(red); the fitted slope is 1.072.
VII
Territory & the path of the day
where the library has been
3.4% of the world touched ·
log-scaled visits · dashed cross = origin
Frontier — Fig. 7. farthest reach in
each compass sector, by the walk's overall lean
origin
death
3290_3919074004_1.wrd
A marathoner of 3,290 steps that
reached radius 149.9, coiled to a gyration radius of
66.1, and wound 0 times around home before trapping — rarer than
4.82σ of its recent peers.
length
3,290 steps
max radius
149.86
gyration
66.09
hull area
21.1K
found
11h ago
VIII
Growth of the library
is the fleet finding better paths, or just more?
Fig. 8. The library over time. The solid ink curve is the running total of
kept paths (left axis); the dashed accent curve is the mean length of each day's arrivals
(right axis). A rising total with a flat mean means the fleet is finding more of the same;
a climbing mean would mean it is finding better. From
2025-12-05 to 2026-07-31,
45.5K paths across
239 days.
IX
Shape space & twins
a map of all paths
1.2K sampled of 45.5K ·
Marathoner
Voyager
Legend
other
≈
The nearest twin. Of the sample, these two lie closest in
standardised shape space (distance 0.349) — walks found
far apart in the library that nonetheless grew up alike.
Compiled from the kept path library · schema v3 ·
figures are drawn from the kept library except §V (population) and §VI (fleet log).
Set in Palatino. All measurement, no decoration.