Parameters
Statistics
Steps taken0
1D ⟨r²⟩—
2D ⟨r²⟩—
3D ⟨r²⟩—
1D √⟨r²⟩—
2D √⟨r²⟩—
3D √⟨r²⟩—
Theory √(Dt)—
⟨r²⟩ = 2d·D·t where d is dimension.
All dims: same ⟨r²⟩ scaling = t.
Return probability:
1D: recurrent (returns a.s.)
2D: recurrent (returns a.s.)
3D: transient (~34% return)
(Pólya 1921)
All dims: same ⟨r²⟩ scaling = t.
Return probability:
1D: recurrent (returns a.s.)
2D: recurrent (returns a.s.)
3D: transient (~34% return)
(Pólya 1921)