Pathfinder · a study of self-avoiding random walks on the 4000 × 4000 lattice
compiled 2026-09-15 22:52
3m ago · refresh 300s
Some 48.7K 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 95.47M
steps — some 95,469 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
48.7K
kept paths
Marathoners
947
length ≥ 3,000 steps
Voyagers
47.9K
reached radius ≥ 300
Legends
142
both at once
Contributors
16
machines in the fleet
I
The anatomy of a walk
48,710 paths · median rule marked
Fig. 1. How many kept paths there are of each length, from the shortest
(1,037 steps) to the longest (4,117).
48,710 paths in 134 equal bins of
23 steps; the dashed rule is the median
(1,920), the mode sits at 1,975.
End-to-end distance R —
med 301.378 · p99 362.287Maximum radius —
med 306.841 · p99 380.112Radius of gyration —
med 93.952 · p99 121.522Tortuosity N/R —
med 6.333 · p99 15.92Fractal dimension —
med 1.297 · p99 1.399Asphericity —
med 0.889 · p99 0.973Convex-hull area —
med 17,664.207 · p99 37,563.318Orbits of home —
med 0.527 · p99 1.813Winding angle —
med -1.058 · p99 443.1Self-contacts —
med 2,824.279 · p99 4,000Choice entropy —
med 2.43 · p99 2.475
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
805
3,129
174.9
211.9
75.3
21.6
Voyagers · radius ≥ 300
47.8K
1,937
303
315.1
95.4
6.4
Legends · both
142
3,138
313
334.9
104.3
10.2
Pearson r across
48.7K 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
3.415% 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
3014_1722828843_4.wrd
A marathoner of 3,014 steps that
reached radius 266.4, coiled to a gyration radius of
83.8, and wound 0 times around home before trapping — rarer than
1.99σ of its recent peers.
length
3,014 steps
max radius
266.37
gyration
83.84
hull area
21.9K
found
4h 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 2026-09-15,
48.7K paths across
285 days.
IX
Shape space & twins
a map of all paths
1.2K sampled of 48.7K ·
Marathoner
Voyager
Legend
other
≈
The nearest twin. Of the sample, these two lie closest in
standardised shape space (distance 0.348) — 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.