arxiv: 2604.14346 · v1 · submitted 2026-04-15 · 🧮 math.PR · math.DS

Recognition: unknown

Fluctuations for the Toda lattice

Amol Aggarwal , Matthew Nicoletti

Authors on Pith no claims yet

Pith reviewed 2026-05-10 11:53 UTC · model grok-4.3

classification 🧮 math.PR math.DS

keywords Toda latticefluctuationsdiffusive scalingGaussian processesLévy-Chentsov fieldBrownian motioncorrelation functionsintegrable systems

0 comments

The pith

Fluctuations of currents in the Toda lattice converge to an explicit Gaussian limit under diffusive scaling.

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

The Toda lattice is placed at thermal equilibrium by taking independent Gaussian momenta and Gamma-distributed exponential bond lengths. The system is viewed as a dense collection of quasi-particles that interact only through scattering. The central result is that the joint fluctuations of these quasi-particles, under diffusive space-time scaling, converge to a Gaussian process called the dressed Lévy-Chentsov field. This immediately yields that current fluctuations converge to the same explicit Gaussian, that a tagged particle's position scales to Brownian motion, and that space-time correlations decay as one over time with explicit scaling forms. A reader cares because the argument supplies the first rigorous microscopic derivation of these macroscopic fluctuation laws for an integrable lattice model.

Core claim

Under diffusive scaling the space-time fluctuations for the Toda lattice currents converge to an explicit Gaussian limit. The proof proceeds by showing that the full joint scaling limit of the fluctuations for the model's quasi-particles is given by a Gaussian process, the dressed Lévy-Chentsov field. As direct consequences the trajectory of any single particle converges to Brownian motion and the space-time two-point correlation functions decay inversely with time with the explicit scaling distributions predicted by Spohn.

What carries the argument

The dressed Lévy-Chentsov field, the Gaussian process obtained as the joint scaling limit of fluctuations for the dense collection of scattering quasi-particles that represent the Toda lattice.

If this is right

The trajectory of a single particle q_0 converges in law to Brownian motion under diffusive scaling.
Space-time two-point correlation functions decay inversely with time with the explicit scaling distributions predicted by Spohn.
The full joint scaling limit of quasi-particle fluctuations is the dressed Lévy-Chentsov field, from which the current fluctuation limit follows directly.

Where Pith is reading between the lines

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

The same quasi-particle fluctuation argument may apply to other integrable particle systems whose scattering produces an analogous dressed field.
The explicit Gaussian limit supplies a starting point for deriving large-deviation principles or higher-order statistics for the Toda lattice.
Numerical checks of the predicted variance growth for tagged-particle position would provide an independent test of the scaling limit.

Load-bearing premise

The initial variables are independent Gaussians for the momenta and independent Gammas for the exponential bond terms, and the lattice dynamics are faithfully captured by the scattering of a dense collection of quasi-particles whose fluctuations form the dressed Lévy-Chentsov field.

What would settle it

Simulate many independent copies of the Toda lattice from the stated equilibrium distribution, compute the diffusively rescaled current field or the rescaled position q_0(t)/sqrt(t) of a tagged particle, and test whether the empirical distribution converges to the predicted Gaussian process (or to a normal random variable with the predicted variance).

read the original abstract

In this paper we consider the Toda lattice $(\mathbf{p}(t);\mathbf{q}(t))$ at thermal equilibrium, meaning that its variables $(p_j)$ and $(e^{q_j - q_{j+1}})$ are independent Gaussian and Gamma random variables, respectively. We show under diffusive scaling that the space-time fluctuations for the model's currents converge to an explicit Gaussian limit. As consequences, we deduce, (i) the scaling limit for the trajectory of a single particle $q_0$ is a Brownian motion; (ii) space-time two-point correlation functions for the model decay inversely with time, with explicit scaling distributions predicted by Spohn (Spohn, J. Phys. A 53 (2020), 265004). Our starting point is the notion that the Toda lattice can be thought of as a dense collection of many ``quasi-particles'' that interact through scattering. The core of our work is to establish that the full joint scaling limit of the fluctuations for these quasi-particles is given by a Gaussian process, called a dressed L\'evy-Chentsov field.

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 derives an explicit Gaussian limit for Toda lattice current fluctuations via a dressed Lévy-Chentsov field from quasi-particle scattering, confirming Spohn predictions, but the mapping from independent equilibrium to the joint process needs tight proof checks.

read the letter

The main takeaway is that Aggarwal and Nicoletti establish convergence of space-time current fluctuations in the Toda lattice to a Gaussian process under diffusive scaling, starting from the product equilibrium with independent Gaussians for momenta and Gammas for the position differences. This yields Brownian motion for a tagged particle trajectory and the inverse-time correlation decay that Spohn predicted from hydrodynamics. The dressed Lévy-Chentsov field is presented as the joint limit of the quasi-particle fluctuations after scattering, and that construction appears new rather than a reduction to earlier processes. The paper states the starting measure clearly and spells out the consequences without extra parameters, which is a clean way to link the microscopic integrable dynamics to the macroscopic limits. The soft spot is the quasi-particle representation: the claim that scattering preserves the exact equilibrium measure and produces the stated Gaussian covariance under scaling rests on controlling the dense collection and the transform without introducing hidden dependencies. The stress-test concern about independence mapping to the dressed field covariance is the central one to verify in the estimates; if those bounds close, the result holds, but any gap there would affect the explicit form. This is for readers working on hydrodynamic limits and fluctuations in integrable particle systems. It would be worth a reading group if the group covers mathematical statistical mechanics or lattice models, since the method could extend. The paper deserves serious referee time because the claims are precise and the approach is direct, even though the technical steps around the scattering limit require scrutiny.

Referee Report

2 major / 2 minor

Summary. The paper considers the Toda lattice at thermal equilibrium, with independent Gaussian momenta p_j and Gamma-distributed e^{q_j - q_{j+1}}. It claims that under diffusive scaling the space-time fluctuations of the currents converge to an explicit Gaussian process (the dressed Lévy-Chentsov field) obtained from a quasi-particle scattering representation. Consequences are that the tagged particle trajectory q_0 converges to Brownian motion and that space-time two-point correlations decay as 1/t with explicit scaling distributions matching Spohn's predictions.

Significance. If the central convergence holds, the work supplies a rigorous microscopic derivation of the fluctuation field for an integrable system from its exact product equilibrium measure, yielding parameter-free predictions for currents, tagged-particle diffusion, and correlations. This strengthens the quasi-particle picture and provides a concrete test of hydrodynamic fluctuation theory for the Toda lattice.

major comments (2)

[Core quasi-particle representation (abstract and main derivation)] The passage from the product initial measure to the dressed Lévy-Chentsov covariance under diffusive scaling and quasi-particle scattering must be shown explicitly; any approximation in the dense limit or scattering transform could alter the claimed Gaussianity and independence properties.
[Consequence (i)] The deduction that q_0 scales to Brownian motion (consequence i) relies on integrating the current fluctuations; the variance computation or the precise use of the limit process to obtain the diffusion coefficient should be isolated and verified independently of the full field convergence.

minor comments (2)

[Notation and definitions] The covariance function of the dressed Lévy-Chentsov field should be written out explicitly in the main text rather than left to the quasi-particle picture.
[Scaling limits] Clarify the precise diffusive scaling (space-time rescaling factors) used for the currents and for the particle trajectory.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and describe the revisions that will be made to clarify the arguments.

read point-by-point responses

Referee: [Core quasi-particle representation (abstract and main derivation)] The passage from the product initial measure to the dressed Lévy-Chentsov covariance under diffusive scaling and quasi-particle scattering must be shown explicitly; any approximation in the dense limit or scattering transform could alter the claimed Gaussianity and independence properties.

Authors: We agree that the transition from the product equilibrium measure to the covariance of the dressed Lévy-Chentsov field deserves a more explicit treatment. The current manuscript derives this limit in Sections 2–3 via the quasi-particle representation, but the steps involving the dense limit and the scattering map can be expanded. In the revision we will insert a dedicated subsection that computes the covariance explicitly from the independent Gaussian and Gamma variables, showing that the diffusive scaling and the scattering transform preserve Gaussianity and the required independence properties without additional approximations. Exact expressions for the dressed covariance will be provided. revision: yes
Referee: [Consequence (i)] The deduction that q_0 scales to Brownian motion (consequence i) relies on integrating the current fluctuations; the variance computation or the precise use of the limit process to obtain the diffusion coefficient should be isolated and verified independently of the full field convergence.

Authors: We accept that the Brownian-motion limit for the tagged particle should be established independently. The manuscript obtains this as a corollary of the full field convergence, but the variance calculation can be isolated. In the revised Section 4 we will add a self-contained argument that integrates the current fluctuations against the limit Gaussian process, computes the resulting variance directly, and extracts the diffusion coefficient without invoking the entire space-time field. This will be presented prior to the general correlation results. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation proceeds from explicit product measure via quasi-particle scattering to Gaussian limit

full rationale

The paper begins with the independent Gaussian-Gamma product equilibrium measure on the Toda variables and invokes the quasi-particle scattering representation to derive the joint diffusive scaling limit of current fluctuations as the dressed Lévy-Chentsov field. From this limit it obtains the Brownian trajectory for q_0 and the inverse-time correlation decay. No equation reduces the target Gaussian process to a fitted parameter, no self-citation supplies a load-bearing uniqueness theorem, and the central claim is not equivalent to the inputs by construction; the steps rely on the integrable dynamics and the initial independence assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the thermal equilibrium measure and the quasi-particle representation; no free parameters are fitted and no new particles are postulated beyond the named limit process.

axioms (1)

domain assumption The variables (p_j) are i.i.d. Gaussian and (e^{q_j - q_{j+1}}) are i.i.d. Gamma, independent of each other.
Stated in the abstract as the definition of thermal equilibrium for the Toda lattice.

invented entities (1)

dressed Lévy-Chentsov field no independent evidence
purpose: The Gaussian process that is the scaling limit of the joint fluctuations of the quasi-particles.
Introduced as the core object whose existence is proved; it is a named limit process rather than a new physical entity.

pith-pipeline@v0.9.0 · 5484 in / 1389 out tokens · 28468 ms · 2026-05-10T11:53:32.636180+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

76 extracted references · 6 canonical work pages · 3 internal anchors

[1]

Asymptotic Scattering Relation for the Toda Lattice

A. Aggarwal. Asymptotic scattering relation for the Toda lattice. Preprint, arXiv:2503.08018

work page internal anchor Pith review Pith/arXiv arXiv
[2]

Aggarwal

A. Aggarwal. Effective velocities in the Toda lattice. To appear inCommun. Pure Appl. Math., Preprint, arXiv:2503.11407

work page arXiv
[3]

Allez, J.-P

R. Allez, J.-P. Bouchaud, and A. Guionnet. Invariant beta ensembles and the Gauss-Wigner crossover.Phys. Rev. Lett., 109(9):094102, 2012

2012
[4]

Berthet and D

P. Berthet and D. M. Mason. Revisiting two strong approximation results of Dudley and Philipp. InHigh dimensional probability, volume 51 ofIMS Lecture Notes Monogr. Ser., pages 155–172. Inst. Math. Statist., Beachwood, OH, 2006

2006
[5]

Bertini, M

B. Bertini, M. Collura, J. De Nardis, and M. Fagotti. Transport in out-of-equilibrium XXZ chains: exact profiles of charges and currents.Physical review letters, 117(20):207201, 2016

2016
[6]

Bhatia.Matrix Analysis, volume 169 ofGraduate Texts in Mathematics

R. Bhatia.Matrix Analysis, volume 169 ofGraduate Texts in Mathematics. Springer-Verlag, New York, NY, 1 edition, 1997

1997
[7]

Bloch, F

A. Bloch, F. Golse, T. Paul, and A. Uribe. Dispersionless Toda and Toeplitz operators.Duke Math. J., 117(1):157–196, 2003

2003
[8]

Bodineau, I

T. Bodineau, I. Gallagher, and L. Saint-Raymond. The Brownian motion as the limit of a deterministic system of hard-spheres.Invent. Math., 203(2):493–553, 2016. 164 AMOL AGGARWAL AND MATTHEW NICOLETTI

2016
[9]

Bodineau, I

T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. Dynamics of dilute gases: a statistical approach. InICM—International Congress of Mathematicians. Vol. 2. Plenary lectures, pages 750–795. EMS Press, Berlin, 2023

2023
[10]

Bodineau, I

T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. Long-time correlations for a hard-sphere gas at equilibrium.Comm. Pure Appl. Math., 76(12):3852–3911, 2023

2023
[11]

Bodineau, I

T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella. Long-time derivation at equilibrium of the fluctuating Boltzmann equation.Ann. Probab., 52(1):217–295, 2024

2024
[12]

Boldrighini, R

C. Boldrighini, R. L. Dobrushin, and Y. M. Sukhov. One-dimensional hard rod caricature of hydrodynamics.J. Statist. Phys., 31(3):577–616, 1983

1983
[13]

Navier-Stokes cor- rection

C. Boldrighini and Y. M. Suhov. One-dimensional hard-rod caricature of hydrodynamics: “Navier-Stokes cor- rection” for local equilibrium initial states.Commun. Math. Phys., 189(2):577–590, 1997

1997
[14]

O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura. Emergent hydrodynamics in integrable quantum systems out of equilibrium.Physical Review X, 6(4):041065, 2016

2016
[15]

N. N. Chentsov. L´ evy Brownian motion for several parameters and generalized white noise.Theory Probab. Appl., 2(2):265–266, 1957

1957
[16]

I. Corwin. The Kardar-Parisi-Zhang equation and universality class.Random Matrices Theory Appl., 1(1):1130001, 76, 2012

2012
[17]

D. A. Croydon and M. Sasada. Generalized hydrodynamic limit for the box–ball system.Commun. Math. Phys., 383(1):427–463, 2021

2021
[18]

De Nardis, D

J. De Nardis, D. Bernard, and B. Doyon. Hydrodynamic diffusion in integrable systems.Phys. Rev. Lett., 121(16):160603, 2018

2018
[19]

De Nardis, D

J. De Nardis, D. Bernard, and B. Doyon. Diffusion in generalized hydrodynamics and quasiparticle scattering. SciPost Phys., 6:049, 2019

2019
[20]

Deift, S

P. Deift, S. Kamvissis, T. Kriecherbauer, and X. Zhou. The Toda rarefaction problem.Commun. Pure Appl. Math., 49(1):35–83, 1996

1996
[21]

Deift and K

P. Deift and K. T.-R. McLaughlin. A continuum limit of the Toda lattice.Mem. Amer. Math. Soc., 131(624):x+216, 1998

1998
[22]

Diederich

S. Diederich. A conventional approach to dynamic correlations in the toda lattice.Phys. Lett. A, 85(4):233–235, 1981

1981
[23]

B. Doyon. Lecture notes on generalised hydrodynamics.SciPost Physics Lecture Notes, page 018, 2020

2020
[24]

Towards an ab initio derivation of generalised hydrodynamics from a gas of interacting wave packets

B. Doyon and F. H¨ ubner. Ab initio derivation of generalised hydrodynamics from a gas of interacting wave packets. Preprint, arXiv:2307.09307

work page internal anchor Pith review Pith/arXiv arXiv
[25]

Doyon, F

B. Doyon, F. H¨ ubner, and T. Yoshimura. New classical integrable systems from generalized tt-deformations. Phys. Rev. Lett., 132(25):251602, 2024

2024
[26]

Doyon, T

B. Doyon, T. Yoshimura, and J.-S. Caux. Soliton gases and generalized hydrodynamics.Phys. Rev. Lett., 120(4):045301, 2018

2018
[27]

T. K. Duy and T. Shirai. The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles.Electron. Commun. Probab., 20:no. 68, 13, 2015

2015
[28]

Dworaczek Guera and R

C. Dworaczek Guera and R. Memin. CLT for realβ-ensembles at high temperature.Electron. J. Probab., 29:Paper No. 171, 45, 2024

2024
[30]

P. A. Ferrari, C. Franceschini, D. G. E. Grevino, and H. Spohn. Hard rod hydrodynamics and the L´ evy Chentsov field. InProbability and statistical mechanics—papers in honor of Errico Presutti, volume 38 ofEnsaios Mat., pages 185–222. Soc. Brasil. Mat., Rio de Janeiro, 2023

2023
[31]

P. A. Ferrari, C. Nguyen, L. T. Rolla, and M. Wang. Soliton decomposition of the box-ball system.Forum Math. Sigma, 9:Paper No. e60, 37, 2021

2021
[32]

P. A. Ferrari and S. Olla. Macroscopic diffusive fluctuations for generalized hard rods dynamics.Ann. Appl. Probab., 35(2):1125–1142, 2025

2025
[33]

P. L. Ferrari and H. Spohn. Scaling limit for the space-time covariance of the stationary totally asymmetric simple exclusion process.Comm. Math. Phys., 265(1):1–44, 2006

2006
[34]

Flaschka

H. Flaschka. The Toda lattice. II. Existence of integrals.Phys. Rev. B, 9(4):1924–1925, 1974

1924
[35]

Gopalakrishnan, D

S. Gopalakrishnan, D. A. Huse, V. Khemani, and R. Vasseur. Hydrodynamics of operator spreading and quasi- particle diffusion in interacting integrable systems.Phys. Rev. B, 98(22):220303, 2018. FLUCTUATIONS FOR THE TODA LATTICE 165

2018
[36]

Gou¨ ezel

S. Gou¨ ezel. Limit theorems in dynamical systems using the spectral method. InHyperbolic dynamics, fluctuations and large deviations, volume 89 ofProc. Sympos. Pure Math., pages 161–193. Amer. Math. Soc., Providence, RI, 2015

2015
[37]

Large deviations of the periodic Toda chain

T. Grava, A. Guionnet, K. K. Kozlowski, and A. Little. Large deviations of the periodic Toda chain. Preprint, arXiv:2604.00635

work page internal anchor Pith review Pith/arXiv arXiv
[38]

Guionnet and R

A. Guionnet and R. M´ emin. Large deviations for Gibbs ensembles of the classical Toda chain.Electron. J. Probab., 27:1–29, 2022

2022
[39]

Johansson

K. Johansson. Shape fluctuations and random matrices.Commun. Math. Phys., 209(2):437–476, 2000

2000
[40]

Kappeler and P

T. Kappeler and P. Topalov. Global wellposedness of KdV inH −1(T,R).Duke Math. J., 135(2):327–360, 2006

2006
[41]

Kardar, G

M. Kardar, G. Parisi, and Y.-C. Zhang. Dynamic scaling of growing interfaces.Phys. Rev. Lett., 56(9):889, 1986

1986
[42]

Killip, J

R. Killip, J. Murphy, and M. Visan. Invariance of white noise for KdV on the line.Invent. Math., 222(1):203–282, 2020

2020
[43]

Kr¨ uger and G

H. Kr¨ uger and G. Teschl. Long-time asymptotics of the Toda lattice for decaying initial data revisited.Rev. Math. Phys., 21(1):61–109, 2009

2009
[44]

Kunz and B

H. Kunz and B. Souillard. Sur le spectre des op´ erateurs aux diff´ erences finies al´ eatoires.Commun. Math. Phys., 78(2):201–246, 1980/81

1980
[45]

L´ evy.Processus Stochastiques et Mouvement Brownien

P. L´ evy.Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Lo` eve. Gauthier-Villars, Paris, 1948

1948
[46]

Lukkarinen and H

J. Lukkarinen and H. Spohn. Weakly nonlinear Schr¨ odinger equation with random initial data.Invent. Math., 183(1):79–188, 2011

2011
[47]

S. V. Manakov. Complete integrability and stochastization of discrete dynamical systems.Soviet Physics JETP 40 (1974), pages 269–274, 1974

1974
[48]

Matetski, J

K. Matetski, J. Quastel, and D. Remenik. Polynuclear growth and the Toda lattice. To appear inJ. Eur. Math. Soc., Preprint, arXiv:2209.02643

work page arXiv
[49]

Matetski, J

K. Matetski, J. Quastel, and D. Remenik. The KPZ fixed point.Acta Math., 227(1):115–203, 2021

2021
[50]

G. Mazzuca. On the mean density of states of some matrices related to the beta ensembles and an application to the Toda lattice.J. Math. Phys., 63(4):Paper No. 043501, 13, 2022

2022
[51]

Mazzuca, T

G. Mazzuca, T. Grava, T. Kriecherbauer, K. T.-R. McLaughlin, C. B. Mendl, and H. Spohn. Equilibrium spacetime correlations of the Toda lattice on the hydrodynamic scale.J. Stat. Phys., 190(8):Paper No. 149, 22, 2023

2023
[52]

Mazzuca and R

G. Mazzuca and R. Memin. CLT forβ-ensembles at high temperature and for integrable systems: a transfer operator approach.Ann. Henri Poincar´ e, 26(1):245–316, 2025

2025
[53]

N. Minami. Local fluctuation of the spectrum of a multidimensional Anderson tight binding model.Commun. Math. Phys., 177(3):709–725, 1996

1996
[54]

J. Moser. Finitely many mass points on the line under the influence of an exponential potential—an integrable system. In A. Deprit and A. Elipe, editors,Dynamical Systems, Theory and Applications, volume 38 ofLecture Notes in Physics, pages 467–497. Springer, Berlin / New York, 1975. Reprinted from the Rencontres/Battelle Recontres (Seattle 1974)

1975
[55]

S. Olla, M. Sasada, and H. Suda. Scaling limits of solitons in the box-ball system. Preprint, arXiv:2411.14818

work page arXiv
[56]

M. Opper. Analytical solution of the classical Bethe ansatz equation for the Toda chain.Phys. Lett. A, 112(5):201–203, 1985

1985
[57]

Pr¨ ahofer and H

M. Pr¨ ahofer and H. Spohn. Exact scaling functions for one-dimensional stationary KPZ growth.J. Statist. Phys., 115(1-2):255–279, 2004

2004
[58]

Presutti and W

E. Presutti and W. D. Wick. Macroscopic stochastic fluctuations in a one-dimensional mechanical system.J. Statist. Phys., 52(1-2):497–502, 1988

1988
[59]

J. Quastel. Introduction to KPZ. In S. Yau, editor,Current Developments in Mathematics, pages 125–194. International Press, 2012. Available as a survey of the KPZ equation and universality class

2012
[60]

Quastel and B

J. Quastel and B. Valk´ o. KdV preserves white noise.Commun. Math. Phys., 277(3):707–714, 2008

2008
[61]

N. Saitoh. A transformation connecting the Toda lattice and the KdV equation.J. Phys. Soc. Japan, 49(1):409– 416, 1980

1980
[62]

Schenker

J. Schenker. Eigenvector localization for random band matrices with power law band width.Commun. Math. Phys., 290(3):1065–1097, 2009

2009
[63]

Schneider

T. Schneider. Classical statistical mechanics of lattice dynamic model systems: transfer integral and molecular- dynamics studies. InStatics and Dynamics of Nonlinear Systems: Proceedings of a Workshop at the Ettore Majorana Centre, Erice, Italy, 1–11 July, 1983, pages 212–241. Springer, 1983. 166 AMOL AGGARWAL AND MATTHEW NICOLETTI

1983
[64]

Schneider and E

T. Schneider and E. Stoll. Excitation spectrum of the Toda lattice: a molecular-dynamics study.Phys. Rev. Lett., 45(12):997, 1980

1980
[65]

H. Spohn. Hydrodynamical theory for equilibrium time correlation functions of hard rods.Ann. Physics, 141(2):353–364, 1982

1982
[66]

H. Spohn. Nonlinear fluctuating hydrodynamics for anharmonic chains.Journal of Statistical Physics, 154(5):1191–1227, 2014

2014
[67]

H. Spohn. Ballistic space-time correlators of the classical Toda lattice.Journal of Physics A: Mathematical and Theoretical, 53(26):265004, jun 2020

2020
[68]

H. Spohn. Generalized Gibbs ensembles of the classical Toda chain.Journal of Statistical Physics, 180(1):4–22, 2020

2020
[69]

Spohn.Hydrodynamic Scales of Integrable Many-Body Systems

H. Spohn.Hydrodynamic Scales of Integrable Many-Body Systems. World Scientific Publishing Co., 2024

2024
[70]

D. W. Stroock.Gaussian measures in finite and infinite dimensions. Universitext. Springer, Cham, 2023

2023
[71]

Takahashi and J

D. Takahashi and J. Satsuma. A soliton cellular automaton.J. Phys. Soc. Japan, 59(10):3514–3519, 1990

1990
[72]

Teschl.Jacobi operators and completely integrable nonlinear lattices, volume 72 ofMathematical Surveys and Monographs

G. Teschl.Jacobi operators and completely integrable nonlinear lattices, volume 72 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2000

2000
[73]

M. Toda. Wave propagation in anharmonic lattices.J. Phys. Soc. Japan, 23(3):501–506, 1967

1967
[74]

L. N. Trefethen.Approximation theory and approximation practice. Society for Industrial and Applied Mathe- matics (SIAM), Philadelphia, PA, 2020

2020
[75]

van Beijeren

H. van Beijeren. Exact results for anomalous transport in one-dimensional Hamiltonian systems.Phys. Rev. Lett., 108:180601, Apr 2012

2012
[76]

Venakides, P

S. Venakides, P. Deift, and R. Oba. The Toda shock problem.Commun. Pure Appl. Math., 44(8–9):1171–1242, 1991

1991
[77]

A. Y. Zaitsev. Estimates of the L´ evy–Prokhorov distance in the multivariate central limit theorem for random variables with finite exponential moments.Theory Probab. Appl., 31(2):203–220, 1987

1987