Exploring the phase transition of planar FK-percolation

Ioan Manolescu

REVIEW 3 major objections 3 minor 61 references

Exploring the phase transition of planar FK-percolation

T0 review · 3 major / 3 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read These lecture notes present the full classification of the planar FK-percolation phase transition and the asymptotic rotational invariance of the critical phase.

desk verdict A mostly clear and honest set of lecture notes on 2D FK-percolation with one load-bearing internal inconsistency in the six-vertex parameter c that needs fixing before it is used for teaching. read the letter →

arxiv 2502.08394 v2 pith:3MJL5U7G submitted 2025-02-12 math.PR

Ioan Manolescu This is my paper · ORCID

classification math.PR MSC 60K3582B4382B27

keywords FK-percolationrandom-clustermodelphasetransitionsix-vertexBetheansatzrotationalinvarianceisoradialgraphsstar-triangletransformation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

These notes give a compact route into the recent description of the phase transition of FK-percolation (the random-cluster model) on the square lattice. The central result they present is a complete classification: for cluster weight $1 \le q \le 4$ the phase transition at the self-dual point is continuous, with a unique infinite-volume measure and polynomial decay of connection probabilities, while for $q > 4$ it is discontinuous, with non-uniqueness at criticality and exponential decay. They also present the result that in the continuous regime the critical loop representation is asymptotically rotationally invariant and universal across isoradial rectangular embeddings, so any subsequential scaling limit inherits these symmetries. The stated aim is pedagogical: geometric proofs using RSW estimates, duality, and the star-triangle transformation are given in streamlined form, while the Bethe ansatz computation of the six-vertex free energy is treated as an external black box.

What carries the argument

The argument rests on three interlocking objects. First, the FK measure on finite graphs with weight $p^{|\omega|}(1-p)^{|E\setminus\omega|}q^{k(\omega)}$, considered at its self-dual point $p=\sqrt{q}/(1+\sqrt{q})$; duality and the RSW crossing estimates for symmetric quads produce the dichotomy theorem separating the continuous (Con) and discontinuous (DisCon) scenarios. Second, the Baxter--Kelland--Wu correspondence translates critical FK-percolation on a torus into the six-vertex model with $a=b=1$, $c=p^2+\sqrt{q}$, so that probabilities of vertical crossing clusters are expressed through the sloped free energy $f_{6V}(\alpha)$; the Bethe ansatz estimate $f_{6V}(0)-f_{6V}(\alpha)\sim C\alpha$ for $c>2$ versus $C\alpha^2$ for $c\le 2$ is then converted into the finiteness or divergence of the correlation length. Third, for rotational invariance, the notes use FK-percolation on isoradial rectangular lattices $L(\alpha)$, where the star-triangle transformation gives a track-exchange operator that preserves connection probabilities; following the extremal coordinates of large clusters through a sequence of track exchanges yields the universality up to linear transformation and, after a drift equality, the full universality and rotational invariance.

What would settle it

Compute numerically, for a six-vertex torus with $a=b=1$, the ratio $\Lambda^{(\alpha L)}/\Lambda^{(0)}$ of leading transfer-matrix eigenvalues for $L$ up to a few hundred and $\alpha$ decreasing to $0$; the logarithm of this ratio should decay linearly in $\alpha$ for $c>2$ (i.e. $q>4$) and quadratically for $c\le 2$. Alternatively, simulate critical FK-percolation for $q=4.1$ and check whether $\phi^0[0\leftrightarrow\partial\Lambda_n]$ decays exponentially in $n$, as the discontinuous prediction requires, rather than polynomially.

Watch

Extended reading notes

Core claim

The core mathematical content, stated as Theorem 4.1, is that planar FK-percolation with $q \ge 1$ has a continuous phase transition for $1 \le q \le 4$ and a discontinuous one for $q > 4$, at $p_c(q)=\sqrt{q}/(1+\sqrt{q})$. The notes also present Theorem 5.2: for $q \in [1,4]$, critical FK-percolation on a rescaled lattice $\delta\mathbb{Z}^2$ and on the same lattice rotated by any angle $\theta$ are within Camia--Newman distance $C\delta^c$, so any subsequential scaling limit is rotationally invariant. The companion universality statement Theorem 5.3 says the critical model on isoradial rectangular lattices with angle $\alpha$ is asymptotically the same as on the square lattice, with quantitative error $C\delta^c$. Around these two results the notes build a self-contained account of the sharp phase transition, the dichotomy between continuous and discontinuous behaviour, and the consequences for correlation length, free energy differentiability, and uniqueness of the infinite-volume measure.

Load-bearing premise

The classification rests on an unproved computational input treated as a black box: the claim that the six-vertex model's free energy gap at small tilt is linear when $q>4$ and quadratic when $q\le 4$; if that estimate is wrong, the continuity/discontinuity conclusion does not follow.

Editorial extensions

If this is right

At $p_c=\sqrt{q}/(1+\sqrt{q})$, for $q\in[1,4]$ the infinite-volume measure at criticality is unique and connection probabilities obey polynomial bounds $c\,n^{-1}\le \phi[0\leftrightarrow \partial\Lambda_n]\le n^{-\alpha}$.
For $q>4$, the transition is first-order: the free and wired measures differ at $p_c$, the order parameter and edge intensity jump, the free energy is not differentiable, and the correlation length stays bounded.
Below and above $p_c$, for every $q\ge 1$ the phase transition is sharp: connection probabilities decay exponentially away from criticality.
Any subsequential scaling limit of critical FK-percolation for $q\in[1,4]$ is invariant under all rotations, and the same limit is obtained from any isoradial rectangular embedding.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the black-box Bethe ansatz estimate is as strong as quoted, the same correspondence should yield quantitative information on critical exponents whenever the estimate can be pushed to small slopes; the notes explicitly leave the $k=1$ case out of reach.
The proof strategy that derives rotational invariance from star-triangle invariance suggests that any family of planar lattice models closed under the star-triangle transformation and satisfying RSW should be asymptotically rotationally invariant, even without exact integrability beyond the star-triangle move.
The uniformity of the constants in $\alpha$ noted in the paper points to a continuum FK model at $\alpha\to 0$ belonging to the same universality class; a numerical check of crossing probabilities on strongly skewed rhombic lattices would test this.
The equality $\mathrm{Drift}_{\mathrm{lat}}=\mathrm{Drift}_{\mathrm{vert}}$ at $\beta/2$ is singled out as the delicate input that selects the isoradial embedding; a direct verification of this drift equality in a simpler exactly solvable case would isolate where the embedding geometry enters.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The notes do exactly what they say: they give a quick, readable tour of the major recent results on planar FK-percolation, with proofs of the classical parts and clear attributions for the rest. The exposition is genuinely good. The DT sharpness proof in Chapter 1 is clean, the RSW-in-strips argument in Chapter 3 is well motivated, and the streamlined presentation of the isoradial universality results in Chapter 5 is a real service to the community, since the original paper is heavy going. The notes are also honest about what is omitted and where the reader should look for details. On the soundness of what is actually proved, I have no substantive concerns.

The problem is in Chapter 4, and it is not cosmetic. In §4.2 the text defines the six-vertex parameter c as e^{lambda/2}+e^{-lambda/2} with e^lambda+e^{-lambda}=sqrt(q), which gives c^2=sqrt(q)+2 and c=2 at q=4. That is the correct value used in the Bethe ansatz regime change. Then Proposition 4.2 states c=p^2+sqrt(q), which at q=4 gives 22/9, placing q=4 on the wrong side of the threshold c=2. Taken literally, Corollary 4.7 would predict a first-order transition at q=4, contradicting both Theorem 4.1 and the earlier definition. This is a genuine internal contradiction in a load-bearing place, not a harmless typo. The sign of the log(1+sqrt(q)) term also differs between Proposition 4.2 and the derivation a few lines later; that one is easier to fix but still worth flagging. The Bethe ansatz estimate itself is a black box, which is fine for lecture notes, but the notes should at least tell the reader which definition of c they are relying on.

Who is this for? Graduate students and probabilists who want the big picture of the continuity/discontinuity classification and the machinery behind the rotational invariance results. As a set of lecture notes it is well above the usual standard, and the errors are local and repairable. The paper deserves a serious referee; in fact it needs one. My recommendation: engage with it, have the author fix the c inconsistency and the sign slip, and after that it would be a solid survey.

J. would_cite: false

Referee Report

3 major / 3 minor

Summary. These lecture notes provide an introduction to FK-percolation on Z^2, with focus on two recent results: the classification of the phase transition as continuous for 1 ≤ q ≤ 4 and discontinuous for q > 4, and the asymptotic rotational invariance and universality of the critical phase for q ∈ [1, 4]. The text develops Bernoulli percolation background, the sharpness proof via φ_p(S), duality, RSW in strips, the renormalisation proof of the dichotomy theorem, the six-vertex/BKW route to the q-classification, and a track-exchange proof of universality up to a linear map and rotational invariance. Most main theorems are explicitly attributed to external papers, especially [DCST17], [DCGH+21], [DCKK+20], and [DCKK+22].

Significance. The notes are a well-structured and largely readable expository account. The proofs that are actually carried out, such as Peierls, sharpness via φ_p(S), self-duality, RSW in strips, and the renormalisation inequality underlying the dichotomy, are standard and appear essentially correct. The exposition is honest about the black-box status of the Bethe ansatz computations, and it gives a valuable streamlined presentation of the track-exchange arguments in Chapter 5. The main obstacles to accepting the manuscript in its current form are internal inconsistencies in Chapter 4, where the six-vertex parameter c is defined in two incompatible ways and where a key proposition contains contradictory sign conventions. These are not cosmetic issues because Chapter 4 is the chapter that derives the headline classification, including the endpoint q = 4.

major comments (3)

[§4.3, Theorem 4.5] The manuscript defines c in two incompatible ways. The paragraph after Proposition 4.2 defines c through λ by e^λ + e^{-λ} = √q and c = e^{λ/2} + e^{-λ/2}, which gives c^2 = √q + 2 and hence c = 2 at q = 4. Proposition 4.2 and Corollary 4.7 instead set c = p^2 + √q, which at q = 4 gives 4/9 + 2 = 22/9 > 2. Taken literally, this would place q = 4 in the linear regime Cα of Theorem 4.5 rather than in the quadratic regime Cα^2, contradicting the continuity statement of Theorem 4.1 at q = 4. This is load-bearing because Corollary 4.7 is the step that converts the six-vertex asymptotics into the FK-percolation estimate used to prove Theorem 4.1. Please adopt a single definition of c, state Theorem 4.5 and Corollary 4.7 in terms of that c, and check every occurrence.
[§4.2, Eq. (4.1) and its proof] The estimate f6V(0) − f6V(α) = Cα + o(α) for c > 2 and Cα^2 + o(α^2) for 0 < c ≤ 2 is quoted from [DCKK+22], but the notes do not prove the condensation of Bethe roots, the identification of the Perron–Frobenius eigenvalue, or the final asymptotic expansion. Since the entire classification of Chapter 4 rests on this estimate, the text should state explicitly at the level of Theorem 4.5 and Corollary 4.7 that these are external results whose proof is outside the scope of the notes. The prose already says this in words, but the formal statements are presented as theorems of the manuscript, which is misleading in a chapter whose conclusion is the paper's headline result.
[§4.2, Eq. (4.1) and its proof] There is a sign inconsistency in the BKW free-energy formula. Proposition 4.2 states fFK = f6V + (1/4) log q + log(1 + √q), while the proof of the proposition concludes with fFK = f6V + (1/4) log q − log(1 + √q). Both statements appear in the same section, and the proof says 'This proves (4.1),' which it does not. The difference does not affect the later threshold argument because only f6V(α) − f6V(0) is used, but the contradiction in a central proposition must be corrected and the signs re-derived.

minor comments (3)

[§2.2, proof of Proposition 2.9] The parametrization p(t) = e^t/(1 − e^t) with t ≥ 0 cannot produce values in (0, 1), and the displayed derivative (2.10) appears to omit the contribution coming from the log(1 − p) term in the definition of g. Please check the intended parametrization and the derivative computation.
[§2.1] In the sentence explaining the use of the Spatial Markov property, 'different boundary conditions ξ' should presumably read 'boundary conditions ζ', and the subsequent reference to '(2.6)' should likely be a reference to the displayed comparison (2.1).
[Throughout] There are several minor typographical slips, including 'curial' for 'crucial', 'repsectively' for 'respectively', and 'finite energy of the FK-percolation' in Proposition 4.2 where 'free energy' is meant. These do not affect the mathematics but should be corrected in a final version.

Circularity Check

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No circularity: the notes are an expository review whose central theorems are quoted and attributed to independent external sources.

full rationale

The paper is lecture notes, not an original derivation. The continuity/discontinuity classification (Theorem 4.1) is explicitly attributed to [DCST17] and [DCGH+21], and the six-vertex computation used in the presented route (Theorem 4.5) is cited as a full proof in [DCKK+22]. No parameter is fitted to the target conclusion, and no equation is defined in terms of the result it is supposed to imply. The BKW correspondence (Proposition 4.2) and the Bethe ansatz estimate are external inputs with stated assumptions independent of the FK-percolation classification; the notes even state that the Bethe ansatz details are not reproduced. Similarly, the rotational invariance and universality results of Chapter 5 are attributed to [DCKK+20] and are proved from explicit track-exchange/star-triangle couplings and RSW estimates, not from the theorems they are meant to establish. There is an internal inconsistency in Section 4.2, where the six-vertex parameter c is first written as c = p^2 + sqrt(q) and later as c = e^{lambda/2} + e^{-lambda/2}; taken literally, this would be a mathematical typo or correctness issue at q = 4, but it is not a circularity because neither definition is derived from the target phase-transition classification. No self-citation chain is load-bearing, no known empirical pattern is merely renamed, and no uniqueness result is imported from the author's own prior work to forbid alternatives. The exposition is self-contained in the sense that each quoted result is properly attributed and the local arguments are genuine proofs conditional on those external theorems.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

Because this is a review, the ledger lists the external results that the notes borrow without proof: monotonicity and FKG, the Graham-Grimmett sharp threshold, the Bethe ansatz estimates for the six-vertex model, isoradial RSW and arm bounds, uniqueness of the infinite-volume measure, the drift equality, and the star-triangle/track-exchange toolbox. There are no fitted free parameters and no newly invented entities; the six-vertex model, IIC measures, and lattices of nails are standard or proof devices.

assumptions (8)

domain assumption FK-percolation with q >= 1 satisfies FKG and stochastic monotonicity (Proposition 2.4).
Used throughout; standard theorem for q >= 1, cited to Holley/FKG theory, not proved in the notes.
domain assumption Graham-Grimmett sharp threshold inequality (Theorem 2.7) for monotone measures.
Stated and referenced to [GG11] without proof; needed to prove the decay in (3.8) in the continuous case.
domain assumption Bethe ansatz analysis of the six-vertex transfer matrix, including the existence and condensation of Bethe roots (Theorem 4.6).
Section 4.3 says 'It is expected that there exists exactly one solution to (4.7)' and defers to [DCGH+18, DCKK+22]. This underpins Theorem 4.5.
domain assumption Asymptotic formula for six-vertex free energy at small slope (Theorem 4.5).
Quoted from [DCKK+22]; the linear versus quadratic behavior in alpha is the exact input that separates q > 4 from q <= 4 in Corollary 4.7.
domain assumption RSW estimates and arm exponent bounds for FK-percolation on isoradial rectangular lattices (Theorem 5.7, Proposition 5.12).
Taken from [DCLM18, DCKK+20]; used throughout Chapter 5 to control crossings, mixing, and IIC measures.
domain assumption Existence and uniqueness of the infinite-volume measure on isoradial rectangular lattices.
Stated as a direct consequence of (RSW) in Section 5.2.3; not proved in the notes.
domain assumption Equality of drifts Driftlat(beta, beta/2) = Driftvert(beta, beta/2) (Proposition 5.26).
Called 'surprisingly profound' and used to deduce M_{pi/2,alpha}=id; the proof is postponed or summarized, making this an unverified input in the notes.
domain assumption Star-triangle transformation preserves connection probabilities (Lemma 5.13) and track-exchange operator exists (Fact 5.15).
Essential to the universality argument; a coupling is sketched but full verification is referenced to [GM13a, DCKK+20].

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Pith. "Pith review of Exploring the phase transition of planar FK-percolation." pith.science (2026). https://pith.science/paper/3MJL5U7G

@misc{pith2026250208394,
  author       = {Pith},
  title        = {Pith review of: Exploring the phase transition of planar FK-percolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MJL5U7G}},
  note         = {Machine review of arXiv:2502.08394}
}

abstract

The aim of these notes is to give a quick introduction to FK-percolation, focusing on certain recent results about the phase transition of the two dimensional model, namely its continuity or discontinuity depending on the cluster weight $q$, and the asymptotic rotational invariance of the critical phase (when the phase transition is continuous). As such, the main focus is on FK-percolation on $\mathbb Z^2$ with $q \geq 1$, but we do mention some important results valid for general dimension. To favour quick access to recent results, the style is minimal, with certain proofs omitted or left as exercises.

Figures

Figures reproduced from arXiv: 2502.08394 by the authors.

**Figure 1.1.** A piece of a square lattice (solid vertices and edges) and its dual (hollow vertices, [PITH_FULL_IMAGE:figures/full_fig_p008_1_1.png] view at source ↗

**Figure 1.2.** Left: An illustration of the argument in the proof of Lemma 1.9: once the cluster C of 0 in S has been explored, for 0 to be connected to distance n, one of the red vertices needs to be connected to distance n − diam(S) in the complement of C. Right: The white region containing 0 is the set of edges S. Here 0 is connected to four vertices on ∂S. When proving Lemma 1.10, S denotes the connected component of 0 in the … view at source ↗

**Figure 1.3.** Left: an (n + 1) × n rectangle is either crossed horizontally in the primal model or vertically in the dual. At the self-dual point, these two events have equal probability 1/2. Right: By symmetry, the probability to connect the lower half of the left side BottomLeft to the right side of the square is at least 1/4. The same holds for connections between the TopRight and the left side. Combining these two crossings w… view at source ↗

Figures from the paper (27 more)

**Figure 1.4.** Figure 1.4: In each square BottomLeft is connected to TopRight. The left and right red paths are the topmost and bottommost, respectively, paths realising these connections in a configuration ω. The grey paths are their reflections σ with respect to the medial line; by construct…

**Figure 1.5.** Figure 1.5: The red path is the leftmost top-bottom crossing of the rectangle. A point on it [PITH_FULL_IMAGE:figures/full_fig_p019_1_5.png]

**Figure 3.1.** Figure 3.1: Left: The events Hn and H∗ n , respectively. Right: If 0 is connected to ∂ΛN , then there exists a path of at least ⌊N/8n⌋ disjoint but neighbouring translates of Λ4n by points of (nZ) 2 for which H∗ n fails. The black annuli are the translates of Λ2n \ Λn for which …

**Figure 3.2.** Figure 3.2: By combining vertical crossings of horizontal translations of [PITH_FULL_IMAGE:figures/full_fig_p034_3_2.png]

**Figure 3.3.** Figure 3.3: The vertical crossings ΓL and ΓR are the leftmost and rightmost crossings of their respective rectangles. Identifying them only has an influence on the state of the edges on them and in the hashed areas. Conditionally on ΓL and ΓR, the part D of Rect(n, n) left betwe…

**Figure 3.4.** Figure 3.4: Left: with v chosen as in (3.16), the probability of obtaining a “diagonal” crossing, i.e., from (vb) to (wd), is bounded away from 0 by a universal multiple of ϕ [PITH_FULL_IMAGE:figures/full_fig_p037_3_4.png]

**Figure 3.5.** Figure 3.5: Left: For m ≤ n/20, combining several instances of Hm produces Hn. Comparing boundary conditions and using (FKG), we conclude that un ≥ ϕ 0 Λ2n\Λn [Hn] ≥ u Cn/m n for some universal constant C. Right: Consider the point x on the right side of ∂Λn that is most likely …

**Figure 3.6.** Figure 3.6: Left: The event C∩E ∩E ∩G ˜ with the elements required by each event highlighted. Its probability may be bounded from above by the probability of the primal circuits of C occurring, conditionally on the dual paths. This produces an upper bound of u k n . Conversely, …

**Figure 4.1.** Figure 4.1: The six possible vertex configurations obeying the ice rule. The weights are [PITH_FULL_IMAGE:figures/full_fig_p044_4_1.png]

**Figure 4.2.** Figure 4.2: See the torus TL,M as a vertical cylinder. The event Ek (here with k = 3) requires the existence of k clusters of ω (in red) between the bottom and top of the cylinder. This leads to the existence of at least 2k paths between the bottom and top of the torus in the lo…

**Figure 4.3.** Figure 4.3: The different steps in the correspondence between the FK-percolation and six [PITH_FULL_IMAGE:figures/full_fig_p046_4_3.png]

**Figure 4.4.** Figure 4.4: The 8 different types of vertices encountered in an oriented loop configuration. Lemma 4.3. For any ω ∈ ΩFK, wFK(ω) = C √ q ℓ(ω ◦) q s(ω) = C √q 2 ℓ0(ω ◦) q s(ω) X ω coherent w. ω w (ω ), (4.3) where ω ◦ is the loop configuration corresponding to ω and the sum is ov…

**Figure 4.5.** Figure 4.5: The unique possible completion between ⃗x (bottom configuration) and ⃗x′ (top configuration) that obeys the ice rule. The two types of c-vertices are marked in red; the matrix entry corresponding to the transition from ⃗x to ⃗x′ is c 4 . Transfer matrix formalism. We…

**Figure 5.1.** Figure 5.1: An example of a graph L(α), where αi is equal to π 2 for i ≤ 3, and some angle α above. The diamond graph is drawn in light black lines; the solid and hollow dots are the vertices of V• and V◦, respectively. The actual isoradial graph G is drawn in thicker black line…

**Figure 5.2.** Figure 5.2: The edge e and its subtended angle θe; the red edges are those of G and the grey ones are those of the diamond graph. The dual graph (blue edges, hollow vertices) is also isoradial. configuration2 ω0 on F c , ϕG[ω on F | ω = ω0 on F c ] = 1 Zω0 (F) Y e∈F p ωe e (1 …

**Figure 5.3.** Figure 5.3: A red cluster C with the cells containing Top(C) and Right(C) marked in blue. Note that there are three topmost cells visited by C, the marked one is the leftmost. intersected by C. Write Top(C) for its associated lattice point. Write Bottom(C) for the lattice point …

**Figure 5.4.** Figure 5.4: The three rhombi together with the drawing, on the left, of the triangle (in which [PITH_FULL_IMAGE:figures/full_fig_p063_5_4.png]

**Figure 5.5.** Figure 5.5: A picture of the transformations in the star-triangle coupling. In the first line, [PITH_FULL_IMAGE:figures/full_fig_p064_5_5.png]

**Figure 5.6.** Figure 5.6: From left to right: The initial lattice L0, with tracks of angle β at the bottom and α at the top; the blue region is the cell (0, 1) of the lattice. Applying S0 ◦ S1 transforms this L0 into L2; the mixed block is red. After more transformations, a block of angle α s…

**Figure 5.7.** Figure 5.7: A step in the transformations Lt . The mixed block is pink, the two other ones are blue; notice that the bottom block is of angle α, which indicates that t > K. The two red clusters are mesoscopic; the extrema are marked, as are the extremum boxes. All extrema of the…

**Figure 5.8.** Figure 5.8: A lattice of nails (in red) is formed of one nail for each point [PITH_FULL_IMAGE:figures/full_fig_p078_5_8.png]

**Figure 5.9.** Figure 5.9: The left boundary Γ of a vertically crossing primal cluster; the conditioning only affects the grey area. Under this conditioning, the cluster of Γ surrounds each one of the grey rhombi D(x) closest to it with uniformly positive probability. depending only on the hom…

**Figure 5.10.** Figure 5.10: A configuration containing a lattice of nails [PITH_FULL_IMAGE:figures/full_fig_p081_5_10.png]

**Figure 5.11.** Figure 5.11: From L on the left, there are two series of transformations leading to L ′ on the right. On the top, we proceed by exchanging horizontal tracks, bringing them down progressively. This produces a sequence of lattices Lt , and a sequence Ct of transformations of the c…

**Figure 5.12.** Figure 5.12: If the angle between u and v is assumed strictly below π/2, then Q is an ellipse. The rotation by π/2 of (u, v) Q←→ (−u, −v) is realised by the red path; it has strictly lower probability than (v, −u) Q←→ (−v, u). We are now ready to conclude. Write θ(α) for the ang…

**Figure 5.13.** Figure 5.13: The rhombi R and T(R) under the assumption that λ > 1. Any crossing in R between the arcs (bc) and (da) produces a crossing in T(R) between (T(a)T(b)) and (T(c)T(d)). However, the latter crossing is strictly easier to achieve, uniformly in the scale δ of the lattic…

**Figure 5.14.** Figure 5.14: A different set of track exchanges than those of Section [PITH_FULL_IMAGE:figures/full_fig_p093_5_14.png]

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

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