pith. sign in

arxiv: 2604.13454 · v1 · submitted 2026-04-15 · 🧮 math.PR

Invariant and periodic measures in classical spin systems on infinite lattices with highly degenerate noise

Tong Lu , Huaizhong Zhao This is my paper

Pith reviewed 2026-05-10 12:59 UTC · model grok-4.3

classification 🧮 math.PR
keywords classical spin systemsinvariant measuresperiodic measuresdegenerate noiseinfinite latticesstochastic differential equationsgeometric ergodicitytightness
0
0 comments X

The pith

Classical spin systems on infinite lattices admit invariant and periodic measures even when noise drives only one particle.

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

This paper establishes that infinite-dimensional stochastic differential equations for classical spin systems on unbounded lattices, with stochastic forcing applied to only a single particle and the rest coupled by nearest-neighbor interactions, possess invariant measures in the time-homogeneous case and periodic measures in the time-periodic case. The argument first obtains existence, uniqueness, and the Markov property for weak martingale solutions under mild assumptions such as weak dissipation in the local interactions. It then shows that the finite-volume restrictions of the system each carry a unique invariant or periodic measure together with geometric ergodicity. These finite-dimensional objects are proved tight, so that weak convergent subsequences produce the desired measures on the full infinite lattice.

Core claim

Under mild assumptions on local interactions such as weak dissipation, the infinite spin system driven by noise at one site admits weak martingale solutions that are Markovian; the finite-volume dynamics possess a unique invariant measure (or periodic measure) with geometric ergodicity; the resulting family of finite-dimensional measures is tight, and every weak limit point of the invariant (respectively averaged periodic) measures is an invariant (respectively periodic) measure for the infinite-dimensional system.

What carries the argument

Tightness of the family of finite-volume invariant (and averaged periodic) measures, followed by weak convergence to obtain invariant or periodic measures on the infinite lattice.

If this is right

  • Weak martingale solutions to the infinite-dimensional SDEs exist and satisfy the Markov property.
  • Every finite sub-lattice carries a unique invariant measure with geometric ergodicity (or a unique periodic measure in the time-periodic setting).
  • Any weak limit of the finite-volume invariant measures is an invariant measure for the infinite system.
  • Any weak limit of the time averages of the lifted periodic measures is a periodic measure for the infinite system.

Where Pith is reading between the lines

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

  • The same finite-volume approximation and tightness route could be tested on other lattice systems whose noise is localized to a single site.
  • Large-scale numerical integration of finite but growing lattices could be used to approximate statistics of the infinite-system measures.
  • The result indicates that the long-time statistics become independent of the precise location of the driving noise once the lattice is connected by the interaction graph.

Load-bearing premise

The local interactions between neighboring particles are strong enough to propagate the effect of the single noise source across the entire infinite lattice.

What would settle it

The family of finite-volume invariant measures fails to be tight as the volume tends to infinity, so that no weakly convergent subsequence exists.

read the original abstract

In this paper, we consider the classical spin systems on unbounded lattices given by infinite-dimensional stochastic differential equations (SDEs). We assume that the stochastic forcing acts only on one particle. The other particles are not subject to stochastic forcing directly, but interact with their nearest neighbouring particles. Under the above highly degenerate noise setting, with some mild assumptions on the local interaction of each particle such as weak dissipation, we obtain the existence, uniqueness and the Markovian property of weak martingale solutions. We prove that the one-dimensional noise can propagate to any spin particle in the system in the sense that there exists a unique invariant/periodic measure and geometric ergodicity holds for the Markovian system when restricted to any finite volume. We then prove the finite-dimensional invariant measure and the average of lifted periodic measure are tight, and weak convergent subsequence gives an invariant and periodic measures of the infinite spin systems, respectively, in the time-homogeneous or time-periodic cases.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies classical spin systems on infinite lattices modeled by infinite-dimensional SDEs with highly degenerate noise acting on a single particle. Under mild assumptions on local interactions (including weak dissipation), it establishes existence, uniqueness, and the Markov property of weak martingale solutions. It proves that the finite-volume restrictions admit unique invariant/periodic measures with geometric ergodicity, then shows tightness of these finite-dimensional measures (and averages of lifted periodic measures) whose weak limits yield invariant and periodic measures on the infinite lattice, in both time-homogeneous and time-periodic cases.

Significance. If the volume-uniform tightness holds, the result would advance the ergodic theory of degenerate stochastic lattice systems by extending finite-volume uniqueness to the infinite setting with minimal noise propagation. The strategy of lifting finite-volume geometric ergodicity via tightness is standard and well-chosen; the paper would benefit from explicit credit to related works on degenerate SDEs and lattice models.

major comments (2)
  1. [Tightness and weak convergence argument (following the finite-volume ergodicity section)] The central existence claim for invariant/periodic measures on the infinite lattice rests on tightness of the family of finite-volume invariant measures (and lifted periodic measures) followed by weak convergence. The manuscript invokes only weak dissipation to obtain the necessary a priori bounds and geometric ergodicity in each finite volume, but the resulting Lyapunov or moment estimates typically carry constants that grow with the number of sites. Without an explicit volume-independent bound (e.g., a uniform-in-N Lyapunov function or moment estimate in the section deriving the infinite-lattice measures), the family need not be tight in the product topology, so the weak-limit step does not necessarily produce a measure on the infinite system.
  2. [Finite-volume uniqueness and geometric ergodicity] The propagation of the one-dimensional noise to all particles is asserted to yield geometric ergodicity in every finite volume under the weak-dissipation assumption. However, the rate of geometric ergodicity and the constants in the Wasserstein or total-variation contraction may deteriorate with volume size; this deterioration must be controlled explicitly to justify passing to the infinite-volume limit while preserving the invariant-measure property.
minor comments (2)
  1. [Introduction] The abstract and introduction would benefit from a brief comparison table or paragraph situating the weak-dissipation assumption against stronger dissipativity conditions used in prior works on non-degenerate lattice SDEs.
  2. [Preliminaries] Notation for the infinite-lattice configuration space and the precise topology in which tightness is proved should be stated explicitly at the first appearance rather than deferred.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concerning the uniformity of bounds and the passage to the infinite-volume limit are important for clarity. We address each major comment below and will revise the manuscript to make the relevant estimates and arguments fully explicit.

read point-by-point responses

  1. Referee: [Tightness and weak convergence argument (following the finite-volume ergodicity section)] The central existence claim for invariant/periodic measures on the infinite lattice rests on tightness of the family of finite-volume invariant measures (and lifted periodic measures) followed by weak convergence. The manuscript invokes only weak dissipation to obtain the necessary a priori bounds and geometric ergodicity in each finite volume, but the resulting Lyapunov or moment estimates typically carry constants that grow with the number of sites. Without an explicit volume-independent bound (e.g., a uniform-in-N Lyapunov function or moment estimate in the section deriving the infinite-lattice measures), the family need not be tight in the product topology, so the weak-limit step does not necessarily produce a measure on the infinite system.

    Authors: We agree that volume-independence must be stated explicitly. The weak dissipation assumption is uniform across lattice sites and independent of volume size N. This permits a Lyapunov function constructed as a sum of local terms whose coefficients do not depend on N, yielding moment bounds that are uniform in N. These bounds are used to establish tightness of the finite-volume invariant measures (and lifted periodic measures) in the product topology, as claimed in the manuscript. We will revise the relevant section to include an explicit lemma or remark isolating the N-independent constants and confirming that the family remains tight. revision: yes

  2. Referee: [Finite-volume uniqueness and geometric ergodicity] The propagation of the one-dimensional noise to all particles is asserted to yield geometric ergodicity in every finite volume under the weak-dissipation assumption. However, the rate of geometric ergodicity and the constants in the Wasserstein or total-variation contraction may deteriorate with volume size; this deterioration must be controlled explicitly to justify passing to the infinite-volume limit while preserving the invariant-measure property.

    Authors: Geometric ergodicity is proved in each fixed finite volume solely to guarantee existence and uniqueness of the invariant (respectively periodic) measure there; the contraction rates may indeed depend on volume size. For the infinite-lattice construction we use only the existence of these measures together with their tightness. The invariance property passes to any weak limit because the finite-volume dynamics are the natural projections of the infinite-dimensional SDE and the topology is the product topology. We do not claim uniqueness or uniform ergodicity rates for the infinite-volume measures. We will add a short clarifying paragraph explaining this distinction and why volume-dependent rates do not obstruct the existence of the limiting measures. revision: yes

Circularity Check

0 steps flagged

No circularity; finite-to-infinite extension via tightness is independent of inputs

full rationale

The derivation begins with existence/uniqueness of weak martingale solutions and finite-volume invariant/periodic measures plus geometric ergodicity under the stated mild/weak dissipation assumptions. It then invokes standard tightness of the family of these finite-volume measures (and lifted periodic measures) followed by weak convergence to obtain the infinite-lattice objects. No quoted step equates a claimed result to its own fitted parameters, self-defines a quantity in terms of the target, or reduces the central existence claim to a self-citation chain. The argument remains self-contained against external benchmarks such as standard tightness criteria in product spaces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claims rest on mild assumptions of weak dissipation in local interactions to ensure solution existence and measure uniqueness; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption mild assumptions on local interaction such as weak dissipation
    Invoked to obtain existence, uniqueness, and Markov property of weak martingale solutions and subsequent ergodicity results.

pith-pipeline@v0.9.0 · 5459 in / 1141 out tokens · 40090 ms · 2026-05-10T12:59:26.164975+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [1]

    D. A. Abanin, W. De Roeck, and F. Huveneers , Exponentially slow heating in periodically driven many-body systems , Phys. Rev. Lett., 115 (2015), p. 256803

    work page 2015

  2. [2]

    Albeverio, Y

    S. Albeverio, Y. Kondratiev, Y. Kozitsky, and M. R \"o ckner , The Statistical Mechanics of Quantum Lattice Systems: A Path Integral Approach , European Mathematical Society, 2009

    work page 2009

  3. [3]

    Albeverio, Y

    S. Albeverio, Y. G. Kondratiev, and T. V. Tsikalenko , Stochastic dynamics for quantum lattice systems and stochastic quantization I : Ergodicity , Random Oper. Stoch. Equ., 2 (1994), pp. 103--139

    work page 1994

  4. [4]

    Albeverio and M

    S. Albeverio and M. R \"o ckner , Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms , Probab. Theory Rel. Fields, 89 (1991), pp. 347--386

    work page 1991

  5. [5]

    G. E. Andrews, R. Askey, and R. Roy , Special functions , Cambridge university press Cambridge, 1999

    work page 1999

  6. [6]

    Bakhtin and H

    Y. Bakhtin and H. Chen , Dynamic polymers: invariant measures and ordering by noise , Probab. Theory Relat. Fields, 183 (2022), pp. 167--227

    work page 2022

  7. [7]

    R. F. Bass , Stochastic Processes , Cambridge University Press, 2011

    work page 2011

  8. [8]

    Brezis , Functional Analysis, Sobolev Spaces and Partial Differential Equations , Springer, 2011

    H. Brezis , Functional Analysis, Sobolev Spaces and Partial Differential Equations , Springer, 2011

    work page 2011

  9. [9]

    Chen , From Markov Chains to Non-Equilibrium Particle Systems, Second Edition , World Scientific, 2004

    M. Chen , From Markov Chains to Non-Equilibrium Particle Systems, Second Edition , World Scientific, 2004

    work page 2004

  10. [10]

    Da Prato and J

    G. Da Prato and J. Zabczyk , Ergodicity for Infinite Dimensional Systems , Cambridge University Press, 1996

    work page 1996

  11. [11]

    height 2pt depth -1.6pt width 23pt, Stochastic Equations in Infinite Dimensions , Cambridge University Press, 2nd ed., 2014

    work page 2014

  12. [12]

    P. H. Damgaard and H. H \"u ffel , Stochastic Quantization , World Scientific, 1988

    work page 1988

  13. [13]

    C. Feng, B. Qu, and H. Zhao , Random quasi-periodic paths and quasi-periodic measures of stochastic differential equations , J. Differ. Equ., 286 (2021), pp. 119--163

    work page 2021

  14. [14]

    Feng and H

    C. Feng and H. Zhao , Random periodic processes, periodic measures and ergodicity , J. Differ. Equ., 269 (2020), pp. 7382--7428

    work page 2020

  15. [15]

    C. Feng, H. Zhao, and J. Zhong , Existence of geometric ergodic periodic measures of stochastic differential equations , J. Differ. Equ., 359 (2023), pp. 67--106

    work page 2023

  16. [16]

    Flandoli , An Introduction to 3D Stochastic Fluid Dynamics , vol

    F. Flandoli , An Introduction to 3D Stochastic Fluid Dynamics , vol. 1942 of Lecture Notes in Mathematics, Springer, Berlin Heidelberg, 2008, pp. 51--150

    work page 1942

  17. [17]

    Flandoli and M

    F. Flandoli and M. Romito , Markov selections for the 3d stochastic Navier–Stokes equations , Probab. Theory Relat. Fields, 140 (2008), pp. 407–--458

    work page 2008

  18. [18]

    Goldman and J

    N. Goldman and J. Dalibard , Periodically driven quantum systems: Effective Hamiltonians and engineered gauge fields , Phys. Rev. X, 4 (2014), p. 031027

    work page 2014

  19. [19]

    An introduction to stochastic PDEs.arXiv preprint arXiv:0907.4178, 2009

    M. Hairer , An I ntroduction to S tochastic PDEs , arXiv preprint arXiv:0907.4178, (2009)

    work page arXiv 2009

  20. [20]

    Hofmanov \'a and J

    M. Hofmanov \'a and J. Seidler , On weak solutions of stochastic differential equations , Stoch. Anal. Appl., 30 (2012), pp. 100--121

    work page 2012

  21. [21]

    Holley and D

    R. Holley and D. Stroock , Diffusions on an infinite dimensional torus , J. Funct. Anal., 42 (1981), pp. 29--63

    work page 1981

  22. [22]

    height 2pt depth -1.6pt width 23pt, Logarithmic sobolev inequalities and stochastic ising models , J. Stat. Phys., 46 (1987), pp. 1159--1194

    work page 1987

  23. [23]

    R. A. Holley and D. W. Stroock , L_2 theory for the stochastic ising model , Z. Wahrsch. Verw. Geb., 35 (1976), pp. 87--101

    work page 1976

  24. [24]

    Ichihara and H

    K. Ichihara and H. Kunita , A classification of the second order degenerate elliptic operators and its probabilistic characterization , Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 30 (1974), pp. 235--254

    work page 1974

  25. [25]

    Kallianpur and J

    G. Kallianpur and J. Xiong , Stochastic D ifferential E quations in I nfinite D imensional S paces , in IMS Lecture Notes—Monograph Series, vol. 26, IMS, 1995

    work page 1995

  26. [26]

    Karatzas and S

    I. Karatzas and S. E. Shreve , Brownian Motion and Stochastic Calculus , Springer-Verlag, 2nd ed., 1998

    work page 1998

  27. [27]

    Khasminskii , Stochastic Stability of Differential Equations , Springer, 2nd ed., 2012

    R. Khasminskii , Stochastic Stability of Differential Equations , Springer, 2nd ed., 2012

    work page 2012

  28. [28]

    T. M. Liggett , Interacting Particle Systems , Classics in Mathematics, Springer Berlin Heidelberg, 1985

    work page 1985

  29. [29]

    height 2pt depth -1.6pt width 23pt, Continuous Time Markov Processes: An Introduction , American Mathematical Society, 2010

    work page 2010

  30. [30]

    Liu and K

    R. Liu and K. Lu , Exponential mixing and limit theorems of quasi-periodically forced 2D stochastic Navier-Stokes equations in the hypoelliptic setting , Comm. Math. Phys., 406 (2025). Paper No. 55, 46 pp

    work page 2025

  31. [31]

    J. C. Mattingly, A. M. Stuart, and D. J. Higham , Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise , Stoch. Process. Appl., 101 (2002), pp. 185--232

    work page 2002

  32. [32]

    S. P. Meyn and R. L. Tweedie , Markov Chains and Stochastic Stability , Cambridge University Press, 2nd ed., 2009

    work page 2009

  33. [33]

    J. R. Munkres , Topology , Prentice Hall, 2nd ed., 2000

    work page 2000

  34. [34]

    Parisi and Y

    G. Parisi and Y. Wu , Perturbation theory without gauge fixing , Sci. Sin., 24 (1980), pp. 483--496

    work page 1980

  35. [35]

    D. W. Stroock and S. R. S. Varadhan , Multidimensional Diffusion Processes , Springer-Verlag, 1979

    work page 1979

  36. [36]

    Y. S. Won , An L 2-approximation method for construction and smoothing estimates of Markov semigroups for interacting diffusion processes on a lattice , Infin. Dimens. Anal. Quantum Probab. Relat. Top., 24 (2021), p. 2150004

    work page 2021

  37. [37]

    Xu and B

    L. Xu and B. Zegarlinski , Existence and exponential mixing of infinite white -stable systems with unbounded interactions , Electron. J. Probab., 15 (2010), pp. 1994--2018

    work page 2010

  38. [38]

    Z \'a k , Existence of equilibrium for infinite system of interacting diffusions , Stoch

    F. Z \'a k , Existence of equilibrium for infinite system of interacting diffusions , Stoch. Anal. Appl., 34 (2016), pp. 1057--1082

    work page 2016

  39. [39]

    Zegarlinski , The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice , Commun

    B. Zegarlinski , The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice , Commun. Math. Phys., 175 (1996), pp. 401–--432

    work page 1996