A study of self-avoiding random walks on the 4,000 × 4,000 lattice: how long they live, how far they reach, what shapes they take, and whether the physics agrees.
Compiled 2026-11-01 13:53 8m ago · refresh 300s
Some 52.3K 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 102.58M
steps — some 102,575 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
52.3K
kept paths
Marathoners
1K
length ≥ 3,000 steps
Voyagers
51.5K
reached radius ≥ 300
Legends
153
both at once
Contributors
16
machines in the fleet
I
The anatomy of a walk
52,346 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).
52,346 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 301.349 · p99 362.114Maximum radius —
med 306.819 · p99 379.98Radius of gyration —
med 93.961 · p99 121.479Tortuosity N/R —
med 6.333 · p99 15.867Fractal dimension —
med 1.297 · p99 1.399Asphericity —
med 0.889 · p99 0.973Convex-hull area —
med 17,659.138 · p99 37,573.116Orbits of home —
med 0.527 · p99 1.811Winding angle —
med -1.004 · p99 442.507Self-contacts —
med 2,823.697 · 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
860
3,128
174.3
211.9
75.1
22
Voyagers · radius ≥ 300
51.3K
1,936
303
315
95.4
6.4
Legends · both
153
3,139
311.8
334.9
104.4
10.2
Pearson r across
52.3K 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
3.435% 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
3458_15278309514_6.wrd
A marathoner of 3,458 steps that
reached radius 257.4, coiled to a gyration radius of
80, and wound 0 times around home before trapping — rarer than
2.66σ of its recent peers.
length
3,458 steps
max radius
257.39
gyration
80.03
hull area
22.6K
found
10h 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 2026-10-31,
52.3K paths across
331 days.
IX
Shape space & twins
a map of all paths
1.2K sampled of 52.3K ·
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.