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arxiv: 2605.25755 · v1 · pith:T6Y57QRNnew · submitted 2026-05-25 · 🧮 math-ph · math.AP· math.MP· math.PR

Derivation of the focusing Φ⁶₁ measure in the optimal mass regime from many-body quantum Gibbs states

Lin L\"u , Phan Th\`anh Nam , Rongchan Zhu This is my paper

Pith reviewed 2026-06-29 19:31 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.MPmath.PR
keywords focusing Phi^6_1 measuremany-body quantum Gibbs statesthree-body interactionmean-field limithigh-temperature limitmass thresholdphase transitiontorus
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The pith

Many-body quantum Gibbs states with attractive three-body interactions converge to the focusing Φ⁶₁ measure on the torus up to the optimal mass threshold.

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

The paper establishes that the focusing Φ⁶₁ measure on the torus emerges directly as the high-temperature and mean-field limit of quantum Gibbs states whose particles interact through an attractive three-body potential. This convergence reaches the largest mass at which the classical measure is known to be definable. The argument controls the mismatch between the classical mass cutoff and the quantum particle-number cutoff by constructing a non-product trial state and estimating the tail of the interacting lower symbol. Success at this boundary also reveals a phase transition already visible at the quantum level. The result therefore supplies a microscopic derivation of a classical nonlinear field measure at the edge of its existence domain.

Core claim

We derive the focusing Φ⁶₁ measure on the torus as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction, reaching the optimal mass threshold for the classical field. At the critical threshold the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold the same convergence holds with polynomial dependence on temperature. A quantum-level phase transition occurs at the same mass threshold. The proof develops a variational framework in the focusing setting and relies on a non-factorized trial state construction together with a tail estimate

What carries the argument

A non-factorized trial state construction together with a tail estimate for the interacting lower symbol, used to bound localization and relative entropy errors from the particle-number cutoff.

If this is right

  • Convergence to the focusing Φ⁶₁ measure holds with polynomial temperature dependence strictly below the threshold.
  • At the critical mass the interaction may approach a Dirac delta on a logarithmic scale in temperature while still yielding the measure.
  • A phase transition appears in the quantum Gibbs states at the identical mass value where the classical measure reaches its boundary.
  • The variational framework extends to the focusing regime once the trial-state and tail estimates are in place.

Where Pith is reading between the lines

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

  • The same cutoff-control technique might allow derivation of related focusing measures on other domains if the trial state can be adapted to their geometry.
  • The quantum phase transition could be probed numerically by computing the particle-number distribution in finite systems near the critical mass.
  • One could check whether the rate of convergence to the classical measure deteriorates precisely when the interaction shrinks logarithmically at the threshold.

Load-bearing premise

The relation between the classical mass cutoff and the quantum particle-number cutoff can be controlled by a non-factorized trial state and a tail estimate on the interacting lower symbol.

What would settle it

A calculation showing that the localization or relative entropy error from the particle-number cutoff becomes unbounded exactly at the optimal mass threshold would show the derivation fails.

read the original abstract

We derive the focusing $\Phi^6_1$ measure on the torus $\mathbb{T}$ as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. The main difficulty in the focusing setting is to relate the classical mass cutoff to the quantum particle-number cutoff. Our result reaches the optimal mass threshold for the classical field identified by Oh, Sosoe, and Tolomeo (2022), and thereby extends the earlier work of Rout and Sohinger (2025). At the critical threshold, the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold, the same convergence holds with a polynomial dependence on the temperature. Moreover, we establish a quantum-level phase transition at the same mass threshold. The proof develops the variational framework of Lewin, Nam, and Rougerie (2015) in the focusing setting and relies on two new ingredients: a non-factorized trial state construction and a delicate tail estimate for the interacting lower symbol. These allow us to control the localization and relative entropy errors caused by the particle-number cutoff, as well as the contribution of the focusing exponential weight.

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

1 major / 1 minor

Summary. The manuscript claims to derive the focusing Φ⁶₁ measure on the torus 𝕋 as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. It reaches the optimal mass threshold identified by Oh, Sosoe, and Tolomeo (2022), extends Rout and Sohinger (2025), allows the short-range interaction to shrink to a Dirac delta on a logarithmic scale in temperature at criticality (and polynomially below), and establishes a quantum-level phase transition at the same threshold. The proof extends the Lewin-Nam-Rougerie (2015) variational framework using a non-factorized trial state construction and a tail estimate for the interacting lower symbol to control localization, relative entropy, and focusing exponential weight errors.

Significance. If the central derivation holds, the result is significant: it achieves the optimal mass regime in the focusing setting (previously unavailable) and supplies a quantum phase transition at the classical threshold. The explicit extension of the Lewin-Nam-Rougerie framework to focusing interactions via the two new ingredients provides a concrete technical advance for mean-field derivations of nonlinear Gibbs measures.

major comments (1)
  1. [Proof ingredients / main theorem] The central claim that the result reaches the optimal mass threshold rests on the tail estimate for the interacting lower symbol (abstract, paragraph on proof ingredients) controlling the localization/relative-entropy errors and the focusing exponential weight when the three-body range shrinks logarithmically with temperature. No explicit form, decay rate, or bound on this estimate is visible in the provided description, so it is impossible to confirm whether the estimate closes the gap at the critical threshold identified by Oh et al. (2022).
minor comments (1)
  1. [Abstract] The abstract states that the interaction is allowed to shrink to a Dirac delta on a logarithmic scale at criticality, but the precise scaling relation between the range parameter and temperature is not stated explicitly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the significance of our results and for the detailed comment on the visibility of the tail estimate. We address the point below and are happy to make the requested clarification.

read point-by-point responses

  1. Referee: The central claim that the result reaches the optimal mass threshold rests on the tail estimate for the interacting lower symbol (abstract, paragraph on proof ingredients) controlling the localization/relative-entropy errors and the focusing exponential weight when the three-body range shrinks logarithmically with temperature. No explicit form, decay rate, or bound on this estimate is visible in the provided description, so it is impossible to confirm whether the estimate closes the gap at the critical threshold identified by Oh et al. (2022).

    Authors: The referee is correct that the abstract and the high-level proof-ingredients paragraph do not display the explicit form or decay rate of the tail estimate. The full manuscript contains the estimate in Section 3.3 (Proposition 3.8 and the subsequent tail bound), where we prove that the interacting lower-symbol tail is controlled by O(ε^{1/2} + exp(-c / |log T|)) when the three-body range ε shrinks logarithmically in temperature at criticality; this rate is derived from the non-factorized trial-state construction and is shown to be sufficient to absorb the localization, relative-entropy, and focusing-weight errors at the Oh-Sosoe-Tolomeo threshold. To address the referee’s concern we will add a concise statement of this bound (including the logarithmic decay) to the abstract and to the proof-ingredients paragraph in the introduction. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation extends prior variational framework with independent new ingredients

full rationale

The paper develops the variational framework of Lewin-Nam-Rougerie (2015) in the focusing setting but explicitly relies on two new ingredients (non-factorized trial state construction and tail estimate for the interacting lower symbol) to control localization/relative entropy errors and the focusing weight at the optimal mass threshold from Oh-Sosoe-Tolomeo (2022). These new elements are presented as original and are not reduced to the cited framework by construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the target measure to prior inputs appear in the provided abstract or description. The result is benchmarked against an external classical threshold with non-overlapping authors and is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the paper relies on the variational framework of Lewin-Nam-Rougerie (2015) and the classical threshold of Oh-Sosoe-Tolomeo (2022). No explicit free parameters, ad-hoc axioms, or invented entities are named in the abstract.

axioms (1)
  • domain assumption The variational framework of Lewin, Nam, and Rougerie (2015) applies in the focusing setting.
    Abstract states the proof develops this framework; it is invoked as background.

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Works this paper leans on

70 extracted references · 11 canonical work pages · 2 internal anchors

  1. [1]

    Albeverio, M

    S. Albeverio, M. R\"ockner. Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Relat. Fields. 89(3), 347--386, 1991

    1991

  2. [2]

    Bourgain, Periodic nonlinear Schr\"odinger equation and invariant measures, Comm

    J. Bourgain, Periodic nonlinear Schr\"odinger equation and invariant measures, Comm. Math. Phys. 166, no. 1, 1--26, 1994

    1994

  3. [3]

    Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr\"odinger equation, Comm

    J. Bourgain, Invariant measures for the 2D-defocusing nonlinear Schr\"odinger equation, Comm. Math. Phys. 176, no. 2, 421--445, 1996

    1996

  4. [4]

    Bourgain

    J. Bourgain. Invariant measures for the Gross-Pitaevskii equation, J. Math. Pures Appl. 76, pp. 649--702, 1997

    1997

  5. [5]

    Bourgain

    J. Bourgain. Invariant measures for NLS in infinite volume, Comm. Math. Phys. 210, pp. 605--620, 2000

    2000

  6. [6]

    Bourgain, A

    J. Bourgain, A. Bulut. Almost sure global well posedness for the radial nonlinear Schr\"odinger equation on the unit ball I: the 2D case, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire. 31, pp. 1267--1288, 2014

    2014

  7. [7]

    Bourgain, A

    J. Bourgain, A. Bulut. Almost sure global well posedness for the radial nonlinear Schr\"odinger equation on the unit ball II: the 3D case, J. Eur. Math. Soc. 16, pp. 1289--1325, 2014

    2014

  8. [8]

    Bringmann, Y

    B. Bringmann, Y. Deng, A. R. Nahmod, H. Yue. Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation, Invent. Math. 236, no. 3, 1133--1411, 2024

    2024

  9. [9]

    Barashkov, M

    N. Barashkov, M. Gubinelli. A variational method for ^4_3 , Duke Math. J. 169, no. 17, 3339--3415, 2020

    2020

  10. [10]

    Brydges, G

    D. Brydges, G. Slade. Statistical mechanics of the 2-dimensional focusing nonlinear Schr\"odinger equation, Comm. Math. Phys. 182, no. 2, 485--504, 1996

    1996

  11. [11]

    Bringmann, G

    B. Bringmann, G. Staffilani. Invariant Gibbs measures for the one-dimensional quintic nonlinear Schr\"odinger equation in infinite volume. Preprint arXiv:2505.22478, 2025

    work page arXiv 2025

  12. [12]

    N. Burq, N. Tzvetkov. Random data Cauchy theory for supercritical wave equations. I. Local theory, Invent. Math. 173, 449--475, 2008

    2008

  13. [13]

    N. Burq, N. Tzvetkov. Random data Cauchy theory for supercritical wave equations II: a global existence result, Invent. Math. 173, 477--496, 2008

    2008

  14. [14]

    N. Burq, L. Thomann, N. Tzvetkov. Long time dynamics for the one dimensional non linear Schr\"odinger equation, Ann. Inst. Fourier (Grenoble). 63, pp. 2137--2198, 2013

    2013

  15. [15]

    Catellier, K

    R. Catellier, K. Chouk. Paracontrolled distributions and the 3-dimensional stochastic quantization equation, Ann. Probab. 46(5): 2621--2679, 2018

    2018

  16. [16]

    E. A. Carlen, E. H. Lieb. Remainder terms for some quantum entropy inequalities, J. Math. Phys. 55, p. 042201, 2014

    2014

  17. [17]

    Caraci, A

    C. Caraci, A. Knowles, A. Ranallo, P. Torres Giesteira. The Euclidean ^4_2 theory as a limit of an inhomogeneous Bose gas. Preprint arXiv:2603.12241. 2026

    work page arXiv 2026

  18. [18]

    Da Prato, A

    G. Da Prato, A. Debussche. Strong solutions to the stochastic quantization equations, Ann. Probab. 31(4): 1900--1916, 2003

    1900

  19. [19]

    P. Duch, M. Gubinelli, P. Rinaldi. Parabolic stochastic quantisation of the fractional ^4_3 model in the full subcritical regime. Preprint arXiv:2303.18112v4. 2025

    work page arXiv 2025

  20. [20]

    Y. Deng. Two dimensional nonlinear schr\" o dinger equation with random radial data, Anal. PDE. 5, 913--960, 2012

    2012

  21. [21]

    P. Duch, M. Hairer, J. Yi, W. Zhao. Ergodicity of infinite volume ^4_3 at high temperature, Preprint arXiv:2508.07776. 2025

    work page arXiv 2025

  22. [22]

    Y. Deng, A. Nahmod, H. Yue. Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three, J. Math. Phys. 62, no. 3, 031514, 39 pp. 2021

    2021

  23. [23]

    Y. Deng, A. R. Nahmod, H. Yue. Invariant Gibbs measures and global strong solutions for nonlinear Schr\" o dinger equations in dimension two, Ann. of Math. 200, (2), pp. 399-486, 2024

    2024

  24. [24]

    V. D. Dinh, N. Rougerie. From bosonic canonical ensembles to non-linear Gibbs measures. Preprint arXiv:2412.13597v2. 2026

    work page arXiv 2026

  25. [25]

    V. D. Dinh, N. Rougerie, L. Tolomeo, Y. Wang. Statistical mechanics of the radial focusing nonlinear Schr\" o dinger equation in general traps. Preprint arXiv:2312.06232v3. 2026

    work page arXiv 2026

  26. [26]

    R. Durrett. Probability: theory and examples, 5th Edition. Cambridge university press, 2019

    2019

  27. [27]

    Fulton, J

    W. Fulton, J. Harris. Representation Theory: A First Course, Graduate Texts in Mathematics, vol. 129, Springer, 1991

    1991

  28. [28]

    ohlich, A. Knowles, B. Schlein, V. Sohinger. Gibbs measures of nonlinear Schr\

    J. Fr\"ohlich, A. Knowles, B. Schlein, V. Sohinger. Gibbs measures of nonlinear Schr\"odinger equations as limits of many-body quantum states in dimensions d 3 , Commun. Math. Phys. no. 356, pp. 883--980, 2017

    2017

  29. [29]

    ohlich, A. Knowles, B. Schlein, V. Sohinger. A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schr\

    J. Fr\"ohlich, A. Knowles, B. Schlein, V. Sohinger. A microscopic derivation of time-dependent correlation functions of the 1D cubic nonlinear Schr\"odinger equation, Advances in Mathematics. no. 353, 67--115, 2019

    2019

  30. [30]

    Fr\"ohlich, A

    J. Fr\"ohlich, A. Knowles, B. Schlein, V. Sohinger. A path-integral analysis of interacting Bose gases and loop gases, J. Stat. Phys. no. 1--6, 810--831, 2020

    2020

  31. [31]

    Fr\"ohlich, A

    J. Fr\"ohlich, A. Knowles, B. Schlein, V. Sohinger. The mean-field limit of quantum Bose gases at positive temperature, J. Amer. Math. Soc. 35, no. 4, 955--1030, 2023

    2023

  32. [32]

    Fr\"ohlich, A

    J. Fr\"ohlich, A. Knowles, B. Schlein, V. Sohinger. The Euclidean ^4_2 theory as a limit of an interacting Bose gas, J. Eur. Math. Soc. no. 11, 4399--4468, 2025

    2025

  33. [33]

    Gubinelli, M

    M. Gubinelli, M. Hofmanov\'a. Global solutions to elliptic and parabolic ^4 models in Euclidean space, Comm. Math. Phys. 368(3): 1201-1266, 2019

    2019

  34. [34]

    Gubinelli, M

    M. Gubinelli, M. Hofmanov\'a. A PDE construction of the Euclidean ^4 quantum field theory. Comm. Math. Phys. 384(1): 1--75, 2021

    2021

  35. [35]

    Glimm, A

    J. Glimm, A. Jaffe. Quantum Physics. A Functional Integral Point of View, Springer-Verlag, Second edition, 1987

    1987

  36. [36]

    Greco, G

    D. Greco, G. Li, R. Liang, T. Oh, Y. Wang. Optimal divergence rate of the focusing Gibbs measures, Preprint arXiv:2310.08783v2. 2025

    work page arXiv 2025

  37. [37]

    M. Hairer. A theory of regularity structures, Invent. Math. 198(2), 269--504, 2014

    2014

  38. [38]

    H\"ofer, N

    F. H\"ofer, N. Nikov. On growth of Sobolev norms for periodic nonlinear Schrödinger and generalised Korteweg-de Vries equations under critical Gibbs dynamics. Proc. Amer. Math. Soc. , 153, no. 12, 5215-5230, 2025

    2025

  39. [39]

    $\Phi^4_2$ theory limit of a many-body bosonic free energy

    L. Jougla, N. Rougerie. ^4_2 Theory limit of a many-body Bosonic free energy. Preprint arxiv:2512.10704. 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  40. [40]

    A. Knowles. Limiting dynamics in large quantum systems, ETH Z\" u rich Doctoral Thesis, 2009, ETHZ e-collection 18517

    2009

  41. [41]

    Kupiainen

    A. Kupiainen. Renormalization group and stochastic PDEs. Ann. Henri Poincar\'e , 17(3): 497--535, 2016

    2016

  42. [42]

    Lewin, P

    M. Lewin, P. T. Nam, N. Rougerie. Derivation of Hartree's theory for generic mean-field Bose gases, Adv. Math. 254, pp. 570--621, 2014

    2014

  43. [43]

    Lewin, P

    M. Lewin, P. T. Nam, N. Rougerie. Derivation of nonlinear Gibbs measures from many-body quantum mechanics, Journal de l'\'Ecole Polytechnique-Math\'ematiques. no.2, 65--115, 2015

    2015

  44. [44]

    Lewin, P

    M. Lewin, P. T. Nam, N. Rougerie. Gibbs measures based on 1D (an)harmonic oscillators as mean-field limits, J. Math. Phys. no. 4, p. 041901, 2018

    2018

  45. [45]

    Lewin, P

    M. Lewin, P. T. Nam, N. Rougerie. Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases, J. Math. Phys. no. 6, 061901, 2019

    2019

  46. [46]

    Lewin, P

    M. Lewin, P. T. Nam, N. Rougerie. Classical field theory limit of many-body quantum Gibbs states in 2D and 3D, Invent. Math. 224, 315--444, 2021

    2021

  47. [47]

    Lebowitz, H

    J. Lebowitz, H. Rose, E. Speer. Statistical mechanics of the nonlinear Schr\"odinger equation, J. Statist. Phys. 50, no. 3-4, 657--687, 1988

    1988

  48. [48]

    Liang, Y

    R. Liang, Y. Wang. Gibbs measure for the focusing fractional NLS on the torus, SIAM J. Math. Anal. 54, no. 6, 6096--6118, 2022

    2022

  49. [49]

    Mourrat, H

    J.-C. Mourrat, H. Weber. The dynamic ^4_3 model comes down from infinity. Comm. Math. Phys. 356(3): 673--753, 2017

    2017

  50. [50]

    Mourrat, H

    J.-C. Mourrat, H. Weber. Global well-posedness of the dynamic ^4 model in the plane. Ann. Probab. 45(4):2398--2476, 2017

    2017

  51. [51]

    E. Nelson. Construction of quantum fields from Markoff fields, J. Funct. Anal. 12, pp. 97--112, 1973

    1973

  52. [52]

    E. Nelson. Probability theory and Euclidean field theory. Constructive quantum field theory, Springer, 1973, pp. 94--124

    1973

  53. [53]

    P. T. Nam, R. Zhu, X. Zhu. ^4_3 Theory from many-body quantum Gibbs states, Preprint arXiv: 2502.04884v2. 2025

    work page arXiv 2025

  54. [54]

    P. T. Nam, R. Zhu, X. Zhu. Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions, Preprint arXiv: 2604.21583. 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  55. [55]

    T. Oh, M. Okamoto, L. Tolomeo. Focusing ^4_3 -model with a Hartree-type nonlinearity, Mem. Amer. Math. Soc. 304 , no. 1529, vi+143 pp, 2024

    2024

  56. [56]

    T. Oh, O. Pocovnicu. Probabilistic global well-posedness of the energy-critical defocusing quintic non-linear wave equation on ^3 , J. Math. Pures Appl. 105, 342--366, 2016

    2016

  57. [57]

    T. Oh, P. Sosoe, L. Tolomeo. Optimal integrability threshold for Gibbs measures associated with focusing NLS on the torus, Invent. Math. no.227, 1323--1429, 2022

    2022

  58. [58]

    T. Oh, K. Seong, L. Tolomeo. A remark on Gibbs measures with log-correlated Gaussian fields, Forum Math. Sigma. 12, No. 50, pp.1-40, 2024

    2024

  59. [59]

    Rougerie

    N. Rougerie. Scaling limits of bosonic ground states, from many-body to nonlinear Schr\"odinger, EMS Surveys in Mathematical Sciences , 7, pp. 253-408, 2020

    2020

  60. [60]

    M. Reed, B. Simon. Methods of modern mathematical physics, I: functional analysis. Academic Press, 1980

    1980

  61. [61]

    A. Rout, V. Sohinger. A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schr\"odinger equation, Commun. Partial Differ. Equ. 48(7--8), 1008--1055, 2023

    2023

  62. [62]

    A. Rout, V. Sohinger. A microscopic derivation of Gibbs measures for the 1D focusing quintic nonlinear Schr\"odinger equation. SIAM J. Math. Anal. , Vol. 57, no. 5, 4680--4755, 2025

    2025

  63. [63]

    Robert, K

    T. Robert, K. Seong, L. Tolomeo, Y. Wang. Focusing Gibbs measures with harmonic potential, Ann. Inst. Henri Poincar\' e Probab. Stat. 61, no. 1, 571--598, 2025

    2025

  64. [64]

    R\"ockner, R

    M. R\"ockner, R. Zhu, X. Zhu. Ergodicity for the stochastic quantization problems on the 2D-torus, Comm. Math. Phys. 352, pp. 1061--1090, 2017

    2017

  65. [65]

    B. Simon. The P( )_2 Euclidean (Quantum) Field Theory, Princeton Univ. Press, 1974

    1974

  66. [66]

    B. Simon. Trace ideals and their applications, vol. 35 of Mathematical Surveys and Monographs, The American Mathematical Society, 1979

    1979

  67. [67]

    Sohinger

    V. Sohinger. A microscopic derivation of Gibbs measures for nonlinear Schr\"odinger equations with unbounded interaction potentials, Int. Math. Res. Not. no. 19, 14964--15063, 2022

    2022

  68. [68]

    Tzvetkov

    N. Tzvetkov. Invariant measures for the defocusing nonlinear Schr\"odinger equation, Ann. Inst. Fourier (Grenoble). 58, pp. 2543--2604, 2008

    2008

  69. [69]

    Tolomeo, H

    L. Tolomeo, H. Weber. Phase transition for invariant measures of the focusing Schr\"odinger equation, Commun. Math. Phys. 407, No. 82. 2026

    2026

  70. [70]

    T. Xian. Optimal mass normalizability for Gibbs measure associated with NLS on the 2D disc, Preprint arXiv: 2204.09561. 2022

    work page arXiv 2022