Pólya recurrence · does a random walk come home? simulation

A drunkard staggering on a grid always finds his way home — does a drunken bird in the air? Does a simple random walk return to where it started, and does the answer depend on dimension?

▶ Run the simulationSee the measured result

Units: dimensionless probability (p₃ = 1 − 1/W₃, W₃ = (√6/32π³)·Γ(1/24)Γ(5/24)Γ(7/24)Γ(11/24) = 1.5163860591519778 — Watson 1939, Glasser & Zucker 1977)

How the lab tests it

Run thousands of independent simple random walks (each step ±1 along one uniformly-chosen axis) on the lattice Zᵈ for d=1,2,3 and record the first-return time of each; build the return probability R_d(N)=P(returned by step N). Three live walkers (a line, a plane, a space) also race home, scoring a RETURN when they hit the origin and an ESCAPE when they wander off.

What it checks

Pólya's theorem: the walk is RECURRENT in d=1,2 (R_d(N)→1, return certain) but TRANSIENT in d≥3 — R_3(N) plateaus at the Pólya constant p₃≈0.3405 (a finite-horizon N^(−½) extrapolation of the upper curve recovers it). One extra dimension turns certain return into probable escape.