Sharp Spectral Zeta Asymptotics on Graphs of Quadratic Growth
Da Xu

TL;DR
This paper establishes sharp asymptotic formulas for spectral zeta functions of Laplacians on large finite subsets of irregular graphs with quadratic volume growth, extending classical results beyond homogeneous lattices.
Contribution
It introduces a novel asymptotic law for spectral zeta values on irregular graphs satisfying certain geometric and analytic conditions, including a new pi-free limit formula for ^2.
Findings
Asymptotic formula for spectral zeta: Z_n(1) l rac{ ext{constant}}{N_n \u2217 \u00a0log N_n}
Identification of the global heat-kernel constant /
Extension of spectral asymptotics to irregular graphs beyond homogeneous lattices
Abstract
We investigate the spectral properties of the Dirichlet Laplacian on large finite metric balls within irregular infinite graphs of quadratic volume growth. We consider an exhaustion and the spectral zeta value of the killed generator . We establish a sharp asymptotic law under the assumptions that the graph satisfies uniform quadratic volume growth (VG(2)) and a Poincare inequality (PI). These analytic-geometric hypotheses imply large-scale regularity. Additionally, we assume a standard quantitative homogenisation property: a uniform local central limit theorem with a polynomial convergence rate. This hypothesis holds for our main example classes and implies the existence of a global heat-kernel constant (independent of ). In particular, the lazy simple random walk (LSRW) satisfies $p_t(x,x) \sim…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsSpectral Theory in Mathematical Physics · Stochastic processes and statistical mechanics · Nonlinear Partial Differential Equations
