Source
Machta, Chachra, Transtrum and Sethna (2013), Science 342:604, LIT-1299, arXiv:1303.6738v1, read in full in NOTE-1018. The model is the main text’s Fig. 2 and supplement §III, Eqs. 14–23.
What was actually shown
The model. Sites j ∈ ℤ. Parameters θ_μ, −N ≤ μ ≤ N, are the probabilities of a hop by μ in one step. They are treated as free: their sum is not fixed, so a direction that changes the number of particles exists. The density after t steps from ρ₀ = δ_{j,0} is ρ_t = θ^{*t}, the t-fold convolution. The data are ρ_t(j) at every j, each with independent Gaussian error of width 1. The Fisher metric is then the least-squares metric g = JᵀJ, J_{jμ} = ∂ρ_t(j)/∂θ_μ (the paper’s Eqs. 10–11).
Re-derivation, step by step.
- The Jacobian. ρ_t is a product of t copies of θ under convolution, so ∂ρ_t/∂θ_μ = t · (δ_μ * θ^{(t−1)}), which gives J_{jμ} = t p_{t−1}(j − μ) with p_{t−1} = θ^{(t−1)}.
- The identity. g_μν = Σ_j J_{jμ}J_{jν} = t² Σ_i p_{t−1}(i) p_{t−1}(i + μ − ν) = t² q_t(μ − ν), where q_t is the autocorrelation of p_{t−1}: the law of X − X′ for two independent (t−1)-step walks. The drift cancels in X − X′, so g does not depend on it. The difference walk has mean 0 and variance 2(t−1)D, with D = Σθ_μ μ² − (Σθ_μ μ)² at Σθ_μ = 1. (The paper’s Eq. 19 contains this, as the integrand |θ̃_k|^{2(t−1)}, but its Eq. 21 is written “for the case where the drift v = 0”, a restriction the identity shows is not needed.)
- Fourier form. q_t(d) = (1/2π) ∫_{−π}^{π} e^{ikd} w_t(k) dk with w_t(k) = |θ̃(k)|^{2(t−1)}, and |θ̃(k)|² = 1 − Dk² + O(k⁴) at Σθ = 1. If the walk is aperiodic, |θ̃(k)| < 1 for k ≠ 0, so all of the weight of w_t outside |k| < t^{−1/2+ε} is exponentially small, and with k = s/√t, w_t → e^{−Ds²} (Laplace’s method).
- Grading. Put M = 2N + 1 and expand e^{ikμ} = Σ_{a<M} (ikμ)^a/a! + r, with |r| ≤ (N|k|)^M/M!. Then g = t² · V S [K_t] S Vᵀ + (remainder), where V_{μa} = μ^a is the M × M Vandermonde matrix on −N, …, N (invertible), S = t^{−1/4} diag(t^{−a/2}), and K_t = (1/2π)∫ u(s) u(s)* w_t(s/√t) ds, u_a(s) = (is)^a/a!. In the graded frame S^{−1}V^{−1}(·)V^{−ᵀ}S^{−1}, the cross terms with r are of order t^{(b−M)/2} ≤ t^{−1/2} and the r r* term of order t^{−1}, so they vanish as t grows. K_t tends to K_∞ = (1/2π)∫ u u* e^{−Ds²} ds, the Gram matrix of the monomials (is)^a/a! under a Gaussian weight. Monomials are linearly independent, so K_∞ is positive definite.
- The eigenvalues. g = X X* with X = t V S K_t^{1/2} (up to the vanishing remainder), so λ_k(g) = σ_k(X)². For invertible A and C, σ_k(B)/(‖A^{−1}‖‖C^{−1}‖) ≤ σ_k(ABC) ≤ ‖A‖‖C‖σ_k(B). With A = V, B = S, C = K_t^{1/2}, whose norms and inverse norms are bounded for large t by step 4, σ_k(S) = t^{−1/4 − k/2} gives λ_k(g) = Θ(t² · t^{−1/2−k}) = Θ(t^{3/2−k}), k = 0, …, 2N. The constants depend on N and on D, not on t.
Checked numerically (a script in the reader’s scratchpad, not the record). For N = 3 and θ_μ = 1/7, the paper’s Fig. 2 case, the local exponents of the seven eigenvalues between t = 40 and t = 80 are 1.501, 0.496, −0.514, −1.517, −2.515 and −3.517 (the seventh is below float precision there); the prediction is 1.5, 0.5, −0.5, −1.5, −2.5, −3.5. λ₂ t^{1/2} is 0.096, 0.094, 0.094 and 0.093 at t = 20, 40, 80, 160. For a random kernel with drift v = 0.60, the identity of step 2 holds to 3 × 10⁻¹⁴ and the exponents are the same. At t = 7 the spectrum spans 7.6 × 10⁹, which matches the figure. The top three eigenvectors at t = 80 are the uniform vector (particle creation), the vector ∝ μ (drift) and a quadratic one (the diffusion constant), as the paper says. When Σθ_μ is held at 1, the creation direction goes and the rest keep their exponents.
What could have come out otherwise. The spectrum could have stayed clustered, as at t = 1 where g is the identity. It could have separated at a rate other than t per order, or depended on the drift. None of these happens.
What this does not say
- Not that information is lost under coarse-graining. The density at time t is not a function of the density at time 1. It is a different experiment with the same error per site, so the data-processing inequality does not apply and the creation and drift eigenvalues grow. This is unlike THEORY-742’s diffusion model, where a trajectory of fixed duration is subsampled and no eigenvalue grows. The two agree in relative terms: here λ_k/λ₁ ∝ t^{1−k}, there g_μ/g₁ = n^{1−μ}, with k and μ both the Hermite order.
- Not a coordinate-free hierarchy. The metric is a least-squares metric with an arbitrary error width (σ = 1, which scales every eigenvalue alike), and its eigenvalues depend on using the θ_μ as coordinates. The paper’s sloppiness is the spread of these eigenvalues. The invariant statement is the relative one: against the metric at t = 1, which is the identity in these coordinates, the generalized eigenvalues are the same numbers.
- Not that the paper’s own derivation is sound as written. Its Eq. 22 expands g in powers of (μ − ν)² and says that “each term in the series contributes a single new non-zero eigenvalue”. But the matrix [(μ − ν)^{2n}] has rank 2n + 1, not 1. The result is right, by the grading argument above, not by that one.
- Nothing about critical systems. Fluctuations here are short-range. The Ising half of the paper is THEORY-777.
Connections
THEORY-742 is the information-geometric reformulation of the same diffusion example by two of the same authors, with stochastic Fisher information of a subsampled trajectory in place of a least-squares metric on densities. It gives exact n^{1−μ} scaling at the Gaussian fixed point. On QUESTION-217, a reader’s remark, not the paper’s: in this model g is exactly the Gauss–Newton Hessian of the sum-of-squares cost, so gradient flow in the θ coordinates near a zero-residual fit converges along eigendirection k at rate λ_k ∝ t^{3/2−k}. Creation, drift, diffusion constant and then the higher moments are then fitted in the order of their relevance. The nested effective models (creation; creation and drift; creation, drift and diffusion) that the paper reads off the same eigenvectors are a refinement chain by construction. All three orders come from one eigenbasis, as in THEORY-745, and depend on the coordinates: natural-gradient flow would fit every direction at the same rate.