The Logarithmic Laplacian on General Graphs

TL;DR

This paper introduces a new integral representation for the logarithmic Laplacian on weighted graphs, deriving explicit formulas, bounds, and asymptotic behaviors, advancing understanding of this operator in graph analysis.

Contribution

The paper provides the first Bochner-type integral representation and explicit pointwise formula for the logarithmic Laplacian on weighted graphs, including bounds and asymptotics.

Findings

Established a Bochner-type integral representation for the logarithmic Laplacian.

Derived explicit pointwise formulas and bounds for weighted lattice graphs.

Analyzed the asymptotic behavior and Fourier multipliers of the operator.

Abstract

We establish, for the first time, a Bochner-type integral representation for the logarithmic Laplacian on weighted graphs. Assuming stochastic completeness of the underlying graph, we further derive an explicit pointwise formula for this operator: \[ \log(-\Delta)\:u(x) =\frac{1}{\mu(x)}\sum_{y\neq x}W_{\log}(x,y)\,(u(x)-u(y)) -\frac{1}{\mu(x)}\sum_{y}W(x,y)\,u(y) +\Gamma'(1)\,u(x). \] In the case of weighted lattice graphs with uniformly positive vertex measures, we obtain sharp two-sided bounds for the associated logarithmic kernel. Additionally, we prove that the logarithmic Laplacian is unbounded on $\ell^{2}$, and we present an alternative derivation of its pointwise form. Moreover, for every $1 < p \leq \infty$ and all $u \in C_c(\mathbb{Z}^{d})$, we establish a strong convergence in $\ell^{p}$: \[\frac{(-\Delta)^{s} u - u}{s} \longrightarrow \log(-\Delta) \:u \quad \text{as } s \to 0^{+}.\]Finally, on the standard lattice $\mathbb{Z}^{d}$, we compute the Fourier multipliers corresponding to both the fractional Laplacian and the logarithmic Laplacian, and derive exact large-time behavior and off-diagonal asymptotics of the associated diffusion kernels, including all sharp asymptotic constants.