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We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points.
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It is sufficient that the moments grow slowly enough such that the power series they generate has a finite radius of convergence, see W. Feller, Introduction to the Theory of Probability and Its Applications , Wiley, New York (1971), vol. 2, p. 514
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To see this, one can write the probability to remain in a given site i i in terms of the eigenmodes and eigenvalues, as: P ^ ( t ) = ∑ λ | ⟨ V λ | i ⟩ | 2 e − λ t \hat{P}(t)=\sum_{\lambda}|\langle V_{\lambda}|i\rangle|^{2}e^{-\lambda t} , with V λ V_{\lambda} the corresponding eigenmodes. Averaging over i i gives P ^ ( t ) = ∑ λ e − λ t \hat{P}(t)=\sum_{\lambda}e^{-\lambda t} , which, in the continuous limit, is the Laplace transform of the probability density
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V. Vitelli et al
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