RANDOM WALK
Pólya's Recurrence · ℤ¹ ℤ² ℤ³
Walk parameters
Steps 5,000
Seed
Controls
Preparing your walks…
Speed ×1.0
Live stats — by dimension
1D
2D
3D
|Xₙ|
0
0
0
Returns
0
0
0
Distinct sites
1
1
1
# Quadrants touched
1/2
1/4
1/8
Pólya's theorem
1D & 2D: walker returns to origin with probability 1.
3D: only p₃ ≈ 0.3405 probability of ever returning.
⟨|Xn|²⟩ = n in every d.
Color = where in space
Same x/y region, same color family. In 3D, the shade changes above (+z) or below (−z) the xy plane. Hover the swatches for details.
+x −x  (1D)
x, y2D walk3D +z3D −z
Older steps fade slightly. The two views show separate walks.
Keyboard
Space+drag pan 3D   R reset
End jump to last
←/→ step ±1   ↑/↓ ±100
1D · ±1 on a line · scroll/drag · dbl-click reset x=0ret=0
2D · ℤ² (N/S/E/W)
ℤ² (N/S/E/W)
Knight's-move
ℤ² + obstacles
Hex lattice
Triangular lattice
Kurt's Step (forage)
Kurt's Forest
Homesick walker
Mood drift
Hungry walker
Momentum walker
Roaming homer
 (0,0)ret=0
3D · drag rotate · space/shift+drag pan · scroll zoom · dbl-click reset (0,0,0)ret=0
Perspective · drag to rotate
0 / 5,000
Built for a bigger screen. This dashboard is dense — try landscape, or open it on a laptop.