--- title: "Sharp variance ordering for rotating Gaussian samplers" date: 2026-11-04 version: "0.2.0-candidate" doi: 10.5281/zenodo.23946160 pdf: https://github.com/ipitchford/rotating-gaussian-variance-ordering/releases/download/v0.2.0-candidate/rotating-gaussian-variance-ordering-0.2.0-candidate.pdf repository: https://github.com/ipitchford/rotating-gaussian-variance-ordering archive: https://zenodo.org/records/23946160 license: CC0-1.0 status: unrefereed (internally replayed; not peer reviewed, not independently reproduced, not formally verified) --- # Sharp variance ordering for rotating Gaussian samplers ## Summary **The setting.** Many statistical and physical computations estimate an average, such as a mean energy or a posterior expectation, by simulating one long random trajectory whose long-run distribution is the target and averaging along it. This is Markov chain Monte Carlo. Successive states are correlated, so the accuracy of the average is measured by its **long-run (asymptotic) variance**: the variance after a run of length $T$, multiplied by $T$. Two samplers with the same target can have very different long-run variances. **The trick being studied.** One way to change those correlations without changing the target is to add a deterministic **rotation** that preserves the target distribution, a *nonreversible perturbation*. For overdamped (first-order) Langevin samplers this is known to help every observable, and the more rotation the better (Hwang, Normand and Wu 2015; Rey-Bellet and Spiliopoulos 2015–2016). **Kinetic** (underdamped) Langevin samplers, widely used in molecular dynamics and Hamiltonian-type MCMC, add a momentum and a friction $\gamma$. There the picture is different. Duncan, Nüsken and Pavliotis (DNP, 2017) proved, for a Gaussian target, that a small matched rotation lowers the variance of linear and quadratic observables when $\gamma>\sqrt2$. They conjectured the same for every square-integrable observable, and their numerics showed rotations can also *raise* the variance. **What the paper shows.** For exact stationary Gaussian dynamics with matched position and momentum rotations, and observables of the position: - **The sharp threshold (Theorem 1).** Every square-integrable observable's long-run variance never increases as the rotation strength grows **if and only if $\gamma\ge\sqrt2$**, the endpoint included. It strictly decreases unless the observable is unaffected by the rotation. This is a global form of DNP's conjecture: it holds at every rotation strength, not just near zero. - **Why √2.** Think of the sampler's position as a damped spring. Its *power spectrum* says how much fluctuation sits at each frequency, $S(\xi)=2\gamma/(\xi^4+(\gamma^2-2)\xi^2+1)$. Rotating at strength $\mu$ shifts the spectrum, so a linear observable's variance becomes $S$ evaluated away from zero. If the spectrum has a single peak at zero, every shift lowers it. A damped spring has such a spectrum exactly when it shows no resonance, which for this oscillator is $\gamma\ge\sqrt2$. Nonlinear observables see convolutions of the same spectrum, and convolution keeps single peaks single (Theorem 2). At $\gamma=1$, for example, the spectrum has a resonance hump: the variance rises from $2$ to $8/3$ before falling. - **Below the threshold, rotation still helps a lot (Theorem 4A).** For *any* friction, observables that are even ($f(-q)=f(q)$) improve at every rotation strength. Only finitely many odd Hermite components can ever be hurt. One finite rotation strength improves every observable at once. Computed crossing points (Table 1 of the paper) show that for $\gamma\ge1$ only the linear part can be hurt, while at $\gamma=0.1$ odd degrees up to 41 can be. - **Matrix version (Theorem 10).** For a stiffness matrix $K$ and friction matrix $\Gamma$ that need not commute, the exact condition is $\Gamma^3-\Gamma K-K\Gamma\succeq0$. The least friction at which rotation is safe for every observable is $(2K)^{1/2}$. Two coupled replicas make the result apply to any Gaussian target. - **Edges of the result.** The literal second-derivative form of the conjecture holds exactly on an explicit class containing every polynomial; rougher observables have sharper peaks (Theorem 5). For large rotations the paper gives sharp rates, including the constant $\sqrt{2\pi/e}$ (Theorem 8). Fixed time horizons and regular sampling intervals can reverse the ordering, while sampling at independent random (Poisson) times preserves it (Section 9). **What it does not claim.** These are exact statements about Gaussian targets in continuous time. Nothing is claimed about step-size discretisation, computational cost or non-Gaussian targets. The threshold $\sqrt2$, and $(2K)^{1/2}$ in matrix form, is a **safety boundary, not a recommended friction**: DNP found small friction (about $0.1$) best in practice, and the below-threshold results say what rotation can still do there. The Hermite expansion, the Breuer–Major covariance identities, closure of single-peaked shapes under convolution, Maini's (2026) convolution-power variance formula, the overdamped uniform convergence and monotonicity of Franke et al. (2021), DNP's large-rotation limit and replica coupling are classical or prior ingredients, credited and not claimed. The paper does not "correct" DNP: it proves their conjecture in a global form and says precisely where its literal derivative wording needs qualification. The sections below move from this overview to a specialist summary, the technical argument, the evidence and its limits, and how to rerun the checks. ## Summary for specialists Model: $dq=(p-\mu Jq)\,dt$, $dp=(-q-\gamma p-\mu Jp)\,dt+\sqrt{2\gamma}\,dW$ in $\mathbb R^d$, $J^\top=-J\ne0$, invariant law $\mathcal N(0,I_{2d})$, $\pi=\mathcal N(0,I_d)$. For real $f\in L^2(\pi)$, $V_f(\mu)=\lim_T T\operatorname{Var}(T^{-1}\int_0^Tf(q_t)\,dt)$, which is twice DNP's $\sigma_f^2$. With $r_\gamma$ the scalar position autocorrelation and $a_{n,\omega}\ge0$ the squared Hermite/angular weights of $f$, $$ V_f(\mu)=\sum_{n\ge1,\ \omega}a_{n,\omega}\,S_n(\mu\omega),\qquad S_n=2\pi p^{*n},\quad p=S_\gamma/(2\pi),\quad S_\gamma(\xi)=\frac{2\gamma}{\xi^4+(\gamma^2-2)\xi^2+1}. $$ - **Theorem 1.** $V_f(b)\le V_f(a)$ for all $0\le aV_f(0)$ for $0<|\mu|<\sqrt{2-\gamma^2}/\lambda$. - **Theorem 2.** For any centred, mean-square continuous stationary Gaussian process with integrable scalar covariance, rotated by $e^{-\mu tJ}$: universal ordering $\iff$ the spectral density is nonincreasing on $[0,\infty)$; strict under strict decrease. - **Theorem 4A.** For every $\gamma>0$: even-chaos observables satisfy $V_f(\mu)N(\gamma)$; and $V_f(\mu)\mu_0(\gamma,J)=\max_{n\le N(\gamma)}\Xi_n(\gamma)/\omega_{\min}$. - **Theorem 5, Proposition 6.** Every $V_f\in C^1$ with $V_f'(0)=0$; $V_f\in C^2$ (equivalently, a finite second derivative at $0$) iff $\mathcal Q_J(f)=\sum a_{n,\omega}\omega^2n^{-3/2}<\infty$; otherwise $(V_f(\mu)-V_f(0))/\mu^2\to-\infty$, with explicit fractional and logarithmic cusps. - **Proposition 7, Theorem 8, Section 7.** $V_f(\mu)-V_{\Pi_Jf}(0)=O(\mu^{-4})$ for polynomials; $D_J(\mu)=\sup_{\|f\|=1}(V_f(\mu)-V_{\Pi_Jf}(0))\sim\sqrt{2\pi/e}/(|\mu|\omega_*)$ when $G_J=\omega_*\mathbb Z$; a Diophantine bound, an explicit $|\mu|^{-1/3}$ example and a Liouville example with no algebraic rate. - **Theorem 10, Corollaries 11–12.** With $[J,K]=[J,\Gamma]=0$ but possibly $K\Gamma\ne\Gamma K$: ordering for every $L^2$ position observable if $\Gamma^3-\Gamma K-K\Gamma\succeq0$, and only if when $J$ is invertible; $\Gamma_*=(2K)^{1/2}$ is the least element in Loewner order; $\Gamma^2\succeq2K$ is insufficient without commutation. Duplication $K\otimes I_2$, $\Gamma\otimes I_2$, $J=I\otimes\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ gives explicit single-copy and replica-average limits. ## Technical account **Covariance reduction (Sections 2–3).** Matched rotations commute with the unrotated drift, so $\operatorname{Cov}(q_t,q_0)=r_\gamma(t)e^{-\mu tJ}$. The Mehler semigroup acts as $\rho^n$ on the $n$-th Hermite chaos and the rotation acts unitarily inside each chaos, so after diagonalising, each chaos component of angular frequency $\omega$ contributes $a_{n,\omega}\int r_\gamma(t)^n\cos(\mu\omega t)\,dt$. This is $a_{n,\omega}S_n(\mu\omega)$ with $S_n$ the Fourier transform of $r_\gamma^n$, a convolution power of the scalar spectrum (formula (6)). This identity is classical in substance (Breuer–Major; Maini 2026, Lemma 3.1, at a fixed frequency). **Criterion (Section 3.2).** The convolution of even, nonnegative, integrable functions that are nonincreasing on $[0,\infty)$ has the same property (Lemma 3, proved via superlevel sets; Lefèvre–Utev discuss the closure). So a single-peaked $S_\gamma$ makes every $S_n$ single-peaked and every term of (6) nonincreasing in $|\mu|$. Conversely a linear observable reads $S_\gamma$ directly. **The kinetic threshold (Section 4).** $S_\gamma'(\xi)=-4\gamma\xi(2\xi^2+\gamma^2-2)/(\cdots)^2$, which is $\le0$ on $[0,\infty)$ iff $\gamma\ge\sqrt2$. At $\gamma=\sqrt2$, $S=2\sqrt2/(1+\xi^4)$ is strictly decreasing, so the endpoint is included. **Below the threshold (Section 4.1).** Even powers $r_\gamma^n\ge0$ make $S_n(0)-S_n(\xi)=2\int r_\gamma^n(1-\cos\xi t)\,dt>0$. For high odd degrees a Gaussian-type lower bound near $t=0$, of order $n^{-3/2}\min(1,\xi^2)$, beats the geometrically small tail. Each low degree has a finite last crossing point by Riemann–Lebesgue. **Matrix case (Section 8).** The physical-position spectrum is $S_q(\xi)=2H_\xi\Gamma H_\xi^*$ with $H_\xi=(K-\xi^2I+i\xi\Gamma)^{-1}$. Without assuming $K\Gamma=\Gamma K$, $S_q=2Q^{-1}$ with $Q(\xi)=K\Gamma^{-1}K+\xi^2B+\xi^4\Gamma^{-1}$ and $B=\Gamma-K\Gamma^{-1}-\Gamma^{-1}K$. Since $Q'(\xi)=2\xi(B+2\xi^2\Gamma^{-1})$ and inversion reverses order, the whitened spectrum decreases in Loewner order iff $B\succeq0$, that is iff $\Gamma B\Gamma=\Gamma^3-\Gamma K-K\Gamma\succeq0$. A tensor-convolution lemma (Lemma 10A) carries Loewner unimodality to all chaoses, including compression to symmetric tensors. ## Evidence, assurance and limitations - **Proofs.** All theorems have written proofs in the manuscript and its appendices. The finite-horizon counterexample of Section 9.2 carries an exact rational certificate. - **Supplementary checks.** `verify.py`: exact rational spectral fingerprints for the first six Hermite orders against cosine quadrature and a complex Laplace sum (optionally simplified to zero by SymPy); covariance against matrix exponentials and a Lyapunov equation; linear and quadratic variances against Poisson-equation and Lyapunov solves; the finite-horizon reversal; aliasing and Poisson-sampling identities; high-degree limits; the figure. `verify_matrix_extension.py`: a noncommuting admissible example, tensor convolution against Lyapunov at three amplitudes, the insufficiency of $\Gamma^2\succeq2K$, the least-friction identity and a precision-mismatch formula. `verify_below_threshold.py`: the Table 1 crossing points from an exact exponential-sum formula cross-checked by quadrature. - **Computed, not proved.** Table 1's crossing points and the observed high-degree error slopes are numerical evidence; the theorems do not depend on them. - **Replay.** All three scripts were rerun from fresh copies with and without Python `-O`. Checks raise errors rather than using `assert`, and a deliberately corrupted least-friction formula is rejected under `-O`. GitHub CI reruns them. - **Limits.** Exact stationary Gaussian dynamics; matched, target-preserving rotations; position observables (momentum observables can get worse). No discretisation, cost or optimal-friction statement. The prior-art searches of 1 and 4 November 2026 covered the originating paper and thesis, the overdamped variance-reduction literature, work citing DNP, related nonreversible-sampling work and formula-level aliases; Maini (2026) and Franke et al. (2021, authors' version) were read in full. No earlier statement of the main results was found. Review was internal and AI-assisted only. ## Who should care, and why | Audience | Potential use | Required caution | | --- | --- | --- | | Designers of kinetic Langevin and molecular-dynamics samplers | A friction level ($\gamma\ge\sqrt2$, or $\Gamma^3-\Gamma K-K\Gamma\succeq0$) above which a target-preserving rotation cannot hurt any position average, and a guarantee that a large enough rotation helps below it | Exact continuous-time Gaussian dynamics; integrator step size, cost and non-Gaussian targets are not covered; the threshold is not a recommended friction | | Researchers in nonreversible MCMC | A global proof of the DNP conjecture's intended content, an exact domain for its derivative wording, and sharp rates | Classical ingredients are credited; no earlier statement of the main results was found | | Gaussian-process and spectral analysts | A clean criterion: universal rotation ordering iff the spectral density is single-peaked | Applies to rotations preserving a Gaussian law, with integrable covariance | ## Why the problem matters Rotations are cheap to add and are used to speed up samplers, but for kinetic samplers they can make some estimates worse. A practitioner needs to know when the trick is safe for every quantity being estimated, not just for a few test functions. The paper turns that question into a one-line spectral test: no resonance peak means no risk. It also explains what remains true at the small frictions people actually use. ## How to inspect or reproduce the recorded checks From the unpacked archive (Python 3.10 or later): ```sh cd research_sampling uv run --with numpy --with scipy --with matplotlib --with sympy python3 -I -B code/verify.py --symbolic --output /tmp/rg/verification_results.json --figures-dir /tmp/rg/figures uv run --with numpy --with scipy python3 -I -B code/verify_matrix_extension.py --output /tmp/rg/matrix_extension_results.json uv run --with numpy --with scipy python3 -I -B code/verify_below_threshold.py --output /tmp/rg/below_threshold_results.json ``` The first two take seconds and the third about two to four minutes. Add `-O` to confirm the checks still run. The reports match the shipped ones in every exact field, with floats equal to round-off. ## The most valuable next projects 1. An exact discrete-time criterion for a specified splitting integrator with rotation, where aliasing (Section 9.3) is the obstacle. 2. A perturbative ordering statement for near-Gaussian targets. 3. A certified value of $N(\gamma)$ and of the Table 1 crossing points. 4. A quantitative comparison of adding rotation with retuning the friction, under a stated cost model. ## What is in the evidence package The archive contains the manuscript (Markdown source, generated LaTeX and PDF), the three verification scripts with their JSON reports, the four-panel figure, the matrix-extension notes, the dated novelty reports and search log, frozen targets and formula fingerprints, a rating-free record of the internal review, the bibliography with the Crossref, DataCite and arXiv records it was generated from, an AI index, research-gate records, CI configuration and SHA-256 manifests. Text and data are CC0-1.0 and code is MIT. This remains an unrefereed candidate. ## Verification status Unrefereed AI-assisted probability and sampling candidate with written proofs. Internal AI-assisted review in five roles, a separate-model adversarial review, one revision round adding the below-threshold theorem, a re-review and a separate-model audit of the revision. External specialist review, independent reproduction and formal verification have not been established. ## References 1. Duncan, Nüsken and Pavliotis (2017), Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions. Journal of Statistical Physics 169. The originating paper: local improvement for linear and quadratic observables of a Gaussian target when γ > √2, the all-observable conjecture proved here in a global form, and small-friction numerics. 2. Nüsken (2018), Topics in sampling schemes based on Markov processes. Doctoral thesis, Imperial College London. Restates the conjecture (p. 62). 3. Hwang, Normand and Wu (2015), Variance reduction for diffusions. Stochastic Processes and their Applications 125. For overdamped samplers, nonreversible perturbations never increase the asymptotic variance. 4. Rey-Bellet and Spiliopoulos (2015), Irreversible Langevin samplers and variance reduction: a large deviations approach. Nonlinearity 28. Overdamped variance reduction, monotone in the perturbation strength. 5. Duncan, Lelièvre and Pavliotis (2016), Variance Reduction Using Nonreversible Langevin Samplers. Journal of Statistical Physics 163. Nonreversible overdamped samplers and their asymptotic variance. 6. Chak, Kantas, Lelièvre and Pavliotis (2023), Optimal friction matrix for underdamped Langevin sampling. ESAIM: Mathematical Modelling and Numerical Analysis 57. Optimises friction for a given observable, a different question from the safety threshold here. 7. Lelièvre, Nier and Pavliotis (2013), Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion. Journal of Statistical Physics 152. Optimises the rate of convergence rather than variance ordering. 8. Breuer and Major (1983), Central limit theorems for non-linear functionals of Gaussian fields. Journal of Multivariate Analysis 13. Hermite covariance and spectral-convolution identities used here, credited. 9. Maini (2026), Spectral delocalization and central limit theorems for Fourier functionals of stationary Gaussian fields. arXiv:2610.37113. Lemma 3.1 and Corollary 3.4 give the same convolution-power variance at a fixed frequency; credited. It compares no frequencies and has no rotated family, Markov dynamics or sampler. 10. Franke, Hwang, Ouled Said and Pai (2021), A note on the asymptotic variance of drift accelerated diffusions. Statistics & Probability Letters. For overdamped diffusions: variance tends to zero uniformly exactly when the drift has no H¹ kernel, and is nonincreasing in the amplitude. Read in the authors' version; compared in Section 1.4. 11. Lefèvre and Utev (2013), Convolution property and exponential bounds for symmetric monotone densities. ESAIM: Probability and Statistics 17. Closure of symmetric unimodality under convolution, credited. 12. Ohzeki and Ichiki (2015), Langevin dynamics neglecting detailed balance condition. Physical Review E 92. Replica duplication with nonreversible coupling, an established device used in Section 8.4. 13. Shapiro and Silverman (1960), Alias-Free Sampling of Random Noise. Journal of the Society for Industrial and Applied Mathematics 8. Classical alias-free Poisson sampling behind Section 9.4.