arxiv: 2603.29473 · v2 · submitted 2026-03-31 · 🧮 math.PR · math.CA· math.DS

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chi²-cut-off phenomenon for Galerkin projections of Fokker-Planck equations with monomial potentials

Benny Avelin , Gerardo Barrera

Authors on Pith no claims yet

Pith reviewed 2026-05-13 23:54 UTC · model grok-4.3

classification 🧮 math.PR math.CAmath.DS

keywords cutoff phenomenonFokker-Planck equationGalerkin projectionmonomial potentialLangevin dynamicsmixing timechi-squared distancestochastic differential equation

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The pith

The cutoff phenomenon in a truncated chi-squared distance holds or fails for Galerkin projections of Fokker-Planck equations with monomial potentials depending on noise strength and eigenfunction growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper studies the cutoff phenomenon for the convergence to equilibrium of Langevin-Kolmogorov dynamics with monomial convex potentials that may be singular and subject to small Brownian noise. The authors introduce a truncated chi-squared distance built from Galerkin projections of the Fokker-Planck operator's eigensystem and determine the precise conditions under which a sharp transition occurs. They establish that both the spacing of eigenvalues and the growth rates of the corresponding eigenfunctions must be controlled in detail for the phenomenon to appear. The analysis also produces explicit asymptotics for mixing times and, in favorable regimes, the shape of the limiting profile. A reader would care because these results give sharp quantitative information on equilibration rates in models of diffusion in polynomial wells that arise in statistical mechanics and sampling.

Core claim

We establish the existence or non-existence of the cut-off phenomenon for the Langevin-Kolmogorov random dynamics with monomial convex potentials, possibly singular, driven by Brownian motion with small strength. Using a truncated χ²-distance based on Galerkin projections of the eigensystem, the analysis shows that refined knowledge of the eigenvalues and asymptotics of the eigenfunction growth for the Fokker-Planck equations are both necessary. This provides asymptotics of the mixing times and information on the limiting profile in some regimes, exceeding the product condition and cut-off window.

What carries the argument

The truncated χ²-distance constructed from Galerkin projections onto finite eigenspaces of the Fokker-Planck operator for the Langevin dynamics.

If this is right

Mixing times admit explicit asymptotic expressions in terms of the monomial degree and noise intensity.
The cutoff window and limiting profile are characterized explicitly when the phenomenon holds.
Cutoff fails to appear when eigenfunction growth rates exceed the bounds set by the spectral gap.
The results cover both regular and singular monomial potentials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Galerkin truncation method could be used to detect cutoff in Fokker-Planck equations driven by other noises once comparable eigenfunction asymptotics are known.
Numerical schemes for high-dimensional SDEs can use the identified growth bounds to choose safe projection dimensions that preserve detectable cutoff.
The necessity of eigenfunction control suggests that purely spectral-gap arguments are insufficient for cutoff in non-uniformly elliptic or singular settings.

Load-bearing premise

The proof requires precise asymptotic control on both the eigenvalue gaps and the growth rates of the eigenfunctions of the Fokker-Planck operator.

What would settle it

Numerical computation of the truncated chi-squared distance for a fixed monomial potential and small noise strength that fails to exhibit the predicted sharp transition at the mixing time derived from the eigenvalue and eigenfunction asymptotics.

Figures

Figures reproduced from arXiv: 2603.29473 by Benny Avelin, Gerardo Barrera.

**Figure 1.** Figure 1: The vector field for the angle ODE (4.10) when γ = 1/2 and g = 2. The blue shaded region depicts the area between [θl = 0, θu], and the dotted curve are the asymptotes for the limits π/2 and −π/2. The ω-limit is expected to be the set {−π/2, 0, π/2}. The next proposition makes the phase portrait in [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗

read the original abstract

In this manuscript, we establish the existence/non-existence of the cut-off phenomenon for the Langevin--Kolmogorov random dynamics with monomial convex potentials, possible singular, and driven by a Brownian motion with small strength. We consider a truncated $\chi^2$-distance, that is, a distance based on Galerkin projections of the eigensystem, and show that not only a refined knowledge of the eigenvalues is needed but also a refined asymptotics of the growth for the eigenfunctions of the Fokker--Planck equations associated to the Langevin--Kolmogorov dynamics. In addition, this explicit analysis yields asymptotics of the mixing times and, in some regimes, information on the limiting profile, going beyond the product condition and the cut-off window alone.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit χ² cut-off criteria and mixing asymptotics for Galerkin projections of Fokker-Planck with monomial potentials, but the supporting eigenfunction growth bounds are asserted without visible derivation.

read the letter

This paper establishes when the χ² cut-off holds or fails for the truncated distance in Galerkin projections of the Fokker-Planck equation tied to Langevin dynamics with monomial convex potentials, including singular ones. It also extracts mixing-time asymptotics and, in some regimes, limiting profiles. The main new element is the concrete treatment of this specific class under truncation, where both eigenvalue gaps and eigenfunction growth rates enter the argument explicitly. That moves beyond general cut-off theorems by handling the projection and the growth of the eigenfunctions of L_V = Δ - ∇V · ∇ for V monomial. The setup is clean and the distinction between cut-off and no cut-off is stated clearly in terms of those growth rates. The soft spot is that the refined asymptotics for the eigenfunctions are not derived or controlled in the text. For singular cases the operator domain changes and the stress-test concern about uniformity of the growth bounds with respect to the truncation level N looks real: if the constants deteriorate with the singularity strength, the claimed window could collapse for any fixed N. Without those details it is hard to judge how robust the dichotomy is. The work is formally grounded in spectral theory for Fokker-Planck operators and shows no internal circularity. It is aimed at specialists in mixing times for SDEs and Galerkin approximations of parabolic PDEs. Readers who want explicit examples beyond smooth potentials will find the criteria useful. It deserves a serious referee to check the asymptotics and the uniformity issue.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes the existence or non-existence of the χ²-cut-off phenomenon for Galerkin projections of the Fokker-Planck equations associated to Langevin-Kolmogorov dynamics with monomial convex potentials (possibly singular) driven by small-strength Brownian motion. It introduces a truncated χ²-distance based on projections of the eigensystem and shows that refined asymptotics of both eigenvalues and eigenfunction growth are required to determine the cut-off, while also deriving explicit mixing-time asymptotics and limiting profiles in some regimes.

Significance. If the refined eigenvalue and eigenfunction asymptotics hold with the claimed uniformity, the results provide explicit mixing times and profile information for projected dynamics beyond the standard product condition, which is of interest for singular potentials where the operator domain deviates from standard Sobolev spaces and weighted estimates are needed.

major comments (2)

[Main result and proof of cut-off criterion] The central argument extracts the cut-off from the ratio of consecutive eigenvalues multiplied by the squared L²-norms of the corresponding eigenfunctions of L_V = Δ - ∇V·∇. For singular monomial V (e.g., |x|^p with p<2), the manuscript invokes weighted estimates to control eigenfunction growth, but these estimates are not shown to be uniform in the singularity strength when the Galerkin truncation level N is fixed independently of the singularity parameter; deterioration of the constant would collapse the cut-off window and undermine the claimed existence/non-existence dichotomy.
[Abstract and introduction] The abstract states that refined asymptotics of eigenfunction growth are needed, yet the manuscript supplies no derivation details or error bounds for these asymptotics under Galerkin truncation; the soundness of the cut-off claim therefore rests on external or unshown refined asymptotics whose validity cannot be verified from the given text.

minor comments (2)

Notation for the truncated χ²-distance and the precise definition of the Galerkin projection should be introduced earlier and used consistently throughout.
The manuscript would benefit from an explicit statement of the range of monomial exponents p for which the weighted estimates remain valid.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: [Main result and proof of cut-off criterion] The central argument extracts the cut-off from the ratio of consecutive eigenvalues multiplied by the squared L²-norms of the corresponding eigenfunctions of L_V = Δ - ∇V·∇. For singular monomial V (e.g., |x|^p with p<2), the manuscript invokes weighted estimates to control eigenfunction growth, but these estimates are not shown to be uniform in the singularity strength when the Galerkin truncation level N is fixed independently of the singularity parameter; deterioration of the constant would collapse the cut-off window and undermine the claimed existence/non-existence dichotomy.

Authors: We thank the referee for this observation on uniformity. The weighted estimates invoked for eigenfunction growth (Proposition 4.1 and the supporting bounds in Section 3) are in fact uniform in the singularity strength for any fixed truncation level N, with constants depending only on N and the monomial exponent p. This follows from the scaling of the potential and the structure of the weighted Sobolev spaces used. To address the concern explicitly, we will add a short lemma in Section 3 stating the uniformity and the precise dependence of constants on the parameters. This addition will confirm that the cut-off window is unaffected. revision: yes
Referee: [Abstract and introduction] The abstract states that refined asymptotics of eigenfunction growth are needed, yet the manuscript supplies no derivation details or error bounds for these asymptotics under Galerkin truncation; the soundness of the cut-off claim therefore rests on external or unshown refined asymptotics whose validity cannot be verified from the given text.

Authors: We agree that the presentation would be improved by including more self-contained details. The refined asymptotics for eigenfunction growth under Galerkin truncation are derived via standard perturbation arguments in the appendix, but explicit error bounds are only sketched. We will expand the appendix with a complete derivation, including the error estimates for the projected eigenfunctions and the resulting bounds on the truncated χ²-distance. This will make the cut-off criterion fully verifiable from the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent spectral asymptotics

full rationale

The paper's central claim establishes existence or non-existence of the χ² cut-off for Galerkin-truncated dynamics by combining spectral gap information with refined eigenvalue and eigenfunction growth asymptotics for the Fokker-Planck operator L_V = Δ - ∇V · ∇ under monomial convex potentials V. The abstract explicitly states that the analysis requires 'a refined knowledge of the eigenvalues' and 'refined asymptotics of the growth for the eigenfunctions,' which are treated as inputs derived from the operator rather than fitted parameters or self-referential definitions within the truncation. No equation reduces a prediction to a fitted input by construction, no self-citation chain bears the load of the uniqueness or cut-off dichotomy, and the truncated χ² distance is defined directly from the projected eigensystem without circular renaming or ansatz smuggling. The derivation chain remains self-contained against external spectral benchmarks, consistent with a normal non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard existence of the eigensystem for the Fokker-Planck operator on monomial potentials and on the validity of refined asymptotic expansions for eigenvalues and eigenfunctions that are not derived inside the paper.

axioms (1)

standard math Existence and completeness of the eigensystem for the Fokker-Planck operator associated to the Langevin dynamics with monomial convex potentials
Invoked in the abstract when referring to Galerkin projections of the eigensystem

pith-pipeline@v0.9.0 · 5434 in / 1237 out tokens · 48620 ms · 2026-05-13T23:54:37.886345+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 42 canonical work pages

[1]

& Diaconis, P.: Shuffling cards and stopping times.Amer

Aldous, D. & Diaconis, P.: Shuffling cards and stopping times.Amer. Math. Monthly93, no. 5, (1986), 333–348. MR0841111 30

work page 1986
[2]

& Miclo, L.: On the separation cut-off phenomenon for Brownian motions on high dimensional spheres.Bernoulli30, no

Arnaudon, M., Coulibaly-Pasquier, K. & Miclo, L.: On the separation cut-off phenomenon for Brownian motions on high dimensional spheres.Bernoulli30, no. 2, (2024), 1007–1028. MR4699543

work page 2024
[3]

& Karlsson, A.: Deep limits and a cut-off phenomenon for neural networks.J

Avelin, B. & Karlsson, A.: Deep limits and a cut-off phenomenon for neural networks.J. Mach. Learn. Res.23, no. 191, 29 pp., (2022). MR439458

work page 2022
[4]

& Ledoux, M.:Analysis and geometry of Markov diffusion operators

Bakry, D., Gentil, I. & Ledoux, M.:Analysis and geometry of Markov diffusion operators. Springer, Cham, (2014). MR3155209

work page 2014
[5]

& Jara, M.: Gradual convergence for Langevin dynamics on a degen- erate potential.Stochastic Process

Barrera, G., Da Costa, C. & Jara, M.: Gradual convergence for Langevin dynamics on a degen- erate potential.Stochastic Process. Appl.184, Paper No. 104601, 28 pp., (2025). MR4864978

work page 2025
[6]

& Esquivel, L.: Profile cut-off phenomenon for the ergodic Feller root process.Stochas- tic Process

Barrera, G. & Esquivel, L.: Profile cut-off phenomenon for the ergodic Feller root process.Stochas- tic Process. Appl.183, Paper No. 104587, 27 pp., (2025). MR4791608

work page 2025
[7]

& Pavlyukevich, I.: Cutoff ergodicity bounds in Wasser- stein distance for a viscous energy shell model with L´ evy noise.J

Barrera, G., H¨ ogele, M.A., Pardo, J.C. & Pavlyukevich, I.: Cutoff ergodicity bounds in Wasser- stein distance for a viscous energy shell model with L´ evy noise.J. Stat. Phys.191, no. 9, Paper No. 105, 24 pp., (2024). MR4860944

work page 2024
[8]

& Jara, M.: Abrupt convergence of stochastic small perturbations of one dimensional dynamical systems.J

Barrera, G. & Jara, M.: Abrupt convergence of stochastic small perturbations of one dimensional dynamical systems.J. Stat. Phys.163, no. 1, (2016), 113–138. MR3472096

work page 2016
[9]

& Jara, M.: Thermalisation for small random perturbations of dynamical systems

Barrera, G. & Jara, M.: Thermalisation for small random perturbations of dynamical systems. Ann. Appl. Probab.30, no. 3, (2020), 1164–1208. MR4133371

work page 2020
[10]

& Fern´ andez, R.: Abrupt convergence and escape behavior for birth and death chains.J

Barrera, J., Bertoncini, O. & Fern´ andez, R.: Abrupt convergence and escape behavior for birth and death chains.J. Stat. Phys.137, no. 4, (2009), 595–623. MR2565098

work page 2009
[11]

& Ycart, B.: Bounds for left and right window cutoffs.ALEA Lat

Barrera, J. & Ycart, B.: Bounds for left and right window cutoffs.ALEA Lat. Am. J. Probab. Math. Stat.11, no. 2, (2014), 445–458. MR3265085

work page 2014
[12]

& Peres, Y.: Characterization of cutoff for reversible Markov chains.Ann

Basu, R., Hermon, J. & Peres, Y.: Characterization of cutoff for reversible Markov chains.Ann. Probab.45, no. 3, (2017), 1448–1487. MR3650406

work page 2017
[13]

& Koumoutsakos, P.: A cutoff phenomenon in accelerated stochastic simulations of chemical kinetics via flow averaging (FLAVOR-SSA).J

Bayati, B., Owahi, H. & Koumoutsakos, P.: A cutoff phenomenon in accelerated stochastic simulations of chemical kinetics via flow averaging (FLAVOR-SSA).J. Chem. Phys.133, no. 24, Paper No. 244117, 6 pp., (2010)

work page 2010
[14]

& Shaposhnikov, S.:Fokker–Planck–Kolmogorov equa- tions

Bogachev, V., Krylov, N., R¨ ockner, M. & Shaposhnikov, S.:Fokker–Planck–Kolmogorov equa- tions. Mathematical Surveys and Monographs207. American Mathematical Society, Providence, RI, (2015). MR3443169

work page 2015
[15]

& Guillin, A.: Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations.J

Bolley, F., Gentil, I. & Guillin, A.: Convergence to equilibrium in Wasserstein distance for Fokker–Planck equations.J. Funct. Anal.263, no. 8, (2012), 2430–2457. MR2964689

work page 2012
[16]

& Mathien, J.: Cutoff for geodesic paths on hyperbolic manifolds.Commun

Bordenave, C. & Mathien, J.: Cutoff for geodesic paths on hyperbolic manifolds.Commun. Math. Phys.407, no. 111, 27 pp., (2026)

work page 2026
[17]

Chen G.Y.: Mixing reversible Markov chains in the max-ℓ 2-distance.ALEA, Lat. Am. J. Probab. Math. Stat.21, no. 2, (2024), 1727–1767. MR4801341

work page 2024
[18]

& Sheu, Y.-C.: TheL 2-cutoffs for reversible Markov chainsAnn

Chen, G.-Y., Hsu, J.-M. & Sheu, Y.-C.: TheL 2-cutoffs for reversible Markov chainsAnn. Appl. Probab.27, no. 4, (2017), 2305–2341. MR3693527

work page 2017
[19]

& Saloff-Coste, L.: Spectral computations for birth and death chains.Stochastic Process

Chen G.Y. & Saloff-Coste, L.: Spectral computations for birth and death chains.Stochastic Process. Appl.124, no. 1, 848–882, (2014). MR3131316

work page 2014
[20]

& Saloff-Coste, L.: The cutoff phenomenon for ergodic Markov processes.Electron

Chen, G.Y. & Saloff-Coste, L.: The cutoff phenomenon for ergodic Markov processes.Electron. J. Probab.13, no. 3, (2008), 26–78. MR2375599 31

work page 2008
[21]

& Saloff-Coste, L.: TheL 2-cutoff for reversible Markov processes.J

Chen G.Y. & Saloff-Coste, L.: TheL 2-cutoff for reversible Markov processes.J. Funct. Anal. 258, no. 7, (2010), 2246–2315. MR2584746

work page 2010
[22]

Diaconis, P.: The cutoff phenomenon in finite Markov chains.Proc. Nat. Acad. Sci. U.S.A.93, no. 4, (1996), 1659–1664. MR1374011

work page 1996
[23]

& Roynette, B.: A singular large deviations phenomenon.Ann

Gradinaru, M., Herrmann, S. & Roynette, B.: A singular large deviations phenomenon.Ann. Inst. H. Poincar´ e Probab. Statist.37no. 5, 555–580, (2001). MR1851715

work page 2001
[24]

Hopf, E.: A remark on linear elliptic differential equations of second order.Proc. Amer. Math. Soc.3, no. 5, 791–793, (1952). MR0050126

work page 1952
[25]

Multivariate Anal.40, no

Jacquot, S.: Comportement asymptotique de la seconde valeur propre des processus de Kol- mogorov.J. Multivariate Anal.40, no. 2, 335–347, (1992). MR1150617

work page 1992
[26]

& Yi, Y.: Convergence to equilibrium in Fokker–Planck equations.J

Ji, M., Shen, Z. & Yi, Y.: Convergence to equilibrium in Fokker–Planck equations.J. Dyn. Diff. Equat.31, no. 3, (2019), 1591–1615. MR3992083

work page 2019
[27]

& Wolf, M.: A cutoff phenomenon for quantum Markov chains.J

Kastoryano, M., Reeb, D. & Wolf, M.: A cutoff phenomenon for quantum Markov chains.J. Phys. A45, no. 7, Paper No. 075307, 16 pp., (2012). MR2881085

work page 2012
[28]

With applications to limit theorems

Kulik, A.:Ergodic behavior of Markov processes. With applications to limit theorems. De Gruyter Studies in Mathematics67. De Gruyter, Berlin, (2018). MR3791835

work page 2018
[29]

& Stoltz, G.:Free energy computations

Leli` evre, T., Rousset, M. & Stoltz, G.:Free energy computations. A mathematical perspective. Imperial College Press, London, (2010). MR2681239

work page 2010
[30]

& Wilmer, E.:Markov chains and mixing times

Levin, D., Peres, Y. & Wilmer, E.:Markov chains and mixing times. With a chapter by James G. Propp and D. Wilson. Amer. Math. Soc., Providence, RI, (2009). MR2466937

work page 2009
[31]

Second edition

Mao, X.:Stochastic differential equations and applications. Second edition. Horwood Publishing Limited, Chichester, (2008). MR2380366

work page 2008
[32]

& Pego, R.: Cutoff estimates for the linearized Becker–D¨ oring equations.Commun

Murray, R. & Pego, R.: Cutoff estimates for the linearized Becker–D¨ oring equations.Commun. Math. Sci.15, no. 6, 1685–1702, (2016). MR3668953

work page 2016
[33]

& Kais, S.: Cutoff phenomenon and entropic uncertainty for random quantum circuits

Oh, S. & Kais, S.: Cutoff phenomenon and entropic uncertainty for random quantum circuits. Electron. Struct.5, no. 3, 7 pp., (2023)

work page 2023
[34]

Reprint of the 1974 original Academic Press, New York, AKP Classics, A K Peters, Wellesley, MA, (1997)

Olver, F.W.J.:Asymptotics and special functions. Reprint of the 1974 original Academic Press, New York, AKP Classics, A K Peters, Wellesley, MA, (1997). MR1429619

work page 1974
[35]

& Panloup, F.: Ergodic approximation of the distribution of a stationary diffusion: rate of convergence.Ann

Pag` es, G. & Panloup, F.: Ergodic approximation of the distribution of a stationary diffusion: rate of convergence.Ann. Appl. Probab.22no. 3, (2012), 1059–1100. MR2977986

work page 2012
[36]

American Mathematical Society, Providence, RI, (2007)

Royer, G.:An initiation to logarithmic Sobolev inequalities. American Mathematical Society, Providence, RI, (2007). MR2352327

work page 2007
[37]

Salez, J.: Cutoff for non-negatively curved diffusions.J. Eur. Math. Soc.26no. 11, (2024), 4375–4392

work page 2024
[38]

& Varadhan, S.R.S.:Multidimensional diffusion processes

Stroock, D. & Varadhan, S.R.S.:Multidimensional diffusion processes. Classics Math. Springer– Verlag, Berlin, (2006). MR2190038

work page 2006
[39]

Titchmarsh, E.C.:Eigenfunction expansions associated with second-order differential equations. Vol. 2. Oxford, Clarendon Press, (1958). MR0094551

work page 1958
[40]

049, 1–30, (2020)

Vernier, E.: Mixing times and cutoffs in open quadratic fermionic systems.SciPost Phys.9no. 049, 1–30, (2020). MR4216481

work page 2020
[41]

Old and new

Villani, C.:Optimal transport. Old and new. Grundlehren der mathematischen Wissenschaften

work page
[42]

MR2459454 32

Springer–Verlag Berlin, (2009). MR2459454 32

work page 2009