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arxiv: 2606.18125 · v1 · pith:2D2JKJWKnew · submitted 2026-06-16 · ❄️ cond-mat.stat-mech · hep-th

Making complex CFTs real: The two-dimensional Potts model for Q>4 and complex Q

Jesper Lykke Jacobsen , Kay Joerg Wiese This is my paper

Pith reviewed 2026-06-26 22:11 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech hep-th
keywords Potts modelloop modelconformal field theoryanalytic continuationcomplex couplingstransfer matrixcritical phenomenaQ-state Potts
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0 comments X

The pith

The Potts model for Q>4 yields complex CFTs whose data continue analytically from the Q≤4 loop model.

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

The paper starts from a triangular-lattice Potts model that includes both two-spin and three-spin interactions, which is equivalent to a loop model in which the parameter Q can vary continuously. At Q=4 a critical fixed point and a tricritical fixed point collide; for Q>4 or for complex Q they reappear as a pair of complex conformally invariant theories once the couplings are allowed to become complex. The central claim is that every piece of conformal data, including the central charge, the critical exponents, and the three-point structure constants, is obtained simply by analytic continuation of the exact results already known for the loop model at Q≤4. The authors test this continuation both for real Q>4 and for selected complex values of Q, using large-scale transfer-matrix calculations and direct comparison with earlier numerical work at Q=5.

Core claim

By studying an equivalent loop model derived from a Potts model with two- and three-spin interactions on the triangular lattice, the authors demonstrate that a critical and a tricritical fixed point collide at Q=4 and then continue as a pair of complex conformally invariant theories for Q>4 or complex Q. They conjecture that the central charge, critical exponents, and three-point structure constants are all obtained by analytic continuation from the Q ≤ 4 case, supported by transfer-matrix computations.

What carries the argument

The loop model with continuously variable Q, obtained from the triangular-lattice Potts model with two- and three-spin interactions, which permits complex couplings that produce a pair of complex fixed points for Q>4.

If this is right

  • All conformal data for the Potts model at real Q>4 are given by the analytic continuation of the known Q≤4 loop-model results.
  • The same continuation supplies the data when Q itself is taken to be complex.
  • The resulting theories remain conformally invariant even though the couplings are complex.
  • The numerical spectra obtained by transfer matrix for Q=5 agree with the continued formulas, confirming the pattern.

Where Pith is reading between the lines

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

  • The same continuation technique may apply to other lattice models whose real-coupling versions exhibit first-order transitions.
  • Transfer-matrix or Monte Carlo methods could be used to test analytic continuations in related statistical-mechanics models with tunable parameters.
  • The construction supplies a concrete route to defining conformal field theories at parameter values where conventional real couplings produce no critical point.

Load-bearing premise

The triangular-lattice Potts model with two- and three-spin interactions remains equivalent to the loop model when Q is continued to values greater than 4 or into the complex plane, provided the couplings are allowed to become complex.

What would settle it

Transfer-matrix computations at some real Q>4 or complex Q that produce central charges or exponents differing from the values predicted by analytic continuation of the Q≤4 formulas would falsify the conjecture.

Figures

Figures reproduced from arXiv: 2606.18125 by Jesper Lykke Jacobsen, Kay Joerg Wiese.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Upper row: paths taken in the analytic continuation, starting at [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The critical domain as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. In yellow we show the domains in which one fixed point is [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The location of the nearest Fisher singularity in the complex [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Top: Singularity of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Top: Singularity in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Left [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Left half of plot: Scaling analysis of [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Left [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Data from Fig [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Error made in the interpolation for the real and imaginary [PITH_FULL_IMAGE:figures/full_fig_p018_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Zeros of [PITH_FULL_IMAGE:figures/full_fig_p019_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Values of [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. All our data. Data for the tricritical fixed point [PITH_FULL_IMAGE:figures/full_fig_p020_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21 [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22 [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. The spectrum for [PITH_FULL_IMAGE:figures/full_fig_p022_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: FIG. 25. Real and imaginary part [PITH_FULL_IMAGE:figures/full_fig_p022_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: FIG. 26. Difference in real and imaginary part for the descendents [PITH_FULL_IMAGE:figures/full_fig_p023_26.png] view at source ↗
Figure 29
Figure 29. Figure 29: FIG. 29. Left: Loci of [PITH_FULL_IMAGE:figures/full_fig_p024_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: FIG. 30. Full spectrum. Note the additional states as compared to [PITH_FULL_IMAGE:figures/full_fig_p024_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: ∆ = 1.00047 + 0.575189i ≈ 1 + i √ 3 , (112) in fine agreement with (h, h¯) = (1, 0) for the field J [cf. Eq. (103)]. As before, we fit the spectrum as a function of y by a poly￾nomial. We can then use this high-quality fit to get the location FIG. 33. Left: real part of the spectrum for L = 15 with the first three primaries as a function of y: ∆2,1 (red, solid, bottom), ∆3,1 (blue dashed, middle) and ∆4,1… view at source ↗
Figure 32
Figure 32. Figure 32: FIG. 32. The location [PITH_FULL_IMAGE:figures/full_fig_p025_32.png] view at source ↗
Figure 36
Figure 36. Figure 36: FIG. 36. Zeros of [PITH_FULL_IMAGE:figures/full_fig_p027_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: FIG. 37. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p027_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: FIG. 38. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p027_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: FIG. 39. Extrapolation for [PITH_FULL_IMAGE:figures/full_fig_p027_39.png] view at source ↗
Figure 42
Figure 42. Figure 42: FIG. 42. Zeros of [PITH_FULL_IMAGE:figures/full_fig_p028_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: FIG. 43. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p028_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: FIG. 44. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p028_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: FIG. 45. Extrapolation for [PITH_FULL_IMAGE:figures/full_fig_p028_45.png] view at source ↗
Figure 48
Figure 48. Figure 48: FIG. 48. Zeros of [PITH_FULL_IMAGE:figures/full_fig_p029_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: FIG. 49. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p029_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: FIG. 50. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p029_50.png] view at source ↗
Figure 51
Figure 51. Figure 51: FIG. 51. Extrapolation for [PITH_FULL_IMAGE:figures/full_fig_p029_51.png] view at source ↗
Figure 52
Figure 52. Figure 52: FIG. 52 [PITH_FULL_IMAGE:figures/full_fig_p030_52.png] view at source ↗
Figure 53
Figure 53. Figure 53: FIG. 53 [PITH_FULL_IMAGE:figures/full_fig_p030_53.png] view at source ↗
Figure 55
Figure 55. Figure 55: FIG. 55 [PITH_FULL_IMAGE:figures/full_fig_p030_55.png] view at source ↗
Figure 61
Figure 61. Figure 61: FIG. 61. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p031_61.png] view at source ↗
Figure 62
Figure 62. Figure 62: FIG. 62. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p031_62.png] view at source ↗
Figure 63
Figure 63. Figure 63: FIG. 63. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p031_63.png] view at source ↗
Figure 67
Figure 67. Figure 67: FIG. 67. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p032_67.png] view at source ↗
Figure 65
Figure 65. Figure 65: FIG. 65. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p032_65.png] view at source ↗
Figure 66
Figure 66. Figure 66: FIG. 66. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p032_66.png] view at source ↗
Figure 72
Figure 72. Figure 72: FIG. 72. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p033_72.png] view at source ↗
Figure 70
Figure 70. Figure 70: FIG. 70. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p033_70.png] view at source ↗
Figure 71
Figure 71. Figure 71: FIG. 71. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p033_71.png] view at source ↗
Figure 74
Figure 74. Figure 74: FIG. 74 [PITH_FULL_IMAGE:figures/full_fig_p034_74.png] view at source ↗
Figure 75
Figure 75. Figure 75: FIG. 75 [PITH_FULL_IMAGE:figures/full_fig_p034_75.png] view at source ↗
Figure 77
Figure 77. Figure 77: FIG. 77 [PITH_FULL_IMAGE:figures/full_fig_p034_77.png] view at source ↗
Figure 83
Figure 83. Figure 83: FIG. 83. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p035_83.png] view at source ↗
Figure 84
Figure 84. Figure 84: FIG. 84. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p035_84.png] view at source ↗
Figure 85
Figure 85. Figure 85: FIG. 85. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p035_85.png] view at source ↗
Figure 89
Figure 89. Figure 89: FIG. 89. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p036_89.png] view at source ↗
Figure 87
Figure 87. Figure 87: FIG. 87. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p036_87.png] view at source ↗
Figure 88
Figure 88. Figure 88: FIG. 88. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p036_88.png] view at source ↗
Figure 94
Figure 94. Figure 94: FIG. 94. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p037_94.png] view at source ↗
Figure 95
Figure 95. Figure 95: FIG. 95. Movement of [PITH_FULL_IMAGE:figures/full_fig_p037_95.png] view at source ↗
Figure 93
Figure 93. Figure 93: FIG. 93. Extrapolation of [PITH_FULL_IMAGE:figures/full_fig_p037_93.png] view at source ↗
Figure 96
Figure 96. Figure 96: FIG. 96. Top line: solutions of [PITH_FULL_IMAGE:figures/full_fig_p038_96.png] view at source ↗
read the original abstract

The two-dimensional $Q$-state Potts model with real couplings has a first-order transition for $Q>4$. Starting from a triangular-lattice Potts model with two- and three-spin interactions, we study an equivalent loop model in which $Q$ is a continuous parameter. By a combination of analytical and numerical arguments, we show that this loop model allows for the collision of a critical and a tricritical fixed point at $Q=4$. These then emerge as a pair of complex conformally invariant theories at $Q>4$, or even complex $Q$, for suitable complex coupling constants. We conjecture that all conformal data (such as the central charge, critical exponents, and three-point structure constants) can be obtained by analytic continuation of known exact results for the loop model with $Q \le 4$. This conjecture is checked, both for real $Q>4$ and for $Q \in \mathbb{C}$, by extensive transfer-matrix computations and comparison to previous studies for $Q=5$.

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 manuscript conjectures that all conformal data (central charge, critical exponents, three-point structure constants) of the two-dimensional Potts model for real Q>4 or complex Q can be obtained by analytic continuation of known exact loop-model results at Q≤4. Starting from a triangular-lattice Potts model with two- and three-spin interactions, the authors argue analytically that a critical and tricritical fixed point collide at Q=4 and then emerge as a complex-conjugate pair for Q>4 (or complex Q) at suitable complex couplings; they support the conjecture with extensive transfer-matrix spectra that match the continued formulas, including checks against prior Q=5 results.

Significance. If the central conjecture holds, the work supplies a concrete route to conformal data for a family of complex CFTs realized by the Potts model, extending the exactly solvable regime beyond Q=4 while preserving the loop-model interpretation. The combination of fixed-point collision analysis and direct numerical verification of multiple independent quantities (spectra, exponents, structure constants) constitutes a non-trivial test; the provision of reproducible transfer-matrix data and explicit comparisons to existing Q=5 literature are positive features.

major comments (2)
  1. [§2] §2 (model definition) and the paragraph following Eq. (loop weight): the claim that the Potts-to-loop mapping survives analytic continuation to complex Q and complex couplings is asserted without a re-derivation or explicit check that no new singularities or redefinitions of the loop fugacity appear; this assumption is load-bearing for equating the continued CFT data to the triangular-lattice model.
  2. [§4.3] §4.3 (transfer-matrix spectra for complex Q): while the numerical eigenvalues are reported to agree with the continued formulas, the manuscript does not quantify the distance to the nearest non-continued singularity or demonstrate that the complex-coupling fixed point remains isolated; without this, the numerical support for the conjecture remains partial.
minor comments (2)
  1. [Figure 3] Figure 3 caption: the legend does not specify the lattice size used for the Q=5 comparison; adding this datum would improve reproducibility.
  2. [§2, §3.2] Notation: the symbol for the three-spin coupling is introduced in §2 but reused with a different normalization in §3.2; a single consistent definition would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, proposing revisions where they strengthen the manuscript without altering its core claims.

read point-by-point responses

  1. Referee: §2 (model definition) and the paragraph following Eq. (loop weight): the claim that the Potts-to-loop mapping survives analytic continuation to complex Q and complex couplings is asserted without a re-derivation or explicit check that no new singularities or redefinitions of the loop fugacity appear; this assumption is load-bearing for equating the continued CFT data to the triangular-lattice model.

    Authors: The Fortuin-Kasteleyn representation rewrites the Potts partition function as a sum over loop configurations whose weight is exactly Q per loop (up to normalization), and this algebraic identity holds for arbitrary complex values of the two- and three-spin couplings because it follows from expanding the local Boltzmann factors without reference to their magnitude or phase. Consequently the loop fugacity remains Q and no additional singularities are introduced by the mapping itself. We nevertheless agree that an explicit statement improves clarity. We will add a short paragraph in §2 that re-derives the loop weight for complex parameters and confirms the absence of redefinitions. revision: yes

  2. Referee: §4.3 (transfer-matrix spectra for complex Q): while the numerical eigenvalues are reported to agree with the continued formulas, the manuscript does not quantify the distance to the nearest non-continued singularity or demonstrate that the complex-coupling fixed point remains isolated; without this, the numerical support for the conjecture remains partial.

    Authors: We concur that a quantitative estimate of the distance to the nearest extraneous singularity would strengthen the numerical case. With transfer-matrix spectra on finite strips, however, systematically locating the closest singularity in the multi-dimensional complex-coupling space is not practicable. We have instead verified agreement for several independent observables (eigenvalues, scaling dimensions, and structure constants) at multiple distinct complex-Q and coupling values; the consistency across these checks provides indirect evidence that the fixed point remains isolated within the explored region. We will insert a paragraph in §4.3 that discusses this robustness while explicitly noting the limitation on quantifying isolation. revision: partial

Circularity Check

0 steps flagged

No significant circularity: conjecture rests on independent exact results plus external numerical checks

full rationale

The paper conjectures that conformal data for Q>4 or complex Q follows by analytic continuation of known exact loop-model results at Q≤4. This continuation is checked by transfer-matrix spectra that constitute an independent numerical test rather than a fit to the same quantities. The Potts-to-loop equivalence for complex parameters is presented as an assumption supported by fixed-point collision arguments at Q=4, but no step reduces by construction to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the Potts model with added three-spin interactions maps exactly onto the loop model for continuous and complex Q, plus the prior exact results for Q≤4 that are taken as given.

axioms (1)
  • domain assumption The triangular-lattice Potts model with two- and three-spin interactions is equivalent to the loop model for continuous real or complex Q.
    This equivalence is the starting point that allows Q to be continued beyond 4.

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discussion (0)

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Reference graph

Works this paper leans on

111 extracted references · 7 linked inside Pith

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    69 to 73) repeats the analysis for the lower half plane

    shows the analysis in the upper half plane, while section D 3 (Figs. 69 to 73) repeats the analysis for the lower half plane. This analysis confirms that theory and transfer matrix give the same value forcwith 4-digits accuracy atL= 15, see Eqs. (D14) and (D19). Another key claim of ours is that in contrast toQ= 5, the Potts model at complexQ= 5 + 2ihas b...

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    Degenerate operatorsV d (r,1) We now study the spectrum atQ= 5in the ground-state sector, using the quotient representation W0,q2 of defect-free FIG. 28. Real and imaginary part∆ (S) r,s of spectrum in the ground- state sector forQ= 5,L= 12, maximally 20 EVs, two consecutive layers. Primaries in dot-dashed, descendents dotted. The dashed vertical lines de...

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    For simplicity we choose the trivial phase factor˜z= 1cor- responding to the spinless non-diagonal operatorsV (0,s)

    Non-diagonal operatorsV (r,s) We now study the non-diagonal operatorsV (r,s) by diago- nalisation in the standard modulesW j,˜zwithj= 2sdefects. For simplicity we choose the trivial phase factor˜z= 1cor- responding to the spinless non-diagonal operatorsV (0,s). We shall use the notationX j(L)for the effective exponent ob- tained from the finite-size scali...

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    From the point of link patterns this means that we now work in a larger space that distin- guishes whether the link between two sites straddles the peri- odic boundary condition

    Additional states in the non-quotient spectrum Rather than diagonalizingT L in the quotient representa- tion W0,q2, as in section VII C 1, we also examined the non- quotient representationW 0,q2. From the point of link patterns this means that we now work in a larger space that distin- guishes whether the link between two sites straddles the peri- odic bo...

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    The dots show the the successive approximations fromc ′(z) = 0

    The contour plot shows|∆−1−i/ √ 3|2. The dots show the the successive approximations fromc ′(z) = 0. with conformal weights(1,0), that satisfy ¯∂J=∂ ¯J̸= 0.(111) and which donotgiving rise to a Kac-Moody algebra. Using here our numerical approach atQ= 5, we find at the critical point, indicated with a red dashed line forL= 12in Fig. 31 ∆ = 1.00047 + 0.575...

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    OPE coefficients: Quotient spectrum corrected byε ′ Up to now we read off exponents as∆ = ∆(z c). When one is not exactly at the fixed point, there are corrections due to subdominant operators. The leading one isε ′ =V d (3,1), in- tegrated over all of space. Denoting bygits effective coupling constant, we expect to see a corrected value∆ corr i,1 (z)[72]...

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    52.c,L= 10, raw data

    Grid FIG. 52.c,L= 10, raw data. FIG. 53.c,L= 10, selected data. FIG. 54.c,L= 10, subtracted data, upper half plane. FIG. 55.c,L= 10, subtracted data, lower half plane. The contour plot 56 gives a good idea about the location of the singularities. Note that the two non-trivial fixed points marked by the red dots in Fig. 56 are not complex conjugate of each...

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    Minigrid, upper half plane Here we use a minigrid of5×5points, centered around the non-trivial fixed point identified in the preceding section. In Fig. 64 we show the area covered by it, as well as the zeroes of the fitting polynomial, for our largest system sizes,L= 15. The non-trivial solution marked in red is clearly visible. -0.4 -0.2 0.2 0.4 x -0.6 -...

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    Minigrid, lower half plane As in subsection D 2, we use a minigrid of5×5points, centered around the non-trivial fixed point identified in the preceding section. In Fig. 69 we show the area covered by it, as well as the zeroes of the fitting polynomial, for our largest system sizesL= 15/16. The non-trivial solution marked in red is clearly visible. -0.4 -0...

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