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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $(S_n)_{n \\ge 0}$ be the simple [[random walk|04VZE3FZXF]] on $\\Z^d$ with $S_0 = 0$: its steps are independent and uniform on the $2d$ vectors $\\pm e_1, \\ldots, \\pm e_d$.\n\n1. If $d = 1$ or $d = 2$, then $P(S_n = 0 \\text{ for some } n \\ge 1) = 1$.\n2. If $d \\ge 3$, then $P(S_n = 0 \\text{ for some } n \\ge 1) \u003c 1$.\n\n## Remarks\n\nThe walk is a [[Markov chain|0G48PYM96Q]], and the origin is [[recurrent|PBZTBWZ4GS]] if and only if $\\sum_{n} P(S_{2n} = 0) = \\infty$.\nFor $d = 1$, $P(S_{2n} = 0) = \\binom{2n}{n} 4^{-n}$, which behaves like $(\\pi n)^{-1/2}$.\nFor $d = 2$, $P(S_{2n} = 0) = \\left(\\binom{2n}{n} 4^{-n}\\right)^2$, which behaves like $(\\pi n)^{-1}$, so the sum still diverges.\nFor general $d$, $P(S_{2n} = 0)$ is of order $n^{-d/2}$, and the sum converges for $d \\ge 3$.\nIn dimension $3$, the probability of return is about $0.34$.\nIn dimensions $1$ and $2$ the walk returns infinitely often, and in dimension $d \\ge 3$ it satisfies $\\abs{S_n} \\to \\infty$ almost surely.\n\n## Explanations\n\n### L1 human\n\nA walker on a grid takes unit steps in random directions.\nOn a line or in the plane, she is sure to come back home eventually.\nIn space of three or more dimensions, there is a fair chance that she wanders off and never returns.\n\n### L2 human\n\nThe proof reduces recurrence to the divergence of the expected number of returns, which local limit estimates for $P(S_{2n} = 0)$ settle.\nAnother proof views the lattice as an electrical network: the walk is recurrent if and only if the effective resistance from the origin to infinity is infinite.\nThe same dichotomy between $d \\le 2$ and $d \\ge 3$ appears for Brownian motion, which is neighbourhood recurrent in the plane and transient in $\\R^3$.\n",
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