I.11 Spectral dimension and continuum diagnostics
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Let \(L\ge0\) be a finite Laplacian on \(N\ge1\) states, with eigenvalues \(\lambda_j\) and at least one zero eigenvalue. For diffusion time \(\tau>0\), define the normalized heat trace
and the scale-dependent spectral dimension
For a finite system, \(d_s(\tau)\) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives
It tends to zero as \(\tau\downarrow0\) and, under the stated assumption of a zero mode, as \(\tau\to\infty\). If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to \(2\tau\lambda_{\min}\). The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controlled refinement limit.
I.11.1 The circle dispersion bound
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For a periodic grid with spacing \(h>0\) and physical wave number \(k\in{\mathbb R}\),
Theorem I.11.1: Low-mode second-order bound
For every real \(k\) and every nonzero real \(h\), define \(\lambda_{k,h}=4h^{-2}\sin^2(kh/2)\). Then
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Proof of Theorem I.11.1
Put \(x=kh/2\). The inequalities \(|\sin x|\le|x|\) and \(|x-\sin x|\le |x|^3/6\), which hold for every real argument, give
Multiplying by \(4/h^2\) proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps \(k\) fixed while \(h\to0\), so it does not assert a uniformly small relative error across all grid modes.
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Text description
Both axes are logarithmic. The measured first-mode dispersion error and its stated quadratic envelope lie nearly together as lattice spacing varies. The displayed finite low-mode comparison illustrates second-order convergence.
Numerical certificate I.11.2: Continuum dispersion
The reported convergence order was \(1.99936\). For every tested mode, the error was below the bound, so none of those tests violated the theorem.
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I.11.2 One- and two-dimensional plateaus
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At intermediate diffusion times, a finite cycle and a finite torus can exhibit plateaus near the dimensions of their underlying geometries. The following figure and numerical certificate show the reported behavior for these two examples. The plateaus occur between the small and large time regimes in which the spectral dimension returns to zero.
Text description
Spectral dimension is plotted against diffusion time. The cycle curve has an intermediate window near one and the torus curve a window near two, followed by decay at long times. Short- and long-time departures reflect the finite models.
Numerical certificate I.11.3: Spectral dimension
The median intermediate plateaus were
These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
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