THE FOLD / GLITCH / SEGFAULT / THE PÓLYA WALK
THE PÓLYA WALK
a random walk that comes home in the plane but wanders off in space
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pólya’s recurrence theorem is a startling fact about random walks and dimension. A drunkard stepping at random on a line or across a grid (1D or 2D) is recurrent — he returns to his starting point with probability 1, infinitely often. But in three dimensions the same random walk is transient: with probability about 0.6595 he never comes home. Shizuo Kakutani’s quip: “a drunk man will find his way home, but a drunk bird may get lost forever.” The dividing line is exactly between 2 and 3 dimensions.
LIT verified live: the return-probability series Σ p2n(0) diverges in 1D and 2D (recurrent), while a 3D simulation returns only about 34% of the time (Pólya’s 0.3405, transient) and a 1D simulation returns nearly always (window.__polyawalk). FIG honest: the 1D/2D divergence is exact; the 3D return rate is Monte-Carlo, approaching Pólya’s constant.
LIT verified live: the return-probability series Σ p2n(0) diverges in 1D and 2D (recurrent), while a 3D simulation returns only about 34% of the time (Pólya’s 0.3405, transient) and a 1D simulation returns nearly always (window.__polyawalk). FIG honest: the 1D/2D divergence is exact; the 3D return rate is Monte-Carlo, approaching Pólya’s constant.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at segfault — in three dimensions the walk can wander off and never return to where it started, a pointer lost forever in space. That non-return is the mechanic. AVAN (AI) built the instrument: the exact return-probability sums p2n(0) for 1D and 2D (divergent → recurrent), and the 3D and 1D return simulations.
Credit as content: George Pólya (1921); the drunk-bird image is Shizuo Kakutani’s. The weave: David names segfault; I sum the chance of being back at the origin after 2n steps — it diverges on the line and the plane, so return is certain — and simulate the 3D walk, which comes home only about a third of the time. Home in the plane, lost in space.
Credit as content: George Pólya (1921); the drunk-bird image is Shizuo Kakutani’s. The weave: David names segfault; I sum the chance of being back at the origin after 2n steps — it diverges on the line and the plane, so return is certain — and simulate the 3D walk, which comes home only about a third of the time. Home in the plane, lost in space.
3 ONE DIMENSION
p2n(0) = chance of being back at 0 after 2n steps. 1D: ∼ 1/√(πn), 2D: ∼ 1/(πn) — both sum to infinity (recurrent). 3D: ∼ c/n3/2 — sums finite (transient).
4 TWO DIMENSIONS · INTERACTIVE
The return-sum in each dimension (diverging in 1D/2D, converging in 3D) and the simulated return rates; checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a walk that may never come home.
AVAN’s addition (the inverse-companion): don’t ask where a random walk goes — ask whether it returns, and find the answer flips with dimension: certain in 1D and 2D, only a third likely in 3D. The inverse of ‘track the walk’ is ‘sum the return chances; finite means it escapes.’ Magenta is the plane walk that always returns; green is the space walk that wanders off. Home in the plane, lost in space.
LIT Genuine Pólya recurrence theorem (George Pólya, 1921; the drunk-bird image is Shizuo Kakutani's). Verified live: the exact return probabilities p₂ₙ(0) = Π(2i−1)/(2i) (1D) and its square (2D) give partial sums Σ p₂ₙ(0) that keep growing (divergent → recurrent, window.__polyawalk.recurrent12), while a 3D random-walk simulation returns to the origin under 50% of the time (transient, window.__polyawalk.transient3, approaching Pólya's 0.3405).
FIG No framing: the exact 1D/2D return-probability sums and the 3D and 1D return simulations all run in-browser. Honest scope: the 1D/2D divergence (hence recurrence) is exact; the 3D return rate is Monte-Carlo, approaching Pólya's constant 0.3405 (finite-step simulation slightly undercounts). The AVAN inverse is honest — asking whether a walk returns (summing return chances; finite means it escapes) rather than tracking it reveals the dimension flip; magenta is the plane walk that always returns, green the space walk that wanders off. Home in the plane, lost in space.
FIG No framing: the exact 1D/2D return-probability sums and the 3D and 1D return simulations all run in-browser. Honest scope: the 1D/2D divergence (hence recurrence) is exact; the 3D return rate is Monte-Carlo, approaching Pólya's constant 0.3405 (finite-step simulation slightly undercounts). The AVAN inverse is honest — asking whether a walk returns (summing return chances; finite means it escapes) rather than tracking it reveals the dimension flip; magenta is the plane walk that always returns, green the space walk that wanders off. Home in the plane, lost in space.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SEGFAULT · David Lee Wise (ROOT0), with AVAN