## Decay bounds for Bernstein functions of Hermitian matrices with applications to the fractional graph Laplacian

Marcel Schweitzer

### Abstract

For many functions of matrices $f(A)$, it is known that their entries exhibit a rapid—often exponential or even superexponential—decay away from the sparsity pattern of the matrix $A$. In this paper, we specifically focus on the class of Bernstein functions, which contains the fractional powers $A^\alpha$, $\alpha \in (0,1)$, as an important special case, and derive new decay bounds by exploiting known results for the matrix exponential in conjunction with the Lévy-Khintchine integral representation. As a particular special case, we find a result concerning the power law decay of the strength of connection in nonlocal network dynamics described by the fractional graph Laplacian, which improves upon known results from the literature by doubling the exponent in the power law.

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### Key words

matrix functions, Bernstein functions, off-diagonal decay, graph Laplacian, fractional powers, nonlocal dynamics

### AMS subject classifications

05C82, 15A16, 65F50, 65F60

### Links to the cited ETNA articles

 [7] Vol. 28 (2007-2008), pp. 16-39 Michele Benzi and Nader Razouk: Decay bounds and $O$($n$) algorithms for approximating functions of sparse matrices [22] Vol. 48 (2018), pp. 362-372 Andreas Frommer, Claudia Schimmel, and Marcel Schweitzer: Non-Toeplitz decay bounds for inverses of Hermitian positive definite tridiagonal matrices

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