For critical long-range percolation on the integers, the quenched and annealed spectral dimensions of the random walk both exist and equal 2/(1 + δ), with δ the effective-resistance exponent from a companion paper.
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Spectral dimensions for one-dimensional critical long-range percolation
For critical long-range percolation on the integers, the quenched and annealed spectral dimensions of the random walk both exist and equal 2/(1 + δ), with δ the effective-resistance exponent from a companion paper.