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This gives the fractal dimension of loop-erased random walks at $n=-2$, self-avoiding walks ($n=0$), Ising lines $(n=1)$, and XY lines ($n=2$), in agreement with numerical simulations. It can be compared to the fractal dimension $d_{\\rm f}^{\\rm tot}$ of all lines, i.e. backbone plus the surrounding loops, identical to $d_{\\rm f}^{\\rm tot} = 1/\\nu$. 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