Mass Gap of Lattice Yang–Mills Theory up to the Dobrushin Ceiling

Introduction

Wilson’s lattice gauge theory [107]¹ is where every attempt to construct four-dimensional Yang–Mills theory begins. Its configurations are assignments of a group element $Q_e$ of a compact group $G$ to each link $e$ of a lattice, its weight is $\exp\left(N\beta\sum_p\operatorname{Re}\operatorname{Tr}Q_p\right)$, a sum over elementary squares (plaquettes), and the inverse coupling $\beta$ is the only parameter.² When $\beta$ is small the theory is understood: the cluster expansion of Osterwalder and Seiler [91, 97] produces a positive gap in the spectrum of the transfer matrix and exponential decay of correlations. Shen, Zhu and Zhu [100] found a second route, through the geometry of the configuration space: they verified the Bakry–Émery criterion [2, 3] on the product manifold $G^{E}$ and obtained log-Sobolev inequalities uniform in the volume, uniqueness of the infinite-volume limit and decay of correlations. The strength of that route is set by one number, the constant in the bound

$$\lvert\operatorname{Hess} S_0(v,v) \rvert\le8(d-1)\lvert v\rvert^2 \tag*{(1)}$$

for the Hessian of the Wilson action $S_0 = \sum_p \operatorname{Re}\operatorname{Tr}Q_p$ in $d$ dimensions. The same constant reappears in later work on Langevin dynamics, on the area law and on the Yang–Mills–Higgs model [27, 87, 99].

This paper has three aims. The first is to find the best constant in (1). The second is to carry that constant, and a sharper analysis of the single-link law of SU(2), through to uniqueness and to a positive lattice mass gap. The third is structural: a criterion, checkable in one finite box, that produces decay at all scales for gauge fields, and an honest account of what still separates all of this from the continuum mass gap.

The Hessian

The bound (1) is obtained plaquette by plaquette. The cross terms between the four links of one plaquette, however, are not independent across the lattice: summed over all plaquettes they reassemble into the square of a covariant divergence, which has a sign. Keeping that square gives the sharp constant.

Theorem A (Sharp Hessian bound). Let $d \ge2$, and let $G$ be one of the groups SU($N$), SO($N$) or U($N$). For every gauge field $Q$ on $\mathbb{Z}^d$ or on a torus and every square-summable tangent vector $v=(X_eQ_e)_e$

$$\pm\operatorname{Hess} S_0(v,v)+\sum_x |D_xX|^2 \le4d|v|^2,$$

where $D_xX$ is the covariant divergence (6). In particular $|\operatorname{Hess} S_0(v,v)| \le4d|v|^2$, and for every $N \ge2$ and $d \ge2$ the constant $4d$ is attained with both signs.

For $d=2$ the two constants coincide; for $d=3$ and $d=4$ they are 12 against 16 and 16 against 24.³ The two extremal fields are drawn in Figure 3.

Strong coupling

The Ricci curvature of SU($N$) in the metric of § 1.1 is $N/2$, and that of SO($N$) is $(N-2)/4$. Subtracting the Hessian bound gives the Bakry–Émery constant directly.

Theorem B (Curvature regime). Let $G=\mathrm{SU}(N)$ and $|\beta|<1/(8d)$, or $G=\mathrm{SO}(N)$ with $N\ge3$ and $|\beta|<(N-2)/(16dN)$. The finite-volume Wilson measures satisfy the Bakry–Émery condition with a constant $K>0$ that depends neither on the volume, nor on the boundary condition, nor on the configuration. They satisfy log-Sobolev and Poincaré inequalities uniform in the volume, and every infinite-volume limit has exponentially decaying correlations of local functions.

Theorem C (The limit of the curvature method). For $G=\mathrm{SU}(N)$ on every torus of even side,

$$\inf_Q \lambda_{\min}\left(\operatorname{Ric}-N\beta\operatorname{Hess}S_0(Q)\right)=\frac{N}{2}-4dN|\beta|.$$

No Bakry–Émery argument with a lower bound uniform over configurations, in the bi-invariant metric, reaches beyond $|\beta|=1/(8d)$.

The curvature bound also controls a coupling of two Langevin dynamics. Shen, Zhu and Zhu built such a coupling and needed a second, separate estimate to make it contract in a weighted metric; that estimate cost a factor of two in the range of $\beta$. Applying Theorem A to the whole weighted sum, rather than link by link, removes the loss.

Theorem D (Uniqueness on the whole curvature range). Under the hypotheses of Theorem B the Langevin dynamics of the Wilson measure on $\mathbb{Z}^d$ has exactly one invariant probability measure $\mu$. The Wilson measures on the periodic tori $(\mathbb{Z}/L\mathbb{Z})^d$ converge to $\mu$ as $L\to\infty$, and every initial law converges to $\mu$ exponentially fast in a weighted Wasserstein distance.

SU(2) beyond the curvature range

For SU(2) the conditional law of one link given all the others is a von Mises–Fisher law on the three-sphere, and its moments are explicit. A Dobrushin argument in the chordal metric then goes past the curvature range.

Theorem E (Uniqueness for SU(2)). Let $G=\mathrm{SU}(2)$ and $|\beta|<1/(6(d-1))$. The Wilson theory on $\mathbb{Z}^d$ has exactly one infinite-volume Gibbs measure, and its truncated correlations of local Lipschitz functions decay exponentially with every rate $\lambda<-\log(6(d-1)|\beta|)$. At $d=4$ the range is $\beta_{\mathrm W}<2/9$. No single-link Wasserstein–Dobrushin argument, in the chordal or the geodesic metric, reaches beyond this range.

Here $\beta_{\mathrm W}=N^2\beta$ is the Wilson coupling of the physics literature.⁴ The range is the largest that a single-link argument can give, and it is reached because of a sharp inequality on the three-sphere, Theorem 7: under every von Mises–Fisher law $\nu_b$, the covariance of a chordally 1-Lipschitz function with a coordinate is at most $1/4$, so that $W_1(\nu_b,\nu_{b'})\le\frac14|b-b'|$. The constant $1/4$ is attained at $b=0$ and fails in the geodesic metric. The proof extends a Lipschitz function from the sphere to the ball it bounds and integrates by parts against a harmonic function; the resulting negative Sobolev norm is a series in modified Bessel functions, and the inequality becomes the positivity of the Taylor coefficients of one entire function.

A lattice mass gap

Reflection positivity turns decay of correlations in the time direction into a gap in the spectrum of the transfer matrix.

Theorem F (Lattice mass gap). *In the range of Theorem B, and for SU(2) in the range of Theorem E, the infinite-volume state is reflection positive and its transfer matrix $T$ has a spectral gap: the spectrum of $T$ on the orthogonal complement of the

vacuum lies in $[0,e^{-m}]$ with $m>0$. For SU(2) one may take $m=-\log(6(d-1)|\beta|)$, which at $d=4$ reads $m=-\log(\frac{9}{2}\beta_{\mathrm W})$ and is positive for $\beta_{\mathrm W}<2/9$.

This is a gap in lattice units at fixed lattice spacing. It is not the mass gap of the continuum theory: our earlier paper [16] shows that no inequality of this kind, uniform in the spacing and stated in lattice units, can produce the continuum gap, because the physical gap in lattice units must vanish like the spacing.

A finite-box criterion

For spin systems, Dobrushin and Shlosman showed that a single finite computation implies uniqueness and decay at all scales [43].⁵ Gauge fields need three changes: boundary data are holonomies, gauge-equivalent boundary conditions must be identified, and disagreement is measured after an interior gauge alignment. With these changes:

Theorem G (Gauge-invariant finite-box criterion). For every compact group $G$ and every finite-range gauge-invariant interaction, the finite-box inequality of Definition 16 implies uniqueness of the infinite-volume Gibbs measure, exponential decay of all truncated correlations, and the same decay on every large torus, uniformly in its size and in every ’t Hooft twist sector. Its box-hereditary form implies Wilson’s area law.

Two-dimensional SU(2) theory meets the criterion at every coupling (Proposition 21). Four-dimensional U(1) theory at weak coupling cannot meet it, because that theory is massless [57, 65]; a criterion that did not fail there would be wrong.

What is proved and what is not

Figure 1 places the thresholds on one axis, and Table 1 in Sec. VIII lists the gaps between this paper and the continuum mass gap, with the status of each. Everything stated as a theorem, proposition or lemma is proved here, by hand, without conditions. The mass gap of the continuum four-dimensional theory is not proved here, at intermediate lattice coupling or in the continuum. What is missing is stated precisely as Problem 27.

Gap	Status	Where, and what remains
Best constant in the Hessian bound	Closed	Theorem A: $4d$, attained with both signs
Uniqueness on the whole curvature range	Closed	Theorem D: removes the factor 2 of [100], $|\beta| < 1/(8d)$
Limit of curvature methods	Closed	Theorem C: no uniform Bakry–Émery bound beyond $1/(8d)$
SU(2) beyond the curvature range	Closed	Theorem E: $\beta_{\mathrm W} < 2/9$, by the sharp quarter, Theorem 7
Limit of single-link methods	Closed	Proposition 13: $\beta_{\mathrm W} = 2/9$
Lattice mass gap at strong coupling	Closed	Theorem F: $m \ge-\log(2\beta_{\mathrm W})$ for SU(2)
Finite-box criterion, twists, area law	Closed	Theorem G, Proposition 18
Two dimensions at every coupling	Closed	Proposition 21
Intermediate fixed coupling, $2/9 \le\beta_{\mathrm W} \lesssim2.2$	Open	No proof of uniqueness or of a gap; Monte Carlo evidence only [37, 75, 86] [76]
Crossover to Bałaban’s regime	Open	Problem 27
Continuum limit with the axioms	Open	Bałaban controls partition functions, not Schwinger functions [12]
Gap in physical units	Open	Equivalent on the lattice to a gap $ma$, which no spacing-uniform estimate gives [16]

Setting

Let $G\in\{\mathrm{SU}(N),\mathrm{SO}(N),\mathrm{U}(N)\}$ with Lie algebra $\mathfrak{g}$. We use the inner product $\langle X,Y\rangle=-\operatorname{Tr}(XY)=\operatorname{Re}\operatorname{Tr}(X^{*}Y)$ on $\mathfrak{g}$ and the bi-invariant metric it induces on $G$. In this metric

$$\operatorname{Ric}_{\mathrm{SU}(N)}=\frac{N}{2},\qquad\operatorname{Ric}_{\mathrm{SO}(N)}=\frac{N-2}{4}. \tag*{(2)}$$

Indeed $\operatorname{Ric}(X,X)=-\frac{1}{4}B(X,X)$ for the Killing form $B$ of a compact group with a bi-invariant metric [84],⁶ and $B(X,X)=2N\operatorname{Tr}X^2$ on $\mathfrak{su}(N)$, $B(X,X)=(N-2)\operatorname{Tr}X^2$ on $\mathfrak{so}(N)$.⁷ With this metric SU(2) is the round three-sphere of radius $\sqrt{2}$; its Hopf fibration by great circles, and the Villarceau circles that link them, are described in [19].

Let $\Lambda\subset\mathbb{Z}^d$ be a finite box or a torus, $E$ its set of positively oriented links and $P$ its set of plaquettes. The plaquette $p$ with base point $x$ in the plane of the directions $\mu<\nu$ has the links

$$e_1=(x,\mu),\qquad e_2=(x+\hat{\mu},\nu),\qquad e_3=(x+\hat{\nu},\mu),\qquad e_4=(x,\nu),$$

References

1. [1]A. Adhikari and S. Cao, Correlation decay for finite lattice gauge theories at weak coupling, Ann. Probab. 53 (2025), 140–174, doi:10.1214/24-AOP1702.DOI
2. [2]D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX, Lecture Notes in Math. 1123, Springer, Berlin, 1985, 177–206, doi:10.1007/BFb0075847.
3. [3]D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren Math. Wiss. 348, Springer, Cham, 2014, doi:10.1007/978-3-319-00227-9.DOI
4. [4]T. Balaban, Propagators and renormalization transformations for lattice gauge theories. I, Comm. Math. Phys. 95 (1984), 17–40, doi:10.1007/BF01215753.DOI
5. [5]T. Balaban, Propagators and renormalization transformations for lattice gauge theories. II, Comm. Math. Phys. 96 (1984), 223–250, doi:10.1007/BF01240221.DOI
6. [6]T. Balaban, Averaging operations for lattice gauge theories, Comm. Math. Phys. 98 (1985), 17–51, doi:10.1007/BF01211042.DOI
7. [7]T. Balaban, Propagators for lattice gauge theories in a background field, Comm. Math. Phys. 99 (1985), 389–434, doi:10.1007/BF01240355.DOI
8. [8]T. Balaban, Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions, Comm. Math. Phys. 109 (1987), 249–301, doi:10.1007/BF01215223.
9. [9]T. Balaban, Renormalization group approach to lattice gauge field theories. II. Cluster expansions, Comm. Math. Phys. 116 (1988), 1–22, doi:10.1007/BF01239022.
10. [10]T. Balaban, Convergent renormalization expansions for lattice gauge theories, Comm. Math. Phys. 119 (1988), 243–285, doi:10.1007/BF01217741.DOI
11. [11]T. Balaban, Large field renormalization. I. The basic step of the R operation, Comm. Math. Phys. 122 (1989), 175–202, doi:10.1007/BF01257412.DOI
12. [12]T. Balaban, Large field renormalization. II. Localization, exponentiation, and bounds for the R operation, Comm. Math. Phys. 122 (1989), 355–392, doi:10.1007/BF01238433.DOI
13. [13]J. Barta, Sur la vibration fondamentale d’une membrane, C. R. Acad. Sci. Paris 204 (1937), 472–473.
14. [14]R. Basu and S. Ganguly, SO(N) lattice gauge theory, planar and beyond, Comm. Pure Appl. Math. 71 (2018), 2016–2064, doi:10.1002/cpa.21788.DOI
15. [15]R. Bauerschmidt and T. Bodineau, A very simple proof of the LSI for high temperature spin systems, J. Funct. Anal. 276 (2019), 2582–2588, doi:10.1016/j.jfa.2019.01.007.DOI
16. [16]D. Bhattacharjee and U. Bhattacharya, Spacing-uniform lattice inequalities cannot produce the Yang–Mills mass gap, Zenodo, release v1.0.0, 2026, doi:10.5281/zenodo.23444338.
17. [17]D. Bhattacharjee, P. Samal and S. Bhattacharya, Geometric structure and renormalization group flow in chiral Yang–Mills–Higgs theory, Preprints (2026), doi:10.20944/preprints202602.2022.v1.DOI
18. [18]D. Bhattacharjee, S. N. Thakur, S. Bhattacharya and P. Samal, SU(n) holonomy deviations in Calabi–Yau manifolds (CY1–CY4), Preprints (2026), doi:10.20944/preprints202602.0023.v1.DOI
19. [19]D. Bhattacharjee, Homotopy groups of spheres, Hopf fibrations and Villarceau circles II, Preprints (2026), doi:10.20944/preprints202602.2038.v1.DOI
20. [20]D. Bhattacharjee, Holonomic quantum computing, SSRN (2026), doi:10.2139/ssrn.6066428.DOI
21. [21]D. Bhattacharjee, S. Singha Roy and P. Samal, Brane-cluster UV completion of quantum gravity, Research Square (2026), doi:10.21203/rs.3.rs-9053205/v1.DOI
22. [22]M. Biernacki and J. Krzyż, On the monotonicity of certain functionals in the theory of analytic functions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 9 (1955), 135–147.
23. [23]J. Bricmont and J. Fröhlich, An order parameter distinguishing between different phases of lattice gauge theories with matter fields, Phys. Lett. B 122 (1983), 73–77, doi:10.1016/0370-2693(83)91171-1.DOI
24. [24]D. Brydges, J. Fröhlich and E. Seiler, On the construction of quantized gauge fields. I. General results, Ann. Physics 121 (1979), 227–284, doi:10.1016/0003-4916(79)90098-8.DOI
25. [25]S. Cao, Wilson loop expectations in lattice gauge theories with finite gauge groups, Comm. Math. Phys. 380 (2020), 1439–1505, doi:10.1007/s00220-020-03912-z.DOI
26. [26]S. Cao, R. Nissim and S. Sheffield, Expanded regimes of area law for lattice Yang–Mills theories, arXiv:2505.16585.arxiv.org/abs/2505.16585
27. [27]S. Cao, R. Nissim and S. Sheffield, Dynamical approach to area law for lattice Yang–Mills, arXiv:2509.04688.arxiv.org/abs/2509.04688
28. [28]S. Cao, M. Park and S. Sheffield, Random surfaces and lattice Yang–Mills, Commun. Amer. Math. Soc. 5 (2025), 774–896, doi:10.1090/cams/59.DOI
29. [29]A. Chandra, I. Chevyrev, M. Hairer and H. Shen, Langevin dynamics for the 2D Yang–Mills measure, Publ. Math. Inst. Hautes Études Sci. 136 (2022), 1–147, doi:10.1007/s10240-022-00132-0.DOI
30. [30]A. Chandra, I. Chevyrev, M. Hairer and H. Shen, Stochastic quantisation of Yang–Mills–Higgs in 3D, Invent. Math. 237 (2024), 541–696, doi:10.1007/s00222-024-01264-2.DOI
31. [31]S. Chatterjee, The leading term of the Yang–Mills free energy, J. Funct. Anal. 271 (2016), 2944–3005, doi:10.1016/j.jfa.2016.04.032.DOI
32. [32]S. Chatterjee, Yang–Mills for probabilists, in Probability and Analysis in Interacting Physical Systems, Springer Proc. Math. Stat. 283, Springer, Cham, 2019, 1–16, doi:10.1007/978-3-030-15338-0_1.DOI
33. [33]S. Chatterjee, Wilson loops in Ising lattice gauge theory, Comm. Math. Phys. 377 (2020), 307–340, doi:10.1007/s00220-020-03738-9.DOI
34. [34]S. Chatterjee, A probabilistic mechanism for quark confinement, Comm. Math. Phys. 385 (2021), 1007–1039, doi:10.1007/s00220-021-04086-y.DOI
35. [35]Y. Chen and R. Eldan, Localization schemes: a framework for proving mixing bounds for Markov chains, Duke Math. J. 174 (2025), 1431–1510, doi:10.1215/00127094-2024-0063.DOI
36. [36]I. Chevreyr and H. Shen, Invariant measure and universality of the 2D Yang–Mills Langevin dynamic, Comm. Pure Appl. Math. 79 (2026), 1973–2102, doi:10.1002/cpa.70043.
37. [37]M. Creutz, Monte Carlo study of quantized SU(2) gauge theory, Phys. Rev. D 21 (1980), 2308–2315, doi:10.1103/PhysRevD.21.2308.DOI
38. [38]J. Dimock, The renormalization group according to Balaban, I. Small fields, Rev. Math. Phys. 25 (2013), 1330010, doi:10.1142/S0129055X13300100.DOI
39. [39]J. Dimock, The renormalization group according to Balaban. II. Large fields, J. Math. Phys. 54 (2013), doi:10.1063/1.4821275.DOI
40. [40]J. Dimock, The renormalization group according to Balaban III. Convergence, Ann. Henri Poincaré 15 (2014), 2133–2175, doi:10.1007/s00023-013-0303-3.DOI
41. [41]R. L. Dobrushin, The description of a random field by means of conditional probabilities and conditions of its regularity, Theory Probab. Appl. 13 (1968), 197–224, doi:10.1137/1113026.
42. [42]R. L. Dobrushin, Prescribing a system of random variables by conditional distributions, Theory Probab. Appl. 15 (1970), 458–486, doi:10.1137/1115049.
43. [43]R. L. Dobrushin and S. B. Shlosman, Constructive criterion for the uniqueness of Gibbs field, in Statistical Physics and Dynamical Systems, Birkhäuser, Boston, 1985, 347–370, doi:10.1007/978-1-4899-6653-7_20.
44. [44]R. L. Dobrushin and S. B. Shlosman, Completely analytical interactions: constructive description, J. Stat. Phys. 46 (1987), 983–1014, doi:10.1007/BF01011153.
45. [45]T. C. Dorlas and B. Savoie, On the Wasserstein distance and the Dobrushin uniqueness theorem, arXiv:2305.19371.DOI
46. [46]B. K. Driver, Convergence of the U(1)4 lattice gauge theory to its continuum limit, Comm. Math. Phys. 110 (1987), 479–501, doi:10.1007/BF01212424.DOI
47. [47]B. K. Driver, YM₂: continuum expectations, lattice convergence, and lassos, Comm. Math. Phys. 123 (1989), 575–616, doi:10.1007/BF01218586.DOI
48. [48]J.-M. Drouffe and J.-B. Zuber, Strong coupling and mean field methods in lattice gauge theories, Phys. Rep. 102 (1983), 1–119, doi:10.1016/0370-1573(83)90034-0.DOI
49. [49]R. Eldan, F. Koehler and O. Zeitouni, A spectral condition for spectral gap: fast mixing in high-temperature Ising models, Probab. Theory Related Fields 182 (2022), 1035–1051, doi:10.1007/s00440-021-01085-x.DOI
50. [50]P. Federbush, A phase cell approach to Yang–Mills theory. I. Modes, lattice-continuum duality, Comm. Math. Phys. 107 (1986), 319–329, doi:10.1007/BF01209397.DOI
51. [51]P. Federbush, A phase cell approach to Yang–Mills theory. III. Local stability, modified renormalization group transformation, Comm. Math. Phys. 114 (1988), 317–343, doi:10.1007/BF01225039.
52. [52]H. Fellner, A covariance estimate for Gibbs measures, J. Funct. Anal. 46 (1982), 387–395, doi:10.1016/0022-1236(82)90053-2.
53. [53]M. P. Forrsström, J. Lenells and F. Viklund, Wilson loops in the abelian lattice Higgs model, Probab. Math. Phys. 4 (2023), 257–329, doi:10.2140/pmp.2023.4.257.
54. [54]K. Fredenhagen and M. Marcu, Confinement criterion for QCD with dynamical quarks, Phys. Rev. Lett. 56 (1986), 223–224, doi:10.1103/PhysRevLett.56.223.DOI
55. [55]S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction, Cambridge University Press, Cambridge, 2017, doi:10.1017/9781316882603.DOI
56. [56]J. Fröhlich and T. Spencer, The Kosterlitz–Thouless transition in two-dimensional abelian spin systems and the Coulomb gas, Comm. Math. Phys. 81 (1981), 527–602, doi:10.1007/BF01208273.DOI
57. [57]J. Fröhlich and T. Spencer, Massless phases and symmetry restoration in abelian gauge theories and spin systems, Comm. Math. Phys. 83 (1982), 411–454, doi:10.1007/BF01213610.DOI
58. [58]C. Garban and A. Sepúlveda, Improved spin-wave estimator for Wilson loops in U(1) lattice gauge theory, Int. Math. Res. Not. IMRN 2023 (2023), 18142–18198, doi:10.1093/imrn/rnac356.DOI
59. [59]C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice, Lecture Notes in Phys. 788, Springer, Berlin, 2010, doi:10.1007/978-3-642-01850-3.DOI
60. [60]H.-O. Georgii, Gibbs Measures and Phase Transitions, 2nd ed., De Gruyter, Berlin, 2011, doi:10.1515/9783110250329.DOI
61. [61]J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd ed., Springer, New York, 1987, doi:10.1007/978-1-4612-4728-9.DOI
62. [62]L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083, doi:10.2307/2373688.DOI
63. [63]L. Gross, Convergence of U(1)3 lattice gauge theory to its continuum limit, Comm. Math. Phys. 92 (1983), 137–162, doi:10.1007/BF01210842.DOI
64. [64]D. J. Gross and E. Witten, Possible third-order phase transition in the large-N lattice gauge theory, Phys. Rev. D 21 (1980), 446–453, doi:10.1103/PhysRevD.21.446.DOI
65. [65]A. H. Guth, Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory, Phys. Rev. D 21 (1980), 2291–2307, doi:10.1103/PhysRevD.21.2291.DOI
66. [66]R. Holley and D. Stroock, Logarithmic Sobolev inequalities and stochastic Ising models, J. Stat. Phys. 46 (1987), 1159–1194, doi:10.1007/BF01011161.DOI
67. [67]A. Jaffe and E. Witten, Quantum Yang–Mills theory, in The Millennium Prize Problems, Clay Mathematics Institute and American Mathematical Society, Cambridge, MA and Providence, RI, 2006, 129–152, https://www.claymath.org/wp-content/uploads/2022/06/yangmills.pdf.
68. [68]R. Jentzsch, Über Integralgleichungen mit positivem Kern, J. Reine Angew. Math. 141 (1912), 235–244, doi:10.1515/crll.1912.141.235.DOI
69. [69]O. Kallenberg, Foundations of Modern Probability, 3rd ed., Springer, Cham, 2021, doi:10.1007/978-3-030-61871-1.DOI
70. [70]J. Kogut and L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975), 395–408, doi:10.1103/PhysRevD.11.395.
71. [71]R. Kotecký and D. Preiss, Cluster expansion for abstract polymer models, Comm. Math. Phys. 103 (1986), 491–498, doi:10.1007/BF01211762.DOI
72. [72]H. Künsch, Decay of correlations under Dobrushin’s uniqueness condition and its applications, Comm. Math. Phys. 84 (1982), 207–222, doi:10.1007/BF01208568.DOI
73. [73]M. Ledoux, The Concentration of Measure Phenomenon, Math. Surveys Monogr. 89, American Mathematical Society, Providence, RI, 2001, doi:10.1090/surv/089.
74. [74]T. Lévy, Yang–Mills measure on compact surfaces, Mem. Amer. Math. Soc. 166 (2003), no. 790, doi:10.1090/memo/0790.DOI
75. [75]B. Lucini and M. Teper, SU(N) gauge theories in four dimensions: exploring the approach to N = ∞, J. High Energy [76] B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: calculations with improved operators, J. High Energy Phys. 2004, no. 06, 012, doi:10.1088/1126-6708/2004/06/012.
76. [77]M. Lüscher, Construction of a selfadjoint, strictly positive transfer matrix for Euclidean lattice gauge theories, Comm. Math. Phys. 54 (1977), 283–292, doi:10.1007/BF01649090.DOI
77. [78]G. Mack and V. B. Petkova, Comparison of lattice gauge theories with gauge groups Z2 and SU(2), Ann. Physics 123 (1979), 442–467, doi:10.1016/0003-4916(79)90346-5.DOI
78. [79]J. Magnen, V. Rivasseau and R. Sénéor, Construction of YM4 with an infrared cutoff, Comm. Math. Phys. 155 (1993), 325–383, doi:10.1007/BF02097397.
79. [80]K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, Chichester, 2000, doi:10.1002/9780470316979.DOI
80. [81]F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case, Comm. Math. Phys. 161 (1994), 447–486, doi:10.1007/BF02101929.DOI
81. [82]E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837–842, doi:10.1090/S0002-9904-1934-05978-0.DOI
82. [83]A. A. Migdal, Recursion equations in gauge field theories, Sov. Phys. JETP 42 (1975), 413–418.
83. [84]J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293–329, doi:10.1016/S0001-8708(76)80002-3.DOI
84. [85]I. Montvay and G. Münster, Quantum Fields on a Lattice, Cambridge University Press, Cambridge, 1994, doi:10.1017/CBO9780511470783.DOI
85. [86]C. J. Morningstar and M. Peardon, Glueball spectrum from an anisotropic lattice study, Phys. Rev. D 60 (1999), 034509, doi:10.1103/PhysRevD.60.034509.DOI
86. [87]R. Nissim, U(N) lattice Yang–Mills in the ’t Hooft regime, arXiv:2510.22788.
87. [88]Y. Ollivier, Ricci curvature of Markov chains on metric spaces, J. Funct. Anal. 256 (2009), 810–864, doi:10.1016/j.jfa.2008.11.001.
88. [89]K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions, Comm. Math. Phys. 31 (1973), 83–112, doi:10.1007/BF01645738.DOI
89. [90]K. Osterwalder and R. Schrader, Axioms for Euclidean Green’s functions. II, Comm. Math. Phys. 42 (1975), 281–305, doi:10.1007/BF01608978.DOI
90. [91]K. Osterwalder and E. Seiler, Gauge field theories on a lattice, Ann. Physics 110 (1978), 440–471, doi:10.1016/0003-4916(78)90039-8.DOI
91. [92]F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361–400, doi:10.1006/jfan.1999.3557.DOI
92. [93]A. M. Polyakov, Compact gauge fields and the infrared catastrophe, Phys. Lett. B 59 (1975), 82–84, doi:10.1016/0370-2693(75)90162-8.DOI
93. [94]A. M. Polyakov, Interaction of Goldstone particles in two dimensions. Applications to ferromagnets and massive Yang–Mills fields, Phys. Lett. B 59 (1975), 79–81, doi:10.1016/0370-2693(75)90161-6.DOI
94. [95]V. Rivasseau, From Perturbative to Constructive Renormalization, Princeton University Press, Princeton, NJ, 1991, doi:10.1515/9781400862085.DOI
95. [96]P. Samal, D. Bhattacharjee and R. Ghora, Renormalization of spacetime dimension, Preprints (2026), doi:10.20944/preprints202602.031.v1.DOI
96. [97]E. Seiler, Gauge Theories as a Problem of Constructive Quantum Field Theory and Statistical Mechanics, Lecture Notes in Phys. 159, Springer, Berlin, 1982, doi:10.1007/3-540-11559-5.DOI
97. [98]A. Sengupta, Gauge theory on compact surfaces, Mem. Amer. Math. Soc. 126 (1997), no. 600, doi:10.1090/memo/0600.DOI
98. [99]H. Shen, R. Zhu and X. Zhu, Langevin dynamics of lattice Yang–Mills–Higgs and applications, Comm. Math. Phys. 407 (2026), doi:10.1007/s00220-025-05528-7.DOI
99. [100]H. Shen, R. Zhu and X. Zhu, A stochastic analysis approach to lattice Yang–Mills at strong coupling, Comm. Math. Phys. 400 (2023), 805–851, doi:10.1007/s00220-022-04609-1.DOI
100. [101]B. Simon, The Statistical Mechanics of Lattice Gases, Volume I, Princeton University Press, Princeton, NJ, 1993, doi:10.1515/9781400863433.DOI
101. [102]E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971, doi:10.1515/9781400883899.
102. [103]D. W. Stroock and B. Zegarlinski, The logarithmic Sobolev inequality for continuous spin systems on a lattice, J. Funct. Anal. 104 (1992), 299–326, doi:10.1016/0022-1236(92)90003-2.DOI
103. [104]G. ’t Hooft, On the phase transition towards permanent quark confinement, Nucl. Phys. B 138 (1978), 1–25, doi:10.1016/0550-3213(78)90153-0.DOI
104. [105]C. Villani, Optimal Transport: Old and New, Grundlehren Math. Wiss. 338, Springer, Berlin, 2009, doi:10.1007/978-3-540-71050-9.
105. [106]G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1944.
106. [107]K. G. Wilson, Confinement of quarks, Phys. Rev. D 10 (1974), 2445–2459, doi:10.1103/PhysRevD.10.2445.DOI
107. [108]L. Wu, Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition, Ann. Probab. 34 (2006), 1960–1989, doi:10.1214/009117906000000368.DOI
108. [109]B. Zegarlinski, The strong decay to equilibrium for the stochastic dynamics of unbounded spin systems on a lattice, Comm. Math. Phys. 175 (1996), 401–432, doi:10.1007/BF02102414.DOI