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In this work we consider deterministic, symmetric matrices with heavy-tailednoise imposed on entries within a fixed distance $K$ to the diagonal. The mostimportant example is discrete 1d random Schr\"odinger operator defined on$0,1,\cdots,N$ where the potentials imposed on the diagonal have heavy-taileddistributions and in particular may not have a finite variance. We assume thenoise is of the form $N^{-\frac{1}{\alpha}}\xi$ where $\xi$ are some i.i.d.random potentials. We investigate the local spectral statistics under variousassumptions on $\xi$: when it has all moments but the moment explodes as $N$gets large; when it has finite $\alpha+\delta$-moment for some $\delta>0$; andwhen it is the $\alpha$-stable law. We prove in the first two cases that alocal law for each element of Green function holds at the almost optimal scalewith high probability. As a bi-product we derive Wegner estimate, eigenvaluerigidity and eigenvector de-localization in the infinity norm. For the case of$\alpha$-stable potentials imposed on discrete 1d Laplacian, we prove that (i)Green function entries are bounded with probability tending to one, implyingeigenvectors are de-localized in the infinity norm; (ii) with positiveprobability some entries of the Green function do not converge to that of thedeterministic matrix; and (iii) the trace of Green function converges to theStieltjes transform of arcsine law with probability tending to one. Thesefindings are in contrast to properties of Levy matrices recently uncovered. Weextend our results to other scaling in front of the noise and derive local lawson the corresponding intermediate scales, and further extend to Wigner matricesperturbed by finite band heavy-tail noise.

Symmetric matrices with banded heavy tail noise: local law and eigenvector delocalization