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  "body": "\n## Definition\n\nLet $G$ be a [[graph|00C0BR7D7H]] and let $\\gamma = (\\gamma_0, \\ldots, \\gamma_n)$ be a finite path in $G$.\nIts *loop erasure* $\\mathrm{LE}(\\gamma) = (\\sigma_0, \\ldots, \\sigma_m)$ is defined inductively.\nLet $s_0 = \\max \\set{k : \\gamma_k = \\gamma_0}$ and $\\sigma_0 = \\gamma_0$.\nIf $s_i \u003c n$, let $\\sigma_{i+1} = \\gamma_{s_i + 1}$ and $s_{i+1} = \\max \\set{k : \\gamma_k = \\sigma_{i+1}}$.\nStop at the first $i = m$ with $s_m = n$.\n\nLet $G$ be connected, let $A$ be a nonempty vertex set, and let $X$ be a simple [[random walk|04VZE3FZXF]] on $G$ from a vertex $x$.\nAssume that the hitting time $\\tau_A = \\min \\set{k \\ge 0 : X_k \\in A}$, a [[stopping time|WND87FBAKC]], is finite almost surely.\nThe *loop-erased random walk* from $x$ to $A$ is the random path $\\mathrm{LE}(X_0, X_1, \\ldots, X_{\\tau_A})$.\n\n## Remarks\n\nThe loop erasure visits each vertex at most once, starts at $\\gamma_0$ and ends at $\\gamma_n$.\nExample: for $\\gamma = (0, 1, 2, 1, 2, 3)$ on $\\Z$, one gets $s_0 = 0$, $\\sigma_1 = 1$, $s_1 = 3$, $\\sigma_2 = \\gamma_4 = 2$, $s_2 = 4$, $\\sigma_3 = \\gamma_5 = 3$, $s_3 = 5$, so $\\mathrm{LE}(\\gamma) = (0, 1, 2, 3)$.\nNon-example: the path $(0, 1, 2, 1, 2, 3)$ itself is not a loop erasure of anything, since it visits $1$ and $2$ twice.\nOn a transient graph, such as $\\Z^d$ with $d \\ge 3$, one also erases the loops of the infinite walk, since each vertex is visited only finitely often.\nWilson's algorithm builds a uniform spanning tree of a finite connected graph from successive loop-erased random walks.\n",
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