The Logarithmic Laplacian on General Graphs
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.