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  "body": "\n## Definition\n\nLet $d \\ge 1$ and $n \\ge 0$ be integers, and let $\\abs{\\cdot}$ be the Euclidean norm on $\\R^d$.\nAn $n$-step *self-avoiding walk* on $\\Z^d$ starting at $0$ is a sequence $\\omega = (\\omega_0, \\omega_1, \\ldots, \\omega_n)$ of points of $\\Z^d$ such that:\n\n1. $\\omega_0 = 0$;\n2. $\\abs{\\omega_i - \\omega_{i-1}} = 1$ for $1 \\le i \\le n$;\n3. $\\omega_i \\ne \\omega_j$ for $0 \\le i \u003c j \\le n$.\n\nLet $c_n$ be the number of $n$-step self-avoiding walks.\nThe uniform probability measure on these $c_n$ walks is also called the self-avoiding walk with $n$ steps.\n\n## Remarks\n\nExample: in $\\Z^2$, the walk $(0,0), (1,0), (1,1), (0,1)$ satisfies all three clauses, and it is one of the $c_3 = 36$ walks with three steps.\nNon-example: $(0,0), (1,0), (0,0)$ satisfies clauses 1 and 2 but not clause 3, since it returns to $0$.\nCutting an $(n + m)$-step walk after $n$ steps gives $c_{n+m} \\le c_n c_m$, so by Fekete's lemma the connective constant $\\mu_d = \\lim_{n \\to \\infty} c_n^{1/n}$ exists.\nWalks that only move in positive coordinate directions are self-avoiding, and walks that never reverse their last step include all self-avoiding walks, so $d \\le \\mu_d \\le 2d - 1$.\n",
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