The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality

The W ulﬀ cry stal of self-dual FK -percolation becomes r ound when approac hing criticality March 18, 2026 Ioan Manolescu † , Maran Mohanarangan ∗ † U niv ersity of Fribourg, ioan.manolescu@unifr.ch ∗ U niv ersity of Fribourg & U niversity of Innsbruck, maran.mohanarangan@unifr.ch Abstract The study of the phase transition in planar FK -percolation on the square lattice has seen signiﬁcant recent breakthroughs. The model undergoes a chang e in the nature of its phase transition at q = 4 , transitioning from a continuous to a discontinuous regime. The aim of this ar ticle is to inv estigate the beha viour of the model in the discontinuous regime as q > 4 approaches the continuous transition point 4 from abo v e, while maintaining the critical parameter p = p c ( q ) . W e pro v e that in this limit, the correlation length becomes isotropic. The core of the proof builds upon the recentl y established rotational in variance of the larg e-scale f eatures of the model at q = 4 [ DCKK + 20 ]. Contents 1 Introduction 1 1.1 Deﬁnition of the model and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Strategy of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Near -critical FK -percolation with q ≤ 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Preliminaries 6 2.1 Isoradial g raphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Deﬁnition and elementar y proper ties of the isoradial random-cluster model . . . . . . . . 6 2.3 Isoradial random-cluster model with q = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Incipient inﬁnite cluster in the half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Half-plane measures 13 3.1 U niqueness of the half-plane free measure . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Rate of decay of connection probabilities in the half-plane . . . . . . . . . . . . . . . . . 15 4 Univ ersality : proof of Theorem 1.3 16 4.1 The coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Expected increment: proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Deducing universality : proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . 25 1 Introduction 1.1 Deﬁnition of the model and main result The r andom-cluster model , introduced b y Fortuin and Kastele yn and hence also ref er red to as F ortuin–Kast eleyn (FK) percolation , is a central model of statistical mechanics. W e consider it here on the tw o-dimensional square lattice Z 2 , where it e xhibits a ﬁrst-order or higher -order phase transition, depending on the cluster -w eight q . W e recall its deﬁnition and a f e w basic proper ties Intr oduction 2 belo w . For more bac kground, we refer the reader to the monograph [ Gri06 ] and, f or an e xposition of more recent results, to the lecture notes [ Man25 ]. W e consider the square lattice ( Z 2 , E ) , that is, the graph with v er te x set Z 2 = { ( n, m ) : n, m ∈ Z 2 } and edges betw een nearest neighbours. W e will slightly abuse notation and denote the graph itself b y Z 2 . Let G = ( V , E ) be a subgraph of the square lattice. A percolation conﬁguration ω on G is an element of { 0 , 1 } E . An edge e ∈ E is said to be open in ω if ω e = 1 and closed otherwise. A conﬁguration ω can be seen as a subgraph of G with v er te x set V and edge set { e ∈ E : ω e = 1 } . When speaking of connections in ω , w e view ω as a graph. A cluster is a maximal connected component of ω (it ma y be an isolated vertex). When G is ﬁnite, let o ( ω ) and c ( ω ) denote the number of open and closed edges in ω , respectiv ely . Fur thermore, let k ( ω ) denote the number of clusters in ω . Then the random-cluster measure on G with parameters p ∈ [ 0 , 1 ] and q > 0 , and free boundar y conditions is a measure on percolation conﬁgurations giv en b y ϕ 0 G,p,q [ ω ] = 1 Z 0 G,p,q p o ( ω ) ( 1 − p ) c ( ω ) q k ( ω ) , where Z 0 G,p,q is a normalising cons tant called the partition function . For q ≥ 1 , the model can be e xtended to inﬁnite volume where it e xhibits a phase transition. More precisely , the famil y of measures ϕ 0 G,p,q con v erg es w eakly as G tends to Z 2 . The limiting measure, denoted by ϕ 0 Z 2 ,p,q , undergoes a phase transition in the sense that there exis ts a cr itical threshold p c : = p c ( q ) ∈ ( 0 , 1 ) such that the probability that there exis ts an inﬁnite cluster under ϕ 0 Z 2 ,p,q is 0 f or all p < p c and 1 if p > p c . It has been shown [ BDC12 ] that the critical threshold is p c = √ q / ( 1 + √ q ) . The beha viour at the cr itical parameter p c is of great interest. One impor tant question is whether the phase transition is continuous or discontinuous , meaning here whether the probability that 0 is contained in an inﬁnite cluster is continuous or discontinuous as a function of p . The phase transition was shown to be continuous f or 1 ≤ q ≤ 4 [ DCST17 ] and discontinuous when q > 4 [ DCGH + 21 ]. Alter nativ e proofs of these tw o regimes were obtained in [ GL25 ] and [ RS20 ], respectiv el y . The behaviour at cr iticality diﬀers dras ticall y betw een the two regimes. In the regime 1 ≤ q ≤ 4 , the measure at the point of phase transition is uniq ue and exhibits proper ties indicative of scale inv ar iance, such as the Russo–Se ymour –W elsh proper ty ( RSW ); it is e xpected to hav e a non-trivial, conf or mall y inv ar iant scaling limit. In contrast, f or q > 4 , there are multiple measures at the point of phase transition, with either sub- or super -cr itical behaviour , but no cr itical measure e xists [ GM23 ]. In par ticular , in the ﬁrst case the cor relation length of the model div erg es when approaching the cr itical point, while in the second case it remains bounded. It was recentl y prov ed in [ DCKK + 20 ] that f or 1 ≤ q ≤ 4 , the model e xhibits asymptotic rotational in variance at criticality , in line with its conjectured conf ormal in variance. More precisely , the model and its rotation b y an arbitrar y angle θ may be coupled so that, at larg e scales, the y produce similar conﬁgurations with high probability . In particular, any subsequential scaling limit of the cr itical model is in variant under all rotations. This result will be instrumental in the proof of our main theorem. In this paper , w e will be par ticularl y interested in the discontinuous phase transition, q > 4 . For the e xposition of our results, w e need to introduce tw o central quantities. F ix θ ∈ [ 0 , 2 π ) and deﬁne e θ as the unit vector with angle θ to the hor izontal axis. For q ≥ 1 and p ≤ p c , the corr elation length of the model in direction θ is deﬁned as the limit ξ p,q ( θ ) : =  lim n →∞ − 1 n log ϕ 0 Z 2 ,p,q  0 ← → ⌊ ne θ ⌋   − 1 , Intr oduction 3 where ⌊ ne θ ⌋ is the v er te x of Z 2 closest to ne θ ∈ R 2 . Similar l y , we deﬁne the point-to-hyperplane decay rat e by ζ p,q ( θ ) : =  lim n →∞ − 1 n log ϕ 0 Z 2 ,p,q  0 ← → H θ ≥ n   − 1 , where H θ ≥ n is the half-space deﬁned by H θ ≥ n : = { x ∈ R 2 : ⟨ x, e θ ⟩ ≥ n } f or n ≥ 0 . The e xistence of both limits f ollo w s from standard sub-additivity arguments. The tw o quantities are related via a so-called conv ex duality (see e.g. [ Ott22 ]). More precisel y , it holds that  ξ p,q ( θ )  − 1 = sup θ ′ ∈ [ 0 , 2 π )  ζ p,q ( θ ′ )  − 1 ⟨ e θ , e θ ′ ⟩ . A hallmark of the regime q > 4 is the ﬁniteness of ξ p c ,q ( θ ) and ζ p c ,q ( θ ) f or all θ ∈ [ 0 , 2 π ) . It should be mentioned that ξ p,q ( θ ) and ζ p,q ( θ ) are both ﬁnite for all p < p c and q ≥ 1 [ BDC12 , DCR T19 ]. Ho we ver , when p = p c and q ∈ [ 1 , 4 ] , both quantities are inﬁnite [ DCST17 ]. The asymptotic rotational inv ar iance of the model in the regime 1 ≤ q ≤ 4 is a strong indication that ξ p c ,q and ζ p c ,q become isotropic as q ↘ 4 . Our main result conﬁrms this. Note that while both ξ p c ,q ( θ ) and ζ p c ,q ( θ ) diver g e f or all θ as q ↘ 4 , our result states that these quantities become asymptoticall y isotropic when renor malised. Theorem 1.1. F or all ε > 0 , ther e exists q 0 > 4 such that for all q ∈ ( 4 , q 0 ] and any θ 1 , θ 2 ∈ [ 0 , 2 π ) , w e have     ξ p c ,q ( θ 1 ) ξ p c ,q ( θ 2 ) − 1     < ε and     ζ p c ,q ( θ 1 ) ζ p c ,q ( θ 2 ) − 1     < ε. Theorem 1.1 ma y also be expressed in ter ms of the W ulﬀ shape , the polar set of the inv erse cor relation length: W q : = \ θ ∈ [ 0 , 2 π ) { x ∈ R 2 : ⟨ x, e θ ⟩ ≤ ξ p c ,q ( θ ) − 1 } . Then Theorem 1.1 implies the f ollowing result. Corollary 1.2. When q ↘ 4 , W q / p V ol ( W q ) tends to the unit disk U = { x ∈ R 2 : | x | ≤ 1 } . The W ulﬀ shape descr ibes the asymptotic shape of a cluster when conditioned to ha v e a large v olume. Indeed, write C f or the cluster of the or igin and denote its cardinality b y | C | . Then, for an y ε > 0 , ϕ Z 2 ,p c ,q  d Hausdorff  1 √ n · C , 1 p V ol ( W q ) · W q  > ε    | C | ≥ n  → 0 as n → ∞ , where d Hausdorff denotes the Hausdor ﬀ distance [ Cer06 ]. Thus, Corollar y 1.2 states that, as q ↘ 4 , the typical shape of a cluster conditioned to be large becomes round. Intr oduction 4 1.2 Strategy of the proof Similar to the rotational inv ar iance pro v ed in [ DCKK + 20 ], our result is accompanied b y a univ er - sality result relating random-cluster models on diﬀerent isoradial graphs. T o that end, we brieﬂy describe an inhomogeneous FK -percolation model on some distor ted embedding of the sq uare lattice Z 2 . A more detailed discussion of this topic will be postponed until Section 2 . Fix q > 4 and α ∈ ( 0 , π ) . Let L ( α ) be the embedding of Z 2 in which horizontal edges ha v e length 2 cos ( α/ 2 ) , v er tical edges ha ve length 2 sin ( α/ 2 ) , and which is rotated by an angle of α/ 2 — see Figure 1 . Deﬁne the inhomogeneous random-cluster model on ﬁnite subg raphs of L ( α ) with cluster w eight q and edg e-parameters p h and p v f or the hor izontal and v er tical edges, respectiv ely , giv en by p h 1 − p h = 1 − p v p v = √ q sinh ( r α ) sinh ( r ( π − α )) with r : = 1 π cosh − 1  √ q 2  , in the same wa y as the model on Z 2 ; we also refer to ( 2.1 ) f or a more general deﬁnition. W r ite ϕ 0 L ( α ) ,q f or the inﬁnite-v olume measure on L ( α ) with the edg e weights p h and p v as abo v e and free boundar y conditions. W e call L ( α ) an isor adial r ectangular lattice and ϕ 0 L ( α ) ,q its associated random-cluster model. Belo w , we will use estimates on lattices L ( α ) f or diﬀerent values of α ∈ ( 0 , π ) . W e will use the phrase “unif or m in α on compacts of ( 0 , π ) ” to mean that estimates are unif orm on any inter v al of the type ( ε, π − ε ) with ε > 0 , but potentially not on ( 0 , π ) . All constants hereafter will be unif or m in α on compacts of ( 0 , π ) . The main results ma y be sho wn to be unif or m o ver the whole interval ( 0 , π ) , but this requires more care and is not essential for our purposes; f or fur ther details, see [ DCLM18 , Sec. 5]. It w as sho wn in [ DCLM18 ] that, f or an y q ≥ 1 ﬁxed, the random-cluster models e xhibit the same type of phase transition across all isoradial rectangular lattices L ( α ) . In par ticular , as f or the critical random-cluster model on Z 2 , the phase transition is discontinuous f or q > 4 . Thus, f or an y direction θ ∈ [ 0 , 2 π ) and any angle α ∈ ( 0 , π ) , the correlation length and the point-to-h yper plane deca y rate may be deﬁned and are bounded, unif or ml y in θ ∈ [ 0 , 2 π ) and α on compacts of ( 0 , π ) : ξ α,q ( θ ) =  lim n →∞ − 1 n log ϕ 0 L ( α ) ,q  0 ← → ⌊ ne θ ⌋   − 1 < ∞ and ζ α,q ( θ ) =  lim n →∞ − 1 n log ϕ 0 L ( α ) ,q  0 ← → H θ ≥ n   − 1 < ∞ . (1.1) Moreo v er, while the con v ex duality in [ Ott22 ] is stated f or the square lattice Z 2 , the same proof still applies to isoradial rectangular lattices L ( α ) , i.e., the relation  ξ α,q ( θ )  − 1 = sup θ ′ ∈ [ 0 , 2 π )  ζ α,q ( θ ′ )  − 1 ⟨ e θ , e θ ′ ⟩ . (1.2) still holds. Las tl y , notice that f or α = π 2 , one obtains a slight modiﬁcation 1 of ξ p c ,q and ζ p c ,q as pre viously deﬁned. The models ϕ 0 L ( α ) ,q f or diﬀerent values of α may be related by modifying the lattice step by step via the so-called star -triang le transf or mation . Features that are stable under these transf or mations can then be transferred from one measure to the other . This strategy w as used successfully for FK - percolation in [ GM14 , DCLM18 , DCKK + 20 ] to transfer RSW estimates and pro v e universality of the models f or q ∈ [ 1 , 4 ] and will be used here to obtain asymptotic univ ersality f or our quantities of interest. 1 Indeed, L ( π / 2 ) is the rotation by π / 4 of √ 2 Z 2 and the measure ϕ 0 L ( α ) ,q is the free measure with p = p c ( q ) on this graph. Intr oduction 5 Theorem 1.3. F or all ε > 0 , ther e exists q 0 > 4 such that for q ∈ ( 4 , q 0 ] , all α ∈ ( ε, π − ε ) and any θ ∈ [ 0 , 2 π ) , it holds that     ξ α,q ( θ ) ξ π / 2 ,q ( θ ) − 1     < ε and     ζ α,q ( θ ) ζ π / 2 ,q ( θ ) − 1     < ε. W e will f ocus on pro ving the statement f or the point-to-hyperplane deca y rate ζ . Indeed, this quantity is simpler to control under the local modiﬁcations w e apply to the model. The analogous statement f or the cor relation length ξ will then be obtained via the con ve x duality ( 1.2 ). W e close this section b y obser ving that Theorem 1.3 directly implies Theorem 1.1 . Pr oof of Theor em 1.1 . W e prov e the statement f or the cor relation length ξ — the analogous result f or ζ is obtained in the same wa y . Fix ε > 0 and 0 < ε ′ < ε 2+ ε . Choose q 0 as in Theorem 1.3 according to ε ′ and let q ∈ ( 4 , q 0 ] . Fix θ 1 ∈ ( 0 , π 4 ] and θ 2 ∈ [ π 4 , π 2 ] , then set α = θ 1 + θ 2 ∈ [ π 4 , 3 π 4 ] . Observ e that e iα/ 2 R is an axis of symmetry of L ( α ) and theref ore, ϕ 0 L ( α ) ,q is inv ar iant under or thogonal reﬂections with respect to said axis. In par ticular , w e hav e ξ α,q ( θ 1 ) = ξ α,q ( θ 2 ) . Theorem 1.3 then implies ξ π / 2 ,q ( θ 1 ) ξ π / 2 ,q ( θ 2 ) = ξ π / 2 ,q ( θ 1 ) /ξ α,q ( θ 1 ) ξ π / 2 ,q ( θ 2 ) /ξ α,q ( θ 2 ) ∈  1 − ε ′ 1 + ε ′ , 1 + ε ′ 1 − ε ′  . In par ticular ,     ξ π / 2 ,q ( θ 1 ) ξ π / 2 ,q ( θ 2 ) − 1     < 2 ε ′ 1 − ε ′ < ε. The abo v e e xtends to all angles θ 1 , θ 2 ∈ [ 0 , 2 π ) using the in variance of ϕ 0 L ( π / 2 ) ,q under re- ﬂections with respect to the horizontal, v er tical, and diagonal ax es. Finall y , note that ξ p c ,q ( θ ) = 1 √ 2 ξ π / 2 ,q ( θ + π 4 ) . 1.3 Near -critical FK -percolation with q ≤ 4 One ma y vie w the topic of the present paper in the conte xt of near -cr itical FK -percolation on Z 2 . Most commonly , the near -cr itical regime refers to FK -percolation on Z 2 with ﬁx ed q ∈ [ 1 , 4 ] and p  = p c , at scales where the model transitions from a cr itical to an oﬀ-cr itical beha viour . W e direct the reader to [ DCM22 ] f or details and only mention here that the near -cr itical FK percolation with 1 ≤ q ≤ 4 e xhibits similar f eatures as the cr itical one, most impor tantl y ( RSW ). It is expected that, as f or the cr itical phase, the near -cr itical regime e xhibits a f or m of asymptotic in variance under rotations. In par ticular , we expect that, f or an y 1 ≤ q ≤ 4 and any two angles θ 1 , θ 2 , ξ p,q ( θ 1 ) ξ p,q ( θ 2 ) → 1 and ζ p,q ( θ 1 ) ζ p,q ( θ 2 ) → 1 as p ↗ p c ( q ) . (1.3) Theorem 1.1 ma y be vie w ed as a manif estation of ( 1.3 ) when the cr itical regime is approached along the line q ↘ 4 and p = p c ( q ) . The authors ha v e not manag ed to adapt the strategy belo w to also pro v e ( 1.3 ) and w e belie v e a ke y ing redient is missing. Indeed, the exact validity of the star -triangle transf ormation (or its manif estation as track -e x chang es; see Proposition 2.2 belo w) is essential to the present argument. These transf or mations are no longer e xact when p  = p c , which w e believ e is a fundamental obstruction to adapting the ar gument. It should be mentioned that [ DC13 ] pro ved ( 1.3 ) f or a variant of Bernoulli percolation (cor re- sponding to q = 1 ) through diﬀerent means. Indeed, this w ork builds on the exis tence and rotational in variance of the near -cr itical scaling limit [ GPS18 ], whic h in turn relies on the descr iption of the scaling limit of the critical phase [ Smi01 , CN06 ]. This approach has not y et been adapted to q > 1 and will lik ely require signiﬁcant new ideas. Preliminaries 6 Organisation of the paper Section 2 contains a revie w of some basic results for FK -percolation on isoradial g raphs. In Section 3 , we deﬁne half-plane measures and track -e x chang es f or such measures, as this framew ork is technicall y more conv enient f or our pur poses. Finall y , in Section 4 , w e pro v e the main result, namel y Theorem 1.3 . 2 Preliminaries In this section, w e introduce isoradial g raphs and the random-clus ter models associated to them. 2.1 Isoradial graphs A rhombic tiling G ⋄ is a tiling of the plane by rhombi of edge-length 1. An y such graph is bipar tite and w e can divide its v er tices in two sets of non-adjacent vertices V • and V ◦ . The isoradial gr aph G associated to G ⋄ is the graph with v er te x set V • and edg e set giv en by the diagonals of the faces of G ⋄ betw een v er tices of V • . If the roles of V • and V ◦ are e xc hanged, w e obtain the dual G ∗ of G , which is also isoradial. The ter m isoradial ref ers to the f act that each face of G can be inscribed in a circle of radius 1. The rhombic tiling G ⋄ is called the diamond gr aph of G . A trac k of G ⋄ is a bi-inﬁnite sequence of adjacent faces ( r i ) i ∈ Z of G ⋄ , such that each pair r i and r i +1 shares an edge, and all suc h edg es are parallel. The angle formed b y an y suc h edg e with the hor izontal axis is called the tr ansv erse ang le of the track. The graphs considered in this paper are of a speciﬁc type: w e assume that all f aces of G ⋄ ha v e horizontal top and bottom edg es — w e call these isor adial rectangular lattices . As a result, the diamond g raphs consist of tw o f amilies of tracks: horizontal trac ks t i with transv erse angles α i ∈ ( 0 , π ) , and vertical trac ks s j , each with transverse angle 0. Each track of one famil y intersects all tracks of the other famil y , but no tw o trac ks from the same famil y intersect. For a sequence of trac k angles α = ( α i ) i ∈ Z ∈ ( 0 , π ) Z , denote by L ( α ) the graph as abov e whose hor izontal tracks ha v e transv erse angles α i in increasing v er tical order . When α i = α f or e v er y i , w e simpl y wr ite L ( α ) = L ( α ) . Note that L ( α ) cor responds to a distorted embedding of the square lattice Z 2 , with e iα/ 2 R and e i ( α + π ) / 2 R as ax es of symmetr y ; see Figure 1 f or an illustration. W e will use the same notation f or ﬁnite or half-inﬁnite sequences α to indicate lattices L ( α ) co v er ing a hor izontal strip or half-plane. For technical reasons (speciﬁcally f or the results of [ DCLM18 ] to adapt readily), we will alwa y s w ork with sequences α containing at most two v alues. This restriction is not essential and may be remov ed by revisiting [ DCLM18 ]. When consider ing isoradial g raphs G = ( V , E ) , we use the notation Λ n = [ − n, n ] 2 and identify it with the subgraph spanned by the v er tices of V contained in Λ n . W e write ∂ Λ n f or the set of v er tices v ∈ V ∩ Λ n that hav e a neighbour in V ∩ Λ c n . Fur thermore, w e write Λ n ( z ) f or the translation of Λ n b y z ∈ R 2 . 2.2 Deﬁnition and elementary properties of the isoradial random-cluster model The isoradial embedding of a graph G produces diﬀerent edge-w eights — called isoradial weights — f or the edg es of G as a function of their length. Indeed if e is an edg e of G and θ e is the angle of the rhombus of G ⋄ containing e and not bisected b y e , w e set p e : =          √ q sin ( r ( π − θ e )) sin ( rθ e ) + √ q sin ( r ( π − θ e )) , q < 4 , 2 π − 2 θ e 2 π − θ e , q = 4 , √ q sinh ( r ( π − θ e )) sinh ( rθ e ) + √ q sinh ( r ( π − θ e )) , q > 4 , where r : =      1 π cos − 1  √ q 2  , q ≤ 4 , 1 π cosh − 1  √ q 2  , q > 4 . Preliminaries 7 p h p v t i s j α Figure 1: Lef t : An isoradial rectangular lattice L ( α ) . The under lying diamond graph is drawn in thin black lines, while the actual lattice is dra wn in red. The white points denote vertices of the dual lattice. T w o trac ks — one hor izontal and one v ertical — are sho wn as dashed grey lines. Note the diﬀerent parameters p v and p h f or the tw o edge or ientations. Right: For α = π / 2 , the diamond graph is Z 2 and the lattice L ( π / 2 ) is the rotation of √ 2 Z 2 b y π 4 . In this case, the model is homogenous, as p v = p h = √ q / ( 1 + √ q ) . For a ﬁnite subgraph G = ( V , E ) of an isoradial g raph G = ( V , E ) , we deﬁne its verte x boundary ∂ G as the set of v er tices in V incident to a v er te x in V \ V . A boundar y condition ξ on G is then given b y a par tition of the set ∂ G . W e sa y that tw o v er tices of G are wir ed if they belong to the same element of the par tition ξ . When no boundar y v er tices are wired tog ether , w e obtain fr ee boundar y conditions, whic h are denoted by 0. Deﬁnition 2.1. For a ﬁnite subgraph G = ( V , E ) of an isoradial g raph G , q ≥ 1 , and a boundary condition ξ , the r andom-clust er measur e on G with cluster w eight q and boundary conditions ξ is the measure ϕ ξ G,q on { 0 , 1 } E giv en by ϕ ξ G,q [ ω ] = 1 Z ξ G,q q k ( ω ξ ) Y e ∈ E p ω e e ( 1 − p e ) 1 − ω e , (2.1) where the p e are the isoradial weights deﬁned abov e, k ( ω ξ ) is the number of connected components in the graph ω ξ which is obtained from ω by identifying v ertices whic h are wired in ξ , and Z ξ G,q is a nor malising constant called the par tition function. W e will mos tly consider the random-cluster model on inﬁnite isoradial graphs G with free boundary conditions obtained by taking the limit of the measures with free boundar y conditions on larger and larg er ﬁnite g raphs G tending to G . W e wr ite ϕ 0 G ,q f or the measure in inﬁnite v olume that is obtained this wa y . Notice that the measures ϕ 0 L ( α ) ,q deﬁned in the introduction are indeed those obtained b y the isoradial w eights abo ve. Let us ﬁnally mention that the basic proper ties of the random-cluster model e xtend readily to this setting and w e refer to [ Gr i06 ] f or proofs in the homog enous setting. The f ollo wing standard proper ties will be used repeatedly without mention. Monotonic properties. F ix a ﬁnite subgraph G = ( V , E ) of an isoradial graph G , q ≥ 1 and a boundary condition ξ . W e say that an ev ent A is incr easing if for any ω ≤ ω ′ 2 , ω ∈ A implies ω ′ ∈ A . The FK G ineq uality s tates that f or an y increasing ev ents A and B , ϕ ξ G,q [ A ∩ B ] ≥ ϕ ξ G,q [ A ] ϕ ξ G,q [ B ] . (FK G) 2 Here, ω ≤ ω ′ ref ers the par tial ordering on { 0 , 1 } E giv en by ω ≤ ω ′ if ω e ≤ ω ′ e f or ev ery edge e ∈ E . Preliminaries 8 For two boundary conditions ξ and ξ ′ such that ξ ≤ ξ ′ , meaning that an y pair of vertices wired in ξ is also wired in ξ ′ , the random-cluster model with boundary conditions ξ is dominated b y the one with boundar y conditions ξ ′ , i.e., f or an y increasing ev ent A , ϕ ξ ′ G,q [ A ] ≥ ϕ ξ G,q [ A ] . (CBC) The latter inequality will be ref er red to as comparison betw een boundar y conditions . Spatial Mark o v property . For G as abov e, a subgraph H of G , q ≥ 1 and a boundar y condition ξ on G , we ha v e ϕ ξ G,q [ ω on H | ω on G \ H ] = ϕ ζ H,q [ ω on H ] , (SMP) where ζ is the boundary condition induced by ω ξ on G \ H — that is, vertices of ∂ H are wired tog ether in ζ if the y are connected in ω ξ in G \ H . A direct consequence of ( SMP ) is the ﬁnite energy property , meaning that the probability of an edg e being open is bounded aw ay from 0 and 1, unif or ml y in the state of all other edges. That is, f or any edge e , there e xists a cons tant c FE > 0 (only depending on the angle θ e ) such that ϕ ξ G,q [ ω e = 1 | ω on G \ { e } ] ∈ [ c FE , 1 − c FE ] . (FE) Moreo v er, when consider ing lattices L ( α ) , c FE is unif or ml y positiv e on compacts of ( 0 , π ) . The track -ex chang e operator . Fix q ≥ 1 and consider a ﬁnite sequence of angles α 1 , . . . , α n and let S denote the strip of an isoradial rectangular lattice with horizontal tracks t 1 , . . . , t n with transv erse angles α 1 , . . . , α n and v er tical trac ks of transv erse angles 0 . Fix 1 < i ≤ n and let S ′ denote the s tr ip obtained from the same sequence of angles, but with α i and α i − 1 e x changed. The boundary v er tices of S and S ′ are those below t 1 and abo ve t n ; they are the same in the tw o str ips. Let ξ be a boundar y condition f or these tw o strips. The measures ϕ ξ S ,q and ϕ ξ S ′ ,q ma y be deﬁned as the unique limits of measures on S ∩ Λ N as N → ∞ . Their uniqueness is an immediate consequence of the ﬁnite energy proper ty ( FE ). Proposition 2.2. There exists a random map T i fr om percolation conﬁgurations on S to per colation conﬁgur ations on S ′ suc h that • if ω ∼ ϕ ξ S ,q , then T i ( ω ) ∼ ϕ ξ S ′ ,q ; • ω and T i ( ω ) agr ee on all edg es outside of t i and t i − 1 ; • on edg es of t i and t i − 1 , T i ( ω ) only depends on the values of ω on edg es of t i and t i − 1 (and potentially on an additional source of randomness, independent of the res t of the conﬁgur ation ω ); • v ertices outside of t i ∩ t i − 1 ar e connected in the same way in ω and T i ( ω ) ; • the modiﬁcation is local: ther e exists c > 0 such that, f or ω ∼ ϕ ξ S ,q , the state of T i ( ω ) on any edg e e is determined 3 by ω r estrict ed to Λ r ( e ) with probability at leas t 1 − e − cr . The map T i will be called the tr ac k-exc hang e operator . The precise constr uction of the coupling between ω and T i ( ω ) is not essential, nor unique. One such coupling may be obtained b y compositions of the star -triangle transformation; w e ref er the reader to [ DCKK + 20 ] f or details. W e w ould merel y like to remark that Proposition 2.2 is an indication of the e xact integ rability of the model deﬁned by the isoradial weights. W e will also use T i as a transf or mation of lattices, and write S ′ = T i ( S ) . 3 The transf or mation may use extra randomness, which ma y be encoded by i.i.d. uniform variables U e on [ 0 , 1 ] f or e ∈ t i − 1 ∪ t i . The precise statement is that ( T i ( ω )) e ma y be determined with probability e xponentially close to 1 from the know ledge of ω and the variables U . on Λ r ( e ) . Preliminaries 9 2.3 Isoradial random-cluster model with q = 4 R ecall from ( 1.1 ) that the random-cluster model on isoradial rectangular lattices with q > 4 shares the f eatures of the model on Z 2 , in par ticular its free inﬁnite-v olume measure exhibits exponential deca y of connection probabilities. Similarl y , for q = 4 , the random-cluster measure on isoradial rectangular lattices behav es like the cr itical measure on Z 2 . W e re view its main characteristics here. While these proper ties hold across the entire regime q ∈ [ 1 , 4 ] , our primar y f ocus will be on the case q = 4 and they will be stated as such. This speciﬁc choice is motiv ated by our ultimate goal to transf er some estimates from q = 4 to the regime q > 4 with q suﬃcientl y close to 4 . In the f ollo wing, let L be a lattice of the type L ( α ) , where α ∈ ( 0 , π ) Z is a sequence containing at mos t tw o v alues. All constants belo w will be unif or m in the v alues of α on compacts of ( 0 , π ) . It was prov ed in [ DCLM18 ] that, f or q = 4 , there e xists a unique inﬁnite-volume measure on L , denoted by ϕ L , 4 . Duality . Recall that if G is an isoradial g raph, then so is its dual G ∗ . T o each conﬁguration ω on G , we associate a dual conﬁguration ω ∗ on G ∗ deﬁned by taking ω ∗ e ∗ = 1 − ω e f or all edges e of G , where e ∗ is the unique edg e of G ∗ intersecting e . The uniqueness of the inﬁnite-volume measure on L implies that the dual of the isoradial measure on L is the isoradial measure on the dual lattice L ∗ , i.e., if ω ∼ ϕ L , 4 , then ω ∗ ∼ ϕ L ∗ , 4 . RSW pr operty . Arguabl y the most useful ing redient in the study of critical planar percolation models is the Russo–Seymour –W elsh (RSW) estimate. It states that the probabilities of crossing rectangles of a given aspect ratio but arbitrar y scale are unif or ml y bounded aw a y from 0 and 1 . Moreo v er, these bounds are uniform, ev en when boundar y conditions are imposed at a macroscopic distance from the rectangle. More precisely , for ρ, ε > 0 , there e xists c = c ( ρ, ε ) > 0 suc h that f or an y ev ent A depending on the edges at distance at least εn from the rectangle [ 0 , ρn ] × [ 0 , n ] , c ≤ ϕ L , 4  C ([ 0 , ρn ] × [ 0 , n ]) | A  ≤ 1 − c, (RSW) where C ([ 0 , ρn ] × [ 0 , n ]) denotes the ev ent that there e xists a path of open edges in [ 0 , ρn ] × [ 0 , n ] from { 0 } × [ 0 , n ] to { ρn } × [ 0 , n ] . W e refer to [ DCLM18 , Thm. 1.1] f or a full proof. W e no w discuss some consequences of ( RSW ). Mixing. One consequence we will use repeatedly is the so-called mixing property . For ev er y ε > 0 , there e xist c mix , C mix ∈ ( 0 , ∞ ) such that for e v ery r ≤ R/ 2 , ev er y e v ent A depending on edg es in Λ r , and e v ery e v ent B depending on edg es outside Λ R , we ha v e that   ϕ L , 4 [ A ∩ B ] − ϕ L , 4 [ A ] ϕ L , 4 [ B ]   ≤ C mix  r R  c mix ϕ L , 4 [ A ] ϕ L , 4 [ B ] . (Mix) This proper ty can be der iv ed from ( RSW ) using the same arguments as in the homog eneous model, see e.g. [ DCM22 , Cor . 2.10]. Arm e v ents and ﬂo w er domains. Fix some angle θ ∈ [ 0 , 2 π ) and deﬁne the f ollo wing arm e v ents. F or r ≤ R and z ∈ L , let A hp ( θ ) 3 ( z ; r, R ) be the ev ent that there exis t three non-intersecting paths γ 1 , γ 2 , γ 3 in the annulus Λ R ( z ) \ Λ r ( z ) , betw een ∂ Λ r ( z ) and ∂ Λ R ( z ) , all contained in the half-space ⟨· , e θ ⟩ ≤ ⟨ z , e θ ⟩ , ar rang ed in clockwise order and such that γ 1 , γ 3 ∈ ω ∗ and γ 2 ∈ ω . W e call this the three-ar m e v ent in the half-plane orthogonal to e θ and call the paths γ 1 , γ 2 , γ 3 arms. W e will now br ieﬂy introduce the notion of ﬂow er domains . These domains are par ticular l y suitable f or studying ar m e v ents. W e ref er the reader to [ DCM22 ] for a more comprehensiv e treatment. Preliminaries 10 Figure 2: Lef t : the ﬂo wer domain from Λ 2 R to Λ R is obtained b y e xploring all pr imal–dual interfaces starting on ∂ Λ 2 R until the y e xit Λ 2 R or reach Λ R . The explored region is gre y and the ﬂo w er domain is the white domain; its boundar y is f or med of one pr imal and one dual petal. Right : a good ﬂo wer domain with the unique pr imal petal contained in the green region. Giv en R ≥ 1 and a conﬁguration ω , let E be the union of Λ c 2 R and all the pr imal/dual inter f aces starting on ∂ Λ 2 R e xplored inw ards until the y either e xit Λ 2 R or enter Λ R . Let F be the connected component of the or igin in E c . The boundary of F is f ormed either of a pr imal or dual circuit, or of an ev en number of alternating primal and dual arcs called petals . W e call F the ﬂo wer domain re v ealed from Λ 2 R to Λ R . See Figure 2 for an e xample. For η > 0 , the ﬂo wer domain F is said to be η - w ell-separated if the endpoints of its petals are at a distance of at least η R from each other . In par ticular , a ﬂo w er domain with a single petal is η -w ell-separated b y default. It was prov ed in [ DCM22 , Lemma 3.2] that the ﬂo w er domain from Λ 2 R to Λ R is well-separated with positiv e probability under ϕ ξ L ∩ Λ 2 R f or an y boundar y condition ξ . This can be e xtended to accommodate measures conditioned on ar m ev ents [ GMM26 ]. Let us giv e a more precise statement in the case of the three-ar m e v ent in the half plane. W e sa y that a ﬂo w er domain is good if it has e xactly two petals (one pr imal and one dual), is 1 4 -w ell- separated, and its pr imal petal is contained in the cone with ape x 0 , bisector − e θ , and aper ture π / 2 (see Figure 2 ). Lemma 2.3 ([ GMM26 ]) . Ther e exists c > 0 suc h that f or every 4 r ≤ R , any z ∈ L , and any conﬁgur ation ω 0 on Λ 4 r ( z ) c that allow s f or the occurrence of A hp ( θ ) 3 ( z ; 1 , R ) , w e have ϕ L , 4  F is good | ω = ω 0 on Λ 4 r ( z ) c and A hp ( θ ) 3 ( z ; 1 , R )  ≥ c, wher e F is the ﬂow er domain r evealed from Λ 2 r ( z ) to Λ r ( z ) . Arm e xponents. Another classical consequence of ( RSW ) is that the probabilities of arm ev ents ma y be bounded by pol ynomials with strictl y positive e xponents. The f ollo wing ar m e xponent bounds will be useful in our arguments. W e call L = L ( α ) periodic if α is a periodic sequence. Proposition 2.4. There exist constants c, C > 0 independent of θ and L suc h that, for all 1 ≤ r < R and z ∈ L , 1 C  r R  2 ≤ ϕ L , 4  A hp ( θ ) 3 ( z ; r, R )  ≤ C  r R  2 , if L is periodic, and (2.2) ϕ L , 4  A hp ( θ ) 3 ( z ; r, R )  ≤ C  r R  1+ c f or all L as abo v e. (2.3) In the ﬁrs t case, C may depend on L , but is unif orm for sequences α of period 2 with v alues in compacts of ( 0 , π ) . Preliminaries 11 Mor eov er , the arm ev ent pr obabilities satisfy a quasi-multiplicativity pr oper ty , i.e., f or any r ≤ ρ ≤ R , w e have 1 C ≤ ϕ L , 4  A hp ( θ ) 3 ( z ; r, R )  ϕ L , 4  A hp ( θ ) 3 ( z ; r, ρ )  ϕ L , 4  A hp ( θ ) 3 ( z ; ρ, R )  ≤ C. (2.4) W e refer to [ DCKK + 20 , Prop. 3.4] f or a proof of ( 2.2 ), ( 2.3 ) and merely note that the y are conseq uences of ( RSW ) and certain symmetries of the lattice. Note that ( 2.2 ) requires the in variance of L under tw o independent translations, without which, onl y the w eaker bound ( 2.3 ) ma y be obtained 4 . The quasi-multiplicativity ( 2.4 ) is a standard consequence of ( RSW ) and the arm-separation property; see [ CDCH16 , GMM26 ] f or details. 2.4 Incipient inﬁnite cluster in the half-plane In this section, we introduce the incipient inﬁnite cluster (hencef or th IIC) measures with three ar ms in the half-plane. Fix tw o angles α , β ∈ ( 0 , π ) and wr ite L mix f or the lattice of type L ( α ) with α ∈ ( 0 , π ) Z alternating betw een α and β , i.e., α = ( α i ) i ∈ Z with α i = β f or i e v en and α i = α f or i odd. Note that the ( RS W ) proper ty applies to L mix . R ecall that the hor izontal tracks of L mix are denoted by ( t k ) k ∈ Z and the v er tical ones by ( s k ) k ∈ Z . W e assume the v er te x between t 0 and t 1 and between s 0 and s 1 to be a pr imal one and consider it to be the or igin of R 2 . For integers i, j , deﬁne the cell ( i, j ) as the set of primal and dual v er tices l ying between the vertical tracks s 2 i − 1 and s 2 i +1 and the horizontal trac ks t 2 j − 1 and t 2 j +1 . Note that the cells are centred around rhombi of angle β . T o eac h cell, we associate its lo wer -left lattice point, which, by our conv ention, is a pr imal v er te x — see Figure 3 . Fix θ ∈ [ 0 , 2 π ) . F or a ﬁnite cluster C of a conﬁguration ω on L mix , let Ext θ ( C ) be the lattice point that maximises the scalar product with e θ and whose associated cell intersects C . If multiple such maximisers e xist, choose the one with the larg est v er tical coordinate. W e call Ext θ ( C ) the extr emum in dir ection e θ of C — note that it is possible f or Ext θ ( C ) to not be par t of C . Finall y , let E θ ( C ) = ⟨ Ext θ ( C ) , e θ ⟩ be the corresponding e xtremal coordinate of C in direction e θ . See Figure 3 . W r ite x n f or the pr imal verte x of L mix closest to − ne θ and let C x n denote the cluster containing x n . W e deﬁne the IIC with three ar ms in the half-plane as the limiting measure ϕ IIC ,θ L mix , 4 [ · ] : = lim n →∞ ϕ L mix , 4  · | Ext θ ( C x n ) = 0  , where the limit uses the weak con ver gence with respect to the product topology . In other words, the probability of an y local ev ent con ver ges. Under ϕ IIC ,θ L mix , 4 , there exis ts an inﬁnite clus ter which is the limit of the clusters C x n . W e call this the incipient inﬁnite cluster; its e xtremal coordinate in the direction e θ is 0 . The measure ϕ IIC ,θ L mix , 4 describes the local en vironment around extrema of lar ge clusters. Indeed, it may be sho wn that the neighbourhood of an e xtremum of a typical larg e cluster is distr ibuted according to ϕ IIC ,θ L mix , 4 , e v en when the cluster is conditioned on (reasonable) larg e scale f eatures. The exis tence of the limit is a manifes tation of the f ollowing mixing proper ty , which in tur n is a consequence of ( RSW ). Proposition 2.5. There exist constants C , c IIC > 0 suc h that the f ollowing holds f or all r ≤ R/ 2 . F or any conﬁgur ations ω 0 on Λ c R , any x ∈ Λ c R and any ev ent A depending only on edg es in Λ r ,    ϕ IIC ,θ L mix , 4 [ A ] − ϕ L mix , 4  A | ω = ω 0 on Λ c R , Ext θ ( C x ) = 0     ≤ C  r R  c IIC , 4 Using the universality result of [ DCKK + 20 ], ( 2.2 ) may be e xtended to all L as abov e, but this requires additional w ork and is not necessary . Preliminaries 12 α β e θ Ext θ ( C ) Figure 3: A cluster C with its e xtremum Ext θ ( C ) in direction e θ . In this ex ample, Ext θ ( C ) does not belong to C itself. The cell associated to the e xtremum is shaded in light blue and the line ⟨· , e θ ⟩ = E θ ( C ) is dra wn in black. as long as the conditioning is non-deg enerat e 5 . A model-speciﬁc construction of the IIC measure can be f ound in [ DCKK + 20 , Oul22 ]; f or its associated polynomial mixing rate, w e refer to [ GPS13 , Prop. 3.1]. T rack -ex chang es and drift. A cr ucial step in the proof of the asymptotic rotational inv ariance f or 1 ≤ q ≤ 4 of [ DCKK + 20 ] is to analy se ho w the extremum of an IIC cluster is aﬀected by track -e x changes. Let ω IIC be sampled from ϕ IIC ,θ L mix , 4 and wr ite C IIC f or the incipient inﬁnite cluster of ω IIC . Deﬁne S ev en as the transformation obtained b y simultaneously appl ying the track -e x chang es T 2 k f or all k ∈ Z , and S odd as the transf or mation obtained by applying the track ex chang es T 2 k − 1 f or k ∈ Z . Since the trac k -ex chang es appearing in the transf or mations act on disjoint tracks and since the inﬁnite-v olume measures on L mix and its transforms are unique, one ma y per f or m these ex chang es simultaneousl y on ω IIC . When S ev en is applied to L mix , each track of angle α is ex chang ed with the track of angle β directl y abov e it. In particular, the v er tices at the bottom of the α -tracks remain ﬁxed. Hence, any cluster containing one of these v er tices admits a well-deﬁned image under S ev en . Analogousl y , when S odd is applied to S ev en ( L mix ) , the tracks of angle α are again e x chang ed with the tracks of angle β directly abo ve them, and clusters containing at least tw o edges thus ha v e a natural imag e after the transformation. Note that ( S odd ◦ S ev en )( L mix ) is a translate of L mix with the same cell structure. Due to larg e clusters being preserved, C IIC has a cor responding cluster in ( S odd ◦ S ev en )( ω IIC ) , which w e denote b y ˜ C IIC . W e then deﬁne the IIC incr ement b y ∆ IIC E θ = E θ ( ˜ C IIC ) − E θ ( C IIC ) . Note that S ev en ( L mix ) is also a translate of L mix . As such, one might expect the application of S ev en and the subsequent application of S odd to ha ve identical eﬀects. This is not the case: S ev en ( L mix ) and L mix are translates of each other , but their par tition into cells diﬀer , which aﬀects the deﬁnition of Ext θ . This is the reason f or appl ying both transf ormations bef ore considering the increment. 5 The conditioning on { ω = ω 0 on Λ c R } is alwa ys deg enerate, but should be understood as imposing cer tain boundar y conditions on the restriction of the measure to Λ R . Here, by non-deg enerate, we mean that there e xists at least one conﬁguration in Λ R such that { Ext θ ( C x ) = 0 } is realised. Half-plane measures 13 The expected increment E [ ∆ IIC E θ ] — where E ref ers to the e xpectation tak en with respect to the coupling betw een ω IIC and ( S odd ◦ S ev en )( ω IIC ) — captures the drif t of the extremum of a larg e cluster under the transf or mation ( S odd ◦ S ev en ) . It was pro ved in [ DCKK + 20 ] that this dr ift v anishes, i.e., f or all θ ∈ [ 0 , 2 π ) , E [ ∆ IIC E θ ] = 0 . (2.5) This fact pla y s a central role in establishing univ ersality among isoradial rectangular lattices f or 1 ≤ q ≤ 4 . 3 Half-plane measures The strategy of pro ving universality by successiv ely transforming one lattice into another via a sequence of track -e x chang es is faced with additional diﬃculties f or q > 4 . In this regime, the boundary conditions inﬂuence the model at inﬁnite distance. Indeed, suppose a lattice L is transf or med into L ′ b y a sequence of track -e x chang es, which w e denote by S . The transf or mation aﬀects the boundar y conditions in an uncontrollable manner , which, in this case, are cr ucial. In particular , it is unclear whether the push-f orward of ϕ 0 L ,q b y S is ϕ 0 L ′ ,q . This problem does not appear in the regime 1 ≤ q ≤ 4 due to the uniqueness of the inﬁnite-volume measure. T o circum v ent this issue, w e will w ork with half-plane measures with free boundar y conditions. From here onw ards, w e consider lattices L ( α ) where α = ( α i ) i ≥ 1 is a half-inﬁnite sequence of angles; as in the previous section, these sequences will contain at mos t tw o distinct v alues. These lattices co v er the upper half-plane R × R ≥ 0 . F or suc h a lattice, wr ite ∂ L ( α ) f or the set of its v ertices l ying on the hor izontal axis R × { 0 } . 3.1 U niqueness of the half-plane free measure W e call a half-inﬁnite sequence of angles α = ( α i ) i ≥ 1 periodic if there exis ts k ∈ N such that α k + i = α i f or all i ≥ 1 . Proposition 3.1. F ix q ≥ 1 and a half-inﬁnite periodic sequence of ang les α . F or any sequence ξ n of boundar y conditions on L ( α ) ∩ Λ n with the pr operty that ξ n is free on ∂ L ( α ) , the measur es ϕ ξ n L ( α ) ∩ Λ n ,q conv erg e as n → ∞ to a measure ϕ 0 L ( α ) ,q , which w e call the free half-plane measure . Pr oof. W r ite ϕ 1 / 0 L ( α ) ∩ Λ n ,q f or the isoradial random-cluster measure on L ( α ) ∩ Λ n with free boundar y conditions on [ − n, n ] × { 0 } and wired boundar y conditions f or the rest of the boundary . Let ϕ 1 / 0 L ( α ) ,q be the half-plane random-cluster measure which is the weak (decreasing) limit of ϕ 1 / 0 L ( α ) ∩ Λ n ,q f or n → ∞ . Similarl y , wr ite ϕ 0 L ( α ) ∩ Λ n ,q f or the random-cluster measure on L ( α ) ∩ Λ n with free boundary conditions ev er ywhere and denote its weak (increasing) limit b y ϕ 0 L ( α ) ,q . By ( CBC ), it suﬃces to sho w that ϕ 1 / 0 L ( α ) ,q = ϕ 0 L ( α ) ,q , which in turn f ollo w s from the fact that, under ϕ 1 / 0 L ( α ) ,q , there exis ts a.s. no inﬁnite cluster . The absence of an inﬁnite cluster in ϕ 1 / 0 L ( α ) ,q f ollow s from the arguments of [ GH00 , GM23 ], with the periodicity of the sequence of angles playing a par ticular role. W e giv e a brief sketc h f or completeness. Consider the increasing limit ϕ of the down ward translates of ϕ 1 / 0 L ( α ) ,q . Then ϕ is a percolation measure on a periodic isoradial lattice L . Fur thermore, ϕ itself is in variant under the translations Half-plane measures 14 which map L to itself. The dual of ϕ produces conﬁgurations on the horizontal translate of L by 1 . Further more, due to the order in which the diﬀerent par ts of the boundary were taken to inﬁnity , ϕ is stochas tically dominated b y its dual (translated by ( − 1 , 0 ) ). Due to the planar ity and periodicity of L , the non-coexis tence theorem of [ She05 , DCR T19 ] applies, and we conclude that, under ϕ , either the primal or the dual conﬁguration contains no inﬁnite cluster . By the domination abo v e, the pr imal conﬁguration contains a.s. no inﬁnite cluster , which e xtends to ϕ 1 / 0 L ( α ) ,q b y stochas tic domination. Corollary 3.2 (T rack -e x change f or half-plane measures) . Fix q ≥ 1 . Let α be a half-inﬁnite periodic sequence of ang les and A ⊂ N be a periodic set containing no consecutive integ ers. Writ e α ′ f or the half-inﬁnite periodic sequence obtained from α by exchanging α i − 1 and α i f or eac h i ∈ A . If ω is sampled according to ϕ 0 L ( α ) ,q and ω ′ is the conﬁguration obtained by applying each ( T i ) i ∈ A to ω , then ω ′ is distributed accor ding to ϕ 0 L ( α ′ ) ,q . Pr oof. Let S K be the str ip of L ( α ) f or med of the hor izontal tracks t 1 , . . . , t K and similar ly let S ′ K be the strip of L ( α ′ ) containing the tracks t 1 , . . . , t K . W r ite ϕ 0 S K ,q and ϕ 0 S ′ K ,q f or the measures on S K and S ′ K , respectivel y , with free boundar y conditions. W r ite S f or the composition of all transf or mations ( T i ) i ∈ A . Assume now that K is such that K + 1 / ∈ A . Then, we ma y apply all trac k e x chang es ( T i ) i ∈ A ; i ≤ K to S K to obtain S ′ K . For simplicity , w e call the composition of these transf or mations also S , since the transf or mations ( T i ) i ∈ A ; i>K do not appl y to S K and hence are considered tr ivial in this setting. If ω is sampled according to ϕ 0 S K ,q , then, by Proposition 2.2 , the law of S ( ω ) is given by ϕ 0 S ′ K ,q . Proposition 3.1 then implies that ϕ 0 L ( α ) ,q = lim K →∞ ϕ 0 S K ,q and ϕ 0 L ( α ′ ) ,q = lim K →∞ ϕ 0 S ′ K ,q . The conclusion f ollo ws. Corollary 3.3. Fix q ≥ 1 and let α and β be two half-inﬁnite periodic sequences of ang les with α i = β i f or all i ≤ n . Then, for any event H depending only on t he edg es on the trac ks t 1 , . . . , t n , ϕ 0 L ( α ) ,q [ H ] = ϕ 0 L ( β ) ,q [ H ] . Pr oof. Let S ( θ 1 , . . . , θ K ) denote the strip with K horizontal tracks of transv erse angles θ 1 , . . . , θ K and v er tical trac ks of transverse angle 0 . Let ϕ ξ S ( θ 1 ,...,θ K ) ,q denote the measure on this strip with boundary conditions ξ . Fix H as in the statement and ε > 0 . Then, b y Proposition 3.1 applied to both L ( α ) and L ( β ) , there exis ts K ≥ n such that   ϕ 0 L ( α ) ,q [ H ] − ϕ ξ S ( α 1 ,...,α K ) ,q [ H ]   ≤ ε and   ϕ 0 L ( β ) ,q [ H ] − ϕ ξ S ( β 1 ,...,β K ) ,q [ H ]   ≤ ε (3.1) f or an y boundar y conditions ξ that are free on the bottom of the strip. Consider now the strips S : = S ( α 1 , . . . , α K , β n +1 , . . . , β K ) and S ′ : = S ( β 1 , . . . , β K , α n +1 , . . . , α K ) . Due to ( 3.1 ) and ( SMP ),   ϕ 0 L ( α ) ,q [ H ] − ϕ 0 S ,q [ H ]   ≤ ε and   ϕ 0 L ( β ) ,q [ H ] − ϕ 0 S ′ ,q [ H ]   ≤ ε. (3.2) Half-plane measures 15 Since α i = β i f or all i ≤ n , there e xists a sequence of trac k e xc hang es that act only abov e the track t n and turn S into S ′ . The conﬁguration on t 1 , . . . , t n does not chang e when applying this sequence of transformations, so w e ﬁnd ϕ 0 S ,q [ H ] = ϕ 0 S ′ ,q [ H ] . Combining this with ( 3.2 ), w e conclude that | ϕ 0 L ( α ) ,q [ H ] − ϕ 0 L ( β ) ,q [ H ] | ≤ 2 ε . Finall y , since ε > 0 is arbitrar y , w e ﬁnd ϕ 0 L ( α ) ,q [ H ] = ϕ 0 L ( β ) ,q [ H ] . 3.2 Rate of deca y of connection probabilities in the half-plane For α ∈ ( 0 , π ) , wr ite L + ( α ) for the half-plane isoradial lattice with constant transverse angle α f or the tracks ( t n ) n ≥ 1 . Since we will ex clusiv el y be working with half-plane measures, we no w introduce the half-plane analogue of ζ . F or θ ∈ [ 0 , 2 π ) , set ζ hp α,q ( θ ) =  lim n →∞ − 1 n log ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n   − 1 . The e xistence of the limit, similarl y to that of ζ α,q ( θ ) , is prov ed using the super -additivity of the sequence max x ∈H θ ≥ n log ϕ 0 L + ( α ) ,q [ 0 ← → x ] . Finall y , observ e that, due to ( 1.1 ) and the ﬁnite energy proper ty ( FE ), 0 < ζ hp α,q ( θ ) < ∞ f or an y q > 4 , α ∈ ( 0 , π ) , and θ  = 3 π / 2 . W e call a direction θ upper -half-plane aiming for L ( α ) if there e xists an inﬁnite number of integ ers n ≥ 1 such that the point x ∈ H θ ≥ n maximising ϕ 0 L ( α ) ,q [ 0 ← → x ] lies in the upper half- plane. The upper -half-plane aiming directions ma y be sho wn to be those dual — in the sense of ( 1.2 ) — to θ ∈ [ 0 , π ] . F or L ( π / 2 ) , all θ ∈ [ 0 , π ] are upper -half-plane aiming due to the inv ar iance of the model with respect to vertical reﬂections. This is not necessar il y the case f or L ( α ) , as this par ticular symmetry ma y be lost. W e note tw o fundamental facts that are due to the inv ar iance of the models ϕ 0 L ( α ) ,q under rotation by π . F or an y θ ∈ [ 0 , π ) and α ∈ ( 0 , π ) , ζ α,q ( θ ) = ζ α,q ( π + θ ) and at least one of θ and π + θ is upper -half-plane aiming . Finall y , w e state the essential link betw een ζ hp α,q ( θ ) and ζ α,q ( θ ) . Proposition 3.4. F ix α ∈ ( 0 , π ) , q > 4 , and θ ∈ [ 0 , 2 π ) . Then ζ α,q ( θ ) ≥ ζ hp α,q ( θ ) , (3.3) with equality if θ is upper -half-plane aiming f or L ( α ) . Pr oof. By ( CBC ) and inclusion of ev ents, ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  ≤ ϕ 0 L ( α ) ,q  0 ← → H θ ≥ n  . T aking the logarithm, dividing b y − n and taking the limit n → ∞ , we obtain ( 3.3 ). Assume no w that θ is upper -half-plane aiming f or L ( α ) . Fix ε > 0 . Then, f or r lar g er than some threshold, ϕ 0 L ( α ) ,q  0 ← → H θ ≥ r  ≥ exp  − r ζ α,q ( θ ) − ε  . Univers ality : proof of Theorem 1.3 16 Consider the point x ∈ H θ ≥ r that maximises ϕ 0 L ( α ) ,q [ 0 ← → x ] . In light of ( 1.1 ), there exis ts a constant C > 0 independent of r such that ϕ 0 L ( α ) ,q [ 0 ← → x ] ≥ 1 C r exp  − r ζ α,q ( θ ) − ε  . By fur ther increasing r and using the fact that θ is upper -half-plane aiming, w e ma y consider x to be in the upper -half-plane and such that ϕ 0 L ( α ) ,q [ 0 ← → x ] ≥ exp  − r ζ α,q ( θ ) − 2 ε  . Finall y , w e ma y ﬁnd R ≥ 1 large enough such that ϕ 0 Λ R ∩ L ( α ) ,q [ 0 ← → x ] ≥ exp  − r ζ α,q ( θ ) − 3 ε  . (3.4) No w ﬁx z ∈ L + ( α ) at a distance at least R from the hor izontal line. Then, f or an y n suﬃciently larg e, b y ( FK G ) and ( CBC ) ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  ≥ ϕ 0 L + ( α ) ,q [ 0 ← → z ] ϕ 0 L + ( α ) ,q [ z ← → z + ℓx ] ≥ ϕ 0 L + ( α ) ,q [ 0 ← → z ] ϕ 0 Λ R ∩ L ( α ) ,q [ 0 ← → x ] ℓ when ℓ is a suﬃcientl y larg e integ er such that z + ℓx ∈ H θ ≥ n . Due to the choice of x , one ma y choose ℓ ≤ n + C 0 r f or some constant C 0 that depends on θ , r , x , and z , but not on n . Using ( 3.4 ), w e conclude the e xistence of a constant c 1 > 0 independent of n suc h that ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  ≥ c 1 exp  − n ζ α,q ( θ ) − 3 ε  . T aking the logarithm, dividing b y − n and taking n to inﬁnity , w e conclude that ζ hp α,q ( θ ) ≥ ζ α,q ( θ ) − 3 ε. Since ε > 0 was chosen arbitrar il y and in light of ( 3.3 ), we conclude that the tw o quantities are equal. 4 U niv ersality : proof of Theorem 1.3 W e brieﬂy outline the idea behind pro ving Theorem 1.3 . Our aim will be to compare ζ hp α,q ( θ ) f or diﬀerent values of α and θ  = 3 π / 2 ﬁx ed. More precisely , we will show the follo wing. Proposition 4.1. F or all ε > 0 , ther e exists q 0 > 4 such that f or q ∈ ( 4 , q 0 ] , all α , β ∈ ( ε, π − ε ) and any θ ∈ [ 0 , 2 π ) with θ  = 3 π / 2 ,     ζ hp α,q ( θ ) ζ hp β ,q ( θ ) − 1     < ε. W e will see in Section 4.3 how to pass from the half-plane connection rates to the full-plane connection rates, and hence pro v e Theorem 1.3 . The rest of the section is dedicated to pro ving Proposition 4.1 . The aim is to show that connection probabilities in ω ∼ ϕ 0 L + ( α ) ,q and ω ′ ∼ ϕ 0 L + ( β ) ,q are close to each other when q > 4 is taken suﬃciently close to 4 . T o achie v e this, w e couple these tw o conﬁgurations and cons truct a sequence of inter mediate conﬁgurations, each obtained from the pre vious one b y a sequence of track -ex chang es. In this coupling, w e keep trac k of the e xtremal coordinate E θ ( C ) of the clus ter C of the or igin in direction e θ . By taking q > 4 close enough to 4 , we argue that the eﬀect of track -e x chang es on E θ ( C ) are almos t identical in la w to ∆ IIC E θ , and theref ore ha v e v anishingly small e xpectation. This will allow us to compare the probabilities in the initial and ﬁnal conﬁgurations ω and ω ′ to ha v e E θ ( C ) ≥ n f or larg e values of n . Ultimatel y , our goal is to sho w that the e xponential rate of deca y of these tw o quantities is almost equal. Univers ality : proof of Theorem 1.3 17 t N +1 t N Figure 4: The track -e x chang es per f or med on a single per iod. From left to r ight: The initial lattice L 0 with trac ks of angle α = π / 2 at the bottom and β < π / 2 at the top. The ﬁrst tw o tracks to be e x changed are marked with dashed grey lines. Appl ying S t ◦ · · · ◦ S 0 transf or ms L 0 into L t +1 and a mix ed block (colored in red) starts appear ing in the middle. The cells in the mixed block at e v en timesteps are centred around rhombi with angle β . After more transf or mations, a β -block starts f or ming at the bottom and an α -bloc k at the top (both marked in blue). By time 2 N , the β -block and the α -bloc k ha v e been ex chang ed completely . The inter f aces of each lattice are marked with thick red lines. 4.1 The coupling W e will now ﬁx some notation necessar y f or the proof of Proposition 4.1 and descr ibe the coupling in detail. Fix tw o angles α, β ∈ ( 0 , π ) and N ≥ 1 e ven. All constants belo w will be unif or m in α and β in compacts of ( 0 , π ) and in N . W r ite L 0 f or the lattice L + ( α ) , where α is the sequence giv en by α i = ( α f or 2 kN < i ≤ ( 2 k + 1 ) N , k ≥ 0 , β other wise. W e can par tition the lattice into blocks of N tracks of constant angle. By means of the track - e x change operator , w e can deﬁne a sequence of lattices ( L t ) t ≥ 0 where the blocks of angle α are e v entually e xc hanged with those of angle β . R ecall that w e write T i f or the track -e x chang e operator e x changing the trac ks t i and t i − 1 . If t i and t i − 1 ha v e the same transverse angle, we set T i = id . For t ≥ 1 , appl y the follo wing sequence of track e x chang es to L t in order to obtain L t +1 : S t = ( T 3 ◦ T 5 ◦ . . . T 2 N − 1 ◦ T 2 N +3 ◦ T 2 N +5 ◦ . . . if t is odd, T 2 ◦ T 4 ◦ . . . if t is ev en. That is, for t ≥ 1 , set L t = S t − 1 ◦ · · · ◦ S 0 ( L 0 ) . See Figure 4 f or an illus tration; note that since N is ev en, S 0 is tr ivial and L 1 = L 0 . The transf ormations S t f or odd times t contains all track -ex chang es T i with i odd, except f or i − 1 ∈ 2 N Z . This ensures that there is nev er any e x chang e of tracks betw een blocks of 2 N successiv e tracks, and ultimately ensures that all lattices L t are 2 N -periodic. Each per iod is f or med of a mixed bloc k of alter nating tracks of angles α and β sandwiched betw een an α -block and a β -block; the sizes of the blocks depend on t and may be null. Lines separating diﬀerent blocks of the lattice (including the hor izontal axis) will be ref er red to as interfaces . After 2 N transf or mations, the mix ed block disappears and the β -block and α -block ha v e been e x changed completel y . W e ref er ag ain to Figure 4 . Fix q > 4 . W e will associate a seq uence of conﬁgurations ( ω t ) t ≥ 0 to the lattices ( L t ) t ≥ 0 with the property that ω t ∼ ϕ 0 L t ,q f or all t ≥ 0 . This produces a coupling of the measures ( ϕ 0 L t ,q ) . The Univers ality : proof of Theorem 1.3 18 coupling is constructed as f ollo w s. F irst, sample ω 0 according to ϕ 0 L 0 ,q . Then, f or t ≥ 0 , assuming that ω 0 , . . . , ω t are already deﬁned, set ω t +1 = S t ( ω t ) . Corollary 3.2 ensures that ω t indeed has law ϕ 0 L t ,q f or all t ≥ 1 . W e wr ite P f or the probability measure gov er ning the random sequence ( ω t ) t ≥ 0 . Fix an angle θ ∈ [ 0 , 2 π ) \ { 3 π / 2 } . For eac h conﬁguration ω t , wr ite C t f or the cluster of the or igin in ω t and recall that Ext θ ( C t ) (resp. E θ ( C t ) ) denotes the e xtremum (resp. e xtremal coordinate) of C t in direction e θ . Since θ remains ﬁxed throughout, w e will hencef or th omit it from the notation. W e will be interested in the ev olution of ( E ( C t )) t ≥ 0 . In order to k eep track of the dynamics of the process, w e deﬁne, f or t ≥ 0 ev en, the increments ∆ t E = E ( C t +2 ) − E ( C t ) . The f ollo wing es timate, bounding the expected increment, conditionally on the e xtremal coor - dinate being lar ge, is a k e y step in the proof of Proposition 4.1 . Proposition 4.2. F or each t ≥ 0 even, ∆ t E is a.s. bounded by 4. Moreo v er , for any δ, ε > 0 , there exists q 0 > 4 such that f or any q ∈ ( 4 , q 0 ] , n, N suﬃciently larg e and any 1 ≤ t ≤ N ( 1 − δ ) or ( 1 + δ ) N ≤ t ≤ 2 N , it holds that E  ∆ t E | E ( C t ) ∈ [ nε, ( n + 1 ) ε )  ≥ − δ. (4.1) Mor eov er , q 0 may be c hosen independently of θ , n and N (bot h suﬃciently larg e) and unif or mly in α, β in compacts of ( 0 , π ) . Note that we do not f ollo w the precise ev olution of E ( C t ) , but rather a “rounding” of E ( C t ) at an arbitrary precision. Indeed, the e xact v alue of E ( C t ) ma y dictate the e xact position of Ext ( C t ) , which may lead to a degenerate conditioning. See the proof of Lemma 4.5 f or the use of this rounding. W e e xcluded t betw een ( 1 − δ ) N and ( 1 + δ ) N from ( 4.1 ) f or conv enience. Indeed, in these time steps, the mixed bloc k comes close to the horizontal axis, which entails some additional complications. While we believ e ( 4.1 ) to also appl y in this case, we omit it as it is not s trictly necessary f or the rest of the proof. As a conseq uence of Proposition 4.2 , w e establish that the coupling is indeed such that the cor responding point-to-h yper plane connection probabilities in L 0 and L 2 N diﬀer only negligibl y if q is chosen suﬃciently close to 4 and the distance n to the h yper plane is large enough. Corollary 4.3. F or any η > 0 and C ≥ 1 , there exis ts q 0 > 4 such t hat for q ∈ ( 4 , q 0 ] and all n suﬃciently larg e, if w e set N = C n , we hav e P  E ( C 2 N ) ≥ ( 1 − η ) n  ≥ η P  E ( C 0 ) ≥ n  . (4.2) Mor eov er , q 0 may be c hosen independently of θ and unif or mly in α and β in compacts of ( 0 , π ) . Pr oof. Fix η > 0 and C ≥ 1 . With no loss of generality , we may assume η < 1 / 2 C . Choose constants ε, δ > 0 so that 5 δ + ε < η 2 C and η ≤ 5 δ + ε 2 ( 1+5 δ + ε ) . Then choose q 0 > 4 , suﬃcientl y close to 4 so that Proposition 4.2 holds for the chosen v alues of δ, ε . F ix no w n suﬃciently larg e that ( 4.1 ) applies f or all values abov e εm ≥ n/ 2 and set N = C n . Univers ality : proof of Theorem 1.3 19 Deﬁne the ε -rounding e t = ⌊ E ( C t ) /ε ⌋ · ε of E ( C t ) and wr ite ∆ t e = e t +2 − e t f or all t ≥ 0 ev en. Then, Proposition 4.2 states that, for an y 0 ≤ t ≤ 2 N e ven, e xcept if ( 1 − δ ) N ≤ t ≤ ( 1 + δ ) N , E  ∆ t e   e t = εm  ≥ − δ − ε (4.3) f or all q ∈ ( 4 , q 0 ] and εm ≥ n/ 2 . Note that ( 4.3 ) does not provide a low er bound on the e xpected increment ∆ t e given the full past of the process. As our ultimate goal is to study the process under the conditioning e 0 ≥ n , this may produce diﬃculties. T o circumv ent them, w e will modify the process ( e 2 t ) t ≥ 0 to render it Marko v , while maintaining the marginal la ws. This is a standard procedure: deﬁne a process ( ˜ e 2 t ) t ≥ 0 on a potentiall y extended probability space as f ollow s. • Let ˜ e 0 = e 0 . • For an y t ≥ 0 , let ω t +1 / 2 be a conﬁguration with the same law as ω t , sampled independently of the pas t, but such that ⌊ E ( C t +1 / 2 ) /ε ⌋ · ε = ˜ e t . • Deﬁne ω t +2 = S t +1 ◦ S t ( ω t +1 / 2 ) and use it to compute ˜ e t +2 . This resampling procedure renders the process ( ˜ e 2 t ) t ≥ 0 Mark o v . Additionall y , it is easy to check that, f or an y ﬁx ed, ev en t , e t and ˜ e t ha v e the same law . Finall y , ( 4.3 ) also applies to the process ( ˜ e t ) t ≥ 0 . Set τ = inf { t ≥ 0 : ˜ e 2 t < n/ 2 } . By ( 4.3 ), w e conclude that  ˜ e 2 ( t ∧ τ ) + ( δ + ε ) t  0 ≤ 2 t ≤ ( 1 − δ ) N and  ˜ e 2 ( t ∧ τ ) + ( δ + ε ) t  ( 1+ δ ) N ≤ 2 t ≤ 2 N are submar ting ales with bounded increments. Further more, ˜ e ( 1+ δ ) N ∧ 2 τ − ˜ e ( 1 − δ ) N ∧ 2 τ ≤ 4 δ N , due to the deterministic bound on ∆ t ˜ e . Thus, by the optional s topping theorem, E  ˜ e 2 ( N ∧ τ ) − ˜ e 0   ˜ e 0  ≥ − ( 5 δ + ε ) N . Ag ain, by the deter ministic bound on ∆ t ˜ e , w e obtain ˜ e 2 ( N ∧ τ ) − ˜ e 0 ≤ 4 N . Appl ying the Mark o v inequality , w e conclude that P  ˜ e 2 ( N ∧ τ ) − ˜ e 0 > − η n   ˜ e 0 ≥ n  ≥ P  ˜ e 2 ( N ∧ τ ) − ˜ e 0 > − 2 ( 5 δ + ε ) N   ˜ e 0 ≥ n  ≥ 5 δ + ε 2 ( 1+5 δ + ε ) ≥ η . Since ηn < n/ 2 , when the e v ent in the ﬁrst line occurs, ˜ e 2 ( N ∧ τ ) > n/ 2 , and thus τ ≥ N . W e conclude from the abo v e that P  ˜ e 2 N > ( 1 − η ) n   ˜ e 0 ≥ n  ≥ η . (4.4) Finall y , since ˜ e and e hav e the same marginals, we ﬁnd that P  E ( C 2 N ) ≥ ( 1 − η ) n  ≥ P  ˜ e 2 N ≥ ( 1 − η ) n  ≥ η P  ˜ e 0 ≥ n  = η P  E ( C 0 ) ≥ n  . In the las t line, w e assumed f or simplicity that n/ε is integer -v alued. This concludes the proof. W e close by remarking that, by using the independence of the increments, it ma y be pro v ed that P [ ˜ e 2 N > ( 1 − η ) n | ˜ e 0 ≥ n ] → 1 as n → ∞ . F or our goal, the w eaker bound ( 4.4 ) suﬃces. Univers ality : proof of Theorem 1.3 20 4.2 Expected increment: pr oof of Pr oposition 4.2 Fix α , β , N , n , θ and t ≥ 0 e v en. All constant belo w are independent of these choices, with α and β being taken in a compact of ( 0 , π ) and n, N being larg e enough. Also ﬁx δ, ε > 0 . Firs t, w e argue that ∆ t E is deterministically bounded b y 4 . Split the vertices of L t and L t +1 into those placed at the top of tracks t i with i ev en and those placed at the bottom of such tracks — this is a bi-partition of L t . The f or mer type of v er te x is not aﬀected by the transf or mation S t , and thus the vertices of this type in C t are the same as those in C t +1 . F inall y , f or both ω t and ω t +1 , an y v ertex of the second category that is connected to 0 is connected to a v er te x of the ﬁrst category b y an open edge whose length is at most 2 . Repeating the same argument when applying S t +1 to ω t +1 (with the roles of the tw o types of v er tices re v ersed), we conclude that C t +2 must contain a v ertex in a cell neighbouring that of Ext ( C t ) , and vice v ersa. Giv en that the diameter of a cell is bounded by 4 , this pro vides the deter minis tic bound ∆ t E ≤ 4 . W e now tur n to ( 4.1 ). Assume no w that 1 ≤ t ≤ N ( 1 − δ ) or ( 1 + δ ) N ≤ t ≤ 2 N . The e xpected increment can be decomposed as f ollo w s: E  ∆ t E | E ( C t ) ∈ [ nε, ( n + 1 ) ε )  = X z E  ∆ t E | Ext ( C t ) = z  P  Ext ( C t ) = z | E ( C t ) ∈ [ nε, ( n + 1 ) ε )  , (4.5) where the sum is taken o ver all points z ∈ L t with ⟨ z , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) . The summands will be controlled in diﬀerent wa ys, depending on the v alue of z . In the f ollowing, z is alwa ys assumed to satisfy ⟨ z , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) . Choose 1 ≤ r ≤ R depending on ε and δ ; the choice of r and R will be explained belo w . The constant q 0 > 4 belo w will depend on ε, δ and on r and R , but not on the angles θ, α , β , nor on n and N abo v e some threshold. W e hencef or th assume εn > R , with further lo wer bounds imposed belo w . W e dis tinguish three scenarios depending on the location of z = Ext ( C t ) . (1) z is in a pure block, at a dis tance at least 4 from an y interface; (2) z is within distance R from an interface, but not as in the ﬁrst case; (3) or z is in the mixed bloc k, at a distance at leas t R from any inter f ace. Bef ore descr ibing ho w to bound the summands in ( 4.5 ), w e s tate a separation lemma that will be useful in the follo wing proofs. R ecall from Section 2.3 the deﬁnition of a good ﬂo wer domain. Lemma 4.4. There exists c > 0 such that, f or any ρ ≥ 1 the follo wing holds. Ther e exists q 0 > 4 suc h that, for any q ∈ [ 4 , q 0 ) and any z at distance at least 4 ρ from the horizontal axis, P  the ﬂow er domain in ω t fr om Λ 2 ρ ( z ) to Λ ρ ( z ) is good   Ext ( C t ) = z  ≥ c. Pr oof. For conﬁgurations sampled according to the measure with q = 4 the statement holds according to Lemma 2.3 . A dapting this to our setting, we ﬁnd that there e xists a constant c > 0 such that, for any ρ ≥ 1 and z at a distance at least 4 ρ from the hor izontal axis, an y t ≥ 0 , and an y conﬁguration ˜ ω on Λ 4 ρ ( z ) c , ϕ 0 L t , 4  F is good   ω = ˜ ω on Λ 4 ρ ( z ) c and Ext ( C t ) = z  ≥ 2 c, (4.6) where we write F for the ﬂo wer domain rev ealed from Λ 2 ρ ( z ) to Λ ρ ( z ) and where the inequality holds as long as the conditioning is non-deg enerate. Univers ality : proof of Theorem 1.3 21 No w ﬁx ρ ≥ 1 . The e v ent in ( 4.6 ) depends only on the conﬁguration in Λ 4 ρ ( z ) with the measure in this region being of the f or m ϕ ξ Λ 4 ρ ( z ) , 4 with boundar y conditions ξ induced b y ˜ ω and a conditioning on an ev ent that depends on ˜ ω . The measures ϕ ξ Λ 4 ρ ( z ) ,q are all continuous in q . As such, there e xists q 0 > 4 suc h that, f or an y q ∈ ( 4 , q 0 ] , any boundary conditions ξ and any non-deg enerate e vent H , d TV  ϕ ξ Λ 4 ρ ( z ) ,q [ · | H ] , ϕ ξ Λ 4 ρ ( z ) , 4 [ · | H ]  ≤ c The abo v e ref ers to the distance in total variation betw een the conditioned measures. Combining the abov e with ( 4.6 ), we ﬁnd ϕ 0 L t ,q  F is good   ω = ˜ ω on Λ 4 ρ ( z ) c and Ext ( C t ) = z  ≥ c, (4.7) f or any conﬁguration ˜ ω on Λ 4 ρ ( z ) c such that the conditioning is non-degenerate. Finall y , as ω t f ollow s the law ϕ 0 L t ,q , ( 4.7 ) directl y implies the desired bound. W e no w return to the proof of Proposition 4.2 and deal with the previousl y deﬁned scenar ios individuall y Case (1): For z in a pure block, at a dis tance at least 4 from an inter f ace, w e ha v e P  ∆ t E ≥ 0 | Ext ( C t ) = z  = 1 . (4.8) Indeed, the cell of z is not aﬀected b y the transformation ( S t +1 ◦ S t ) and theref ore intersects C t +2 . The ineq uality stems from the f act that some other point ma y o v er take the e xtremum and thus increase the e xtremal coordinate. Case (2): W e argue that the probability f or Ext ( C t ) to be in case 2 under the conditioning E ( C t ) ∈ [ nε, ( n + 1 ) ε ) is small. Combining this with the deter ministic lo w er bound ∆ t E ≥ − 4 sho w s that the contr ibution of points z in the second case to ( 4.5 ) is not substantially negativ e. Lemma 4.5. Ther e exist constants C > 1 and q 0 > 4 suc h that, for any q ∈ ( 4 , q 0 ] , if n, N are suﬃciently larg e, P  Ext ( C t ) is at a distance at most R fr om an int er f ace, but at least C R from the horizontal axis   E ( C t ) ∈ [ nε, ( n + 1 ) ε )  < δ. (4.9) Pr oof of Lemma 4.5 . Fix C ≥ 1 ; we will see belo w ho w to choose it. W e hencef or th assume N > 2 C R . Additionall y , w e will assume Consider a potential realisation z of Ext ( C t ) with ⟨ z , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) and z at a distance at least 4 C R from the horizontal axis and with the minimal distance betw een z and an interface being smaller than R . Call the latter distance d . Consider the f ollo wing e xploration procedure of the unconditioned conﬁguration ω t . Explore the ﬂo w er domain F from Λ 2 C R ( z ) to Λ C R ( z ) and rev eal the conﬁguration on F c ; w e call this the e xplored region. Conditionall y on the exploration, w e distinguish three scenarios: (i) F is not good; (ii) the conﬁguration in the e xplored region is such that, f or any conﬁguration in F , w e hav e Ext ( C t )  = z ; (iii) the conﬁguration in the e xplored region is such that the clus ter of the origin is connected to the pr imal petal of F and is contained in H θ 0 be the constant given b y Lemma 4.4 . Then there e xists ˜ q 0 > 4 (depending on c good and C R , but not on z or an y other ﬁx ed quantities) such that P  Case (i) | Ext ( C t ) = z  ≤ 1 − c good f or an y q ∈ ( 4 , ˜ q 0 ] . In case (ii), w e ha ve P  Ext ( C t ) = z | F  = 0 , where the conditioning is that F is the result of the exploration procedure descr ibed abo v e. W e conclude that P  Case (iii) and Ext ( C t ) = z  ≥ c good · P  Ext ( C t ) = z  . (4.10) No w ﬁx a realisation of F as in case (iii). The measure for ω t inside F is then giv en b y ϕ ξ F ,q , where ξ are the boundar y conditions induced b y the e xploration. For conﬁgurations in F , wr ite Ext ( ξ ) f or the e xtremum of the cluster of the pr imal petal in the direction e θ . If ω t is such that ⟨ Ext ( ξ ) , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) , then Ext ( C t ) = Ext ( ξ ) . This applies in par ticular to Ext ( ξ ) = z , but also to other points z ′ in the strip ⟨ z ′ , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) . Consider no w 4 < q 0 ≤ ˜ q 0 (depending on C R ) such that, f or an y 4 ≤ q ≤ q 0 and an y F and z ′ as abov e, 1 2 ≤ ϕ ξ F ,q [ Ext ( ξ ) = z ′ ] ϕ ξ F , 4 [ Ext ( ξ ) = z ′ ] ≤ 2 . The exis tence of suc h a v alue q 0 is guaranteed b y the continuity in q of the measures ϕ ξ F ,q . Standard applications of ( RSW ), ( Mix ) and ( 2.4 ) sho w that, for any z ′ ∈ Λ C R/ 2 ( z ) with ⟨ z ′ , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) , if we write d ′ f or the distance between z ′ and the interface 6 c 0 ≤ ϕ ξ F , 4 [ Ext ( ξ ) = z ′ ] ϕ L mix , 4  A hp ( θ ) 3 ( z ′ ; 1 , d ′ )  ϕ 0 L t , 4  A hp ( θ ) 3 ( z ′ ; d ′ , C R )  ≤ 1 c 0 , f or some univ ersal cons tant c 0 > 0 . Appl ying the abo v e to z ′ = z and using Proposition 2.4 , w e ﬁnd ϕ ξ F ,q [ Ext ( ξ ) = z ] ≤ C 0 d − 2  d C R ) 1+ c = C 0 d − 1+ c ( C R ) − 1 − c . (4.11) f or some univ ersal cons tant C 0 > 0 and with the cons tant c > 0 giv en by ( 2.3 ). Con v ersel y , for any ﬁxed z ′ as abov e with d ′ ≥ C R/ 4 , w e ha ve ϕ ξ F ,q  Ext ( ξ ) = z ′  ≥ c 1 ( C R ) − 2 . f or some universal cons tant c 1 > 0 . There e xists a constant 7 c 2 = c 2 ( ε ) > 0 , independent of an y other quantity e x cept ε , such that the number of vertices z ′ ∈ Λ C R/ 2 ( z ) with ⟨ z ′ , e θ ⟩ ∈ [ nε, ( n +1 ) ε ) and d ′ ≥ C R/ 4 is at least c 2 C R . Thus, summing ov er all such z ′ , we conclude that ϕ ξ F ,q  ⟨ Ext ( ξ ) , e θ ⟩ ∈ [ nε, ( n + 1 ) ε )  ≥ c 2 c 1 ( C R ) − 1 . (4.12) 6 W e assume here that there is a unique inter f ace intersecting F . Due to the assumption that N ≥ 2 C R , there exis t at most two such inter f aces. The case of two inter f aces may be treated in a similar wa y , but is omitted here for simplicity . 7 It is here that it is essential that we condition on a rounding of e t and not its exact value. Univers ality : proof of Theorem 1.3 23 Combining ( 4.11 ) and ( 4.12 ), we conclude that P  Ext ( C t ) = z | F  = ϕ ξ F ,q  Ext ( ξ ) = z  ≤ c 3 d − 1+ c ( C R ) − c ϕ ξ F ,q  ⟨ Ext ( ξ ) , e θ ⟩ ∈ [ nε, ( n + 1 ) ε )  = c 3 d − 1+ c ( C R ) − c P  E ( C t ) ∈ [ nε, ( n + 1 ) ε ) | F  , where c 3 = C 0 c 1 c 2 > 0 is a univ ersal cons tant. The conditioning is ag ain that F is the result of the e xploration procedure described abo v e. As the abo ve is valid f or an y explored ﬂow er domain F in case (iii), w e conclude that P  Case (iii) and Ext ( C t ) = z  ≤ c 3 d − 1+ c ( C R ) − c P  E ( C t ) ∈ [ nε, ( n + 1 ) ε )  . Finall y , combining this with ( 4.10 ), we ha v e P  Ext ( C t ) = z   E ( C t ) ∈ [ nε, ( n + 1 ) ε )  ≤ 1 c good c 3 d − 1+ c ( C R ) − c . Summing ov er z with d ≤ R , ⟨ z , e θ ⟩ ∈ [ nε, ( n + 1 ) ε ) , and which are at a distance at least 4 C R from the horizontal axis, w e ﬁnd P  Ext ( C t ) is at a distance at most R from an inter f ace, but at leas t 4 C R from the horizontal axis   E ( C t ) ∈ [ nε, ( n + 1 ) ε )  ≤ c 4 C − c , where c 4 is a universal constant and c > 0 is given b y ( 2.3 ). By choosing C suﬃciently larg e, w e may render the right-hand side of the abo v e smaller than δ , thus pro ving ( 4.9 ) with 4 C instead of C . W e retur n no w to the analy sis of case (2). Let C ≥ 1 and q 0 > 4 be the constants giv en by Lemma 4.5 . Assume hencef or th that 4 < q ≤ q 0 , and that N is such that δ N sin α > C R + 4 and δ N sin β > C R + 4 . For t ≤ ( 1 − δ ) N the trac ks t 0 , . . . , t δ N are part of the frozen bloc k of angle α ; f or t ≥ ( 1 + δ ) N , they are par t of the frozen block of angle β . In both cases, due to our assumption on N , these bloc ks hav e a height at least C R . Thus, case (2) implies that the extremum is within distance R of an inter f ace, but also at a distance at least C R from the hor izontal axis. Combining the deterministic bound ∆ t E ≥ − 4 with ( 4.9 ), we ﬁnd X z in case (2) E  ∆ t E | Ext ( C t ) = z  P  Ext ( C t ) = z | E ( C t ) ∈ [ nε, ( n + 1 ) ε )  ≥ − 4 δ, (4.13) where the sum is taken ov er all possible realisations z of Ext ( C t ) included in case (2). Case (3): The remaining points z are in the mix ed block, at a distance at least R from an y interface. In this scenar io, w e want to relate the increment ∆ t E to the increment ∆ IIC E of an IIC (see Section 2.4 ). T o that end, w e ﬁrst prov e that the local environment around an e xtremum is indistinguishable from that of an IIC e xtremum (up to an arbitrar il y small er ror). Lemma 4.6. F or any r ≥ 1 , there exists a c hoice of R ≥ r and q 0 > 4 suc h that, for any q ∈ [ 4 , q 0 ) , any t ≥ 0 ev en, and any z ∈ L t in the mixed bloc k , at a distance at least R fr om an interface, d TV  P  · | Ext ( C t ) = z  , ϕ IIC L mix , 4  ≤ δ, wher e the two measures ref er to the conﬁguration in Λ r ( z ) and the latter is translat ed by z . Univers ality : proof of Theorem 1.3 24 Note here that R and q 0 depend on r and δ , but not on z , t or an y other quantity previousl y ﬁx ed. Pr oof. Fix r ≥ 1 and let R ≥ r be a constant to be ﬁxed belo w . Fix a point z as in the statement. Firs t, obser v e that, in L t , the lattice in Λ R ( z ) is identical to L mix — including its partition into cells. Thus, Proposition 2.5 ensures that, b y c hoosing R = R ( r, δ ) ≥ r suﬃciently larg e, d TV  ϕ 0 L t , 4  · | ω = ω 0 on Λ R ( z ) c , Ext ( C t ) = z  , ϕ IIC L mix , 4  ≤ δ 2 f or an y conﬁguration ω 0 on Λ R ( z ) c f or which the conditioning is not deg enerate, where the two measures refer to the conﬁguration in Λ r ( z ) and the latter is translated by z . No w , b y continuity of the measures ϕ ξ Λ R ( z ) ,q , there e xists q 0 = q 0 ( R, δ ) > 4 suc h that, f or an y 4 < q ≤ q 0 , d TV  ϕ 0 L t , 4 [ · | ω = ω 0 on Λ R ( z ) c , Ext ( C t ) = z ] , ϕ 0 L t ,q [ · | ω = ω 0 on Λ R ( z ) c , Ext ( C t ) = z ]  ≤ δ 2 f or an y ω 0 as abov e, with both measures ref er ring to the conﬁguration in the full bo x Λ R ( z ) . Combining the tw o display s abo v e, and keeping in mind that the la w of ω t is ϕ 0 L t ,q , w e obtain the desired conclusion. W e no w aim to control the ter ms E [ ∆ t E | Ext ( C t ) = z ] f or z deep within the mix ed block. Sample ω IIC according to ϕ IIC L mix , 4 and translate it by z . Include this sample under the measure P so as to maximise the probability under P [ · | Ext ( C t ) = z ] that ω t and ω IIC are identical in Λ r ( z ) . W r ite C IIC f or the cluster of z in ω IIC and ˜ C IIC f or the cor responding cluster in ( S t +1 ◦ S t )( ω IIC ) . Notice that the eﬀect of ( S t +1 ◦ S t ) on the local environment of Ext ( C IIC ) is identical to that of ( S odd ◦ S ev en ) . In par ticular , we ha v e that E ( ˜ C IIC ) − E ( C IIC ) = ∆ IIC E . There are multiple reasons why the increment ∆ t E might diﬀer from ∆ IIC E . W e sa y a coupling err or occurs if the conﬁgurations ω t and ω IIC are not identical in Λ r ( z ) . An increment error occurs if the conﬁgurations ( S t +1 ◦ S t )( ω IIC ) and ω t +2 = ( S t +1 ◦ S t )( ω t ) are not identical on Λ r/ 2 ( z ) . Finall y , an IIC error occurs if the e xtremum of ˜ C IIC is not contained in Λ r/ 2 ( z ) . If none of these er rors occur , then ω t +2 is identical to ( S t +1 ◦ S t )( ω IIC ) on Λ r/ 2 ( z ) and said bo x contains Ext ( ˜ C IIC ) . In par ticular , Ext ( ˜ C IIC ) ∈ C t +2 and theref ore E ( C t +2 ) ≥ E ( ˜ C IIC ) . The inequality comes from the case where C t +2 has an extremum outside of Λ r/ 2 ( z ) . While w e e xpect this to be unlik ely under P [ · | Ext ( C t ) = z ] , we will not endeav our to bound its probability . By collecting the er ror ter ms, w e obtain E  ∆ t E | Ext ( C t ) = z  ≥ E  ∆ IIC E − 4 1 error | Ext ( C t ) = z  ≥ − 4 P  er ror | Ext ( C t ) = z  , (4.14) where the ﬁrst inequality is due to the deter minis tic bound on increments and the second one to the fact that that E [ ∆ IIC E ] = 0 — see ( 2.5 ). W e will now bound the probabilities of each type of er ror occur ring. W e star t with the IIC and increment er rors, as these are controlled by taking r lar g e enough. Indeed, since the track e x changes are local transformations, the probability under P [ · | Ext ( C t ) = z ] that an increment Univers ality : proof of Theorem 1.3 25 er ror occurs is bounded b y C e − cr f or universal cons tants C and c — see Proposition 2.2 . For an IIC er ror to occur , C IIC should contain a point z ′ / ∈ Λ r/ 2 ( z ) with ⟨ z ′ , e θ ⟩ ≥ ⟨ z , e θ ⟩ − 2 . By a union bound on the possible v alues of z ′ and using Proposition 2.4 , the probability of an IIC error occur ring may be bounded by C r − 1 f or some universal constant C . This computation is identical to the cor responding one in [ DCKK + 20 ]. Thus, by taking r ≥ 1 suﬃciently larg e w e ma y ensure that P  increment or IIC er ror | Ext ( C t ) = z  ≤ δ. Finall y , with r ﬁx ed, Lemma 4.6 states that one ma y c hoose R ≥ r larg e enough and q 0 > 4 such that, P  coupling er ror | Ext ( C t ) = z  ≤ δ. Inser ting the last tw o bounds in ( 4.14 ), we ﬁnd that, f or R ≥ r ≥ 1 chosen as abo v e, and f or q ∈ ( 4 , q 0 ] , with q 0 as abov e, E  ∆ t E | Ext ( C t ) = z  ≥ − 8 δ. (4.15) Conclusion of pr oof of ( 4.1 ) : T ake R ≥ r ≥ 1 as dictated b y case (3), so that ( 4.15 ) holds. Consider no w q 0 > 4 suﬃciently close to 4 so that ( 4.8 ), ( 4.13 ), and ( 4.15 ) hold f or all 4 < q < q 0 , with the v alues of r , R c hosen abov e. Then, summing these bounds we ﬁnd E  ∆ t E | E ( C t ) ∈ [ nε, ( n + 1 ) ε )  ≥ − 12 δ, as required. □ 4.3 Deducing universality : proof of Theorem 1.3 With Corollar y 4.3 at hand, w e are almos t ready to pro ve our main result. W e will pro v e Proposi- tion 4.1 and then see ho w it implies Theorem 1.3 . Pr oof of Pr oposition 4.1 . Fix ε > 0 and α, β ∈ ( ε, π − ε ) and θ  = 3 π 2 . All constants belo w will be independent of α , β and θ , but may depend on ε . Due to ( 1.1 ), w e ma y ﬁx C such that, f or an y q > 4 and an y n ≥ 1 larg e enough 8 , ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  ≤ 2 ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n belo w t C n  and the same f or L + ( β ) . Set N = C n and deﬁne the lattices L t accordingl y . Then, by the abov e and Corollary 3.3 , P  E ( C 0 ) ≥ n  ≥ 1 2 ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  and P  E ( C 2 N ) ≥ n ( 1 − ε )  ≤ 2 ϕ 0 L + ( β ) ,q  0 ← → H θ ≥ n ( 1 − ε )  . (4.16) Let q 0 > 4 be given by Corollar y 4.3 f or η = ε and C ﬁxed as abo v e. F ix q ∈ ( 4 , q 0 ] . By ( 4.2 ) and ( 4.16 ), w e ha v e ϕ 0 L + ( β ) ,q  0 ← → H θ ≥ n ( 1 − ε )  ≥ ε 4 ϕ 0 L + ( α ) ,q  0 ← → H θ ≥ n  (4.17) 8 The low er bound on n ma y depend on q , as we will take n to inﬁnity ﬁrst. Univers ality : proof of Theorem 1.3 26 f or all n suﬃcientl y larg e. By taking the logarithm, dividing b y − n ( 1 − ε ) , and taking the limit n → ∞ in ( 4.17 ), we obtain ( 1 − ε ) · ζ hp β ,q ( θ ) − 1 ≤ ζ hp α,q ( θ ) − 1 . By in verting the roles of α and β in the coupling and repeating the same argument, we obtain the opposite bound. Pr oof of Theor em 1.3 . W e star t with proving the statement about ζ . Fix ε > 0 and let q 0 > 4 be given by Proposition 4.1 . Hencef or th consider q ∈ ( 4 , q 0 ] . Let θ ∈ [ 0 , 2 π ) and choose ˜ θ ∈ { θ, θ + π } (mod 2 π ) to be upper -half-plane aiming f or L ( α ) . Then, by Proposition 3.4 and Proposition 4.1 applied to ˜ θ , ζ α,q ( θ ) = ζ α,q ( ˜ θ ) = ζ hp α,q ( ˜ θ ) ≤ ( 1 + ε ) ζ hp π 2 ,q ( ˜ θ ) ≤ ( 1 + ε ) ζ π 2 ,q ( ˜ θ ) = ( 1 + ε ) ζ π 2 ,q ( θ ) . Appl ying the same reasoning with the roles of α and π / 2 e x chang ed w e obtain the opposite bound. Note that the v alue of ˜ θ ma y a prior i chang e for this second case. Finall y , the statement about ξ is readil y deduced from that about ζ using ( 1.2 ). A ckno wledgements. The authors thank Sébas tien Ott f or useful discussions. This w ork w as suppor ted b y the Swiss National Science Foundation and the swissuniv ersities “cotutelle de thèse ” grant. Ref erences [BDC12] Vincent Beﬀara and Hugo Duminil-Copin. The self-dual point of the two-dimensional random-cluster model is cr itical f or q ≥ 1 . Probab. Theor y Relat ed Fields , 153(3-4):511–542, 2012. [CDCH16] Dmitry Chelkak, Hugo Duminil-Copin, and Clément Hongler . Crossing probabilities in topological rectangles f or the cr itical planar FK-Ising model. Electron. J. Probab. , 21:Paper No. 5, 28, 2016. [Cer06] Raphaël Cerf. The Wulﬀ cr ystal in Ising and percolation models , v olume 1878 of Lecture No tes in Mathematics . Springer - V erlag, Berlin, 2006. Lectures from the 34th Summer School on Probability Theor y held in Saint-Flour , July 6–24, 2004, With a f orew ord b y Jean Picard. [CN06] F ederico Camia and Charles M. N ewman. T w o-dimensional critical percolation: the full scaling limit. Comm. Math. Phys. , 268(1):1–38, 2006. [DC13] Hugo Duminil-Copin. Limit of the Wulﬀ crys tal when approaching critically f or site perco- lation on the tr iangular lattice. Electron. Commun. Probab. , 18:no. 93, 9, 2013. [DCGH + 21] Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu, and Vincent T assion. Discontinuity of the phase transition for the planar random-clus ter and Potts models with q > 4 . Ann. Sci. Éc. N orm. Supér . (4) , 54(6):1363–1413, 2021. [DCKK + 20] Hugo Duminil-Copin, Karol Kajetan Kozlo wski, Dmitry Krachun, Ioan Manolescu, and Mendes Oulamara. R otational inv ariance in cr itical planar lattice models, 2020. Prepr int [DCLM18] Hugo Duminil-Copin, Jhih-Huang Li, and Ioan Manolescu. Univ ersality f or the random- cluster model on isoradial g raphs. Electron. J. Probab. , 23:Paper No. 96, 70, 2018. [DCM22] Hugo Duminil-Copin and Ioan Manolescu. Planar random-clus ter model: scaling relations. F orum Math. Pi , 10:Paper No. e23, 83, 2022. [DCR T19] Hugo Duminil-Copin, Aran Raouﬁ, and Vincent T assion. Shar p phase transition f or the random-cluster and Potts models via decision trees. Ann. of Math. (2) , 189(1):75–99, 2019. Univers ality : proof of Theorem 1.3 27 [DCST17] Hugo Duminil-Copin, Vladas Sidoravicius, and Vincent T assion. Continuity of the phase transition for planar random-cluster and Potts models with 1 ≤ q ≤ 4 . Comm. Math. Phy s. , 349(1):47–107, 2017. [GH00] Hans-Otto Geor gii and Y asunari Higuchi. Percolation and number of phases in the tw o- dimensional Ising model. J. Math. Phy s. , 41(3):1153–1169, 2000. Probabilistic techniques in equilibr ium and nonequilibrium statistical ph ysics. [GL25] Ale xander Glazman and Piet Lammers. Delocalisation and Continuity in 2D: Loop O ( 2 ) , Six-Vertex, and Random-Cluster Models. Comm. Math. Phys. , 406(5):Paper No. 108, 2025. [GM14] Geoﬀre y R. Grimmett and Ioan Manolescu. Bond percolation on isoradial graphs: cr iticality and universality . Probab. Theor y Related Fields , 159(1-2):273–327, 2014. [GM23] Ale xander Glazman and Ioan Manolescu. Structure of Gibbs measure f or planar FK- percolation and Potts models. Probab. Math. Phys. , 4(2):209–256, 2023. [GMM26] Loïc Gassmann, Ioan Manolescu, and Maran Mohanarang an. Gener ic ar m separation f or planar percolation. In preparation, 2026. [GPS13] Christophe Garban, Gábor Pete, and Oded Schramm. Pivotal, cluster , and interface measures f or cr itical planar percolation. J. Amer . Math. Soc. , 26(4):939–1024, 2013. [GPS18] Christophe Garban, Gábor P ete, and Oded Sc hramm. The scaling limits of near -cr itical and dynamical percolation. J. Eur . Math. Soc. (JEMS) , 20(5):1195–1268, 2018. [Gri06] Geoﬀre y Gr immett. The random-clust er model , volume 333 of Gr undlehr en der mathematis- c hen Wissensc haf ten [F undamental Principles of Mathematical Sciences] . Springer - V erlag, Berlin, 2006. [Man25] Ioan Manolescu. Explor ing the phase transition of planar FK-percolation, 2025. Prepr int [Ott22] Sébastien Ott. Exis tence and proper ties of connections deca y rate f or high temperature percolation models. Electron. J. Probab. , 27:Paper No. 100, 19, 2022. [Oul22] Mendes Oulamara. Random g eometr y and fr ee ener gy of critical planar lattice models . PhD thesis, Univ ersité Paris-Saclay, June 2022. [RS20] Gourab Ray and Yinon Spinka. A short proof of the discontinuity of phase transition in the planar random-cluster model with q > 4 . Comm. Math. Phys. , 378(3):1977–1988, 2020. [She05] Scott Sheﬃeld. Random sur f aces. Astérisq ue , (304):vi+175, 2005. [Smi01] Stanisla v Smir no v . Cr itical percolation in the plane: conf or mal inv ar iance, Cardy’ s f or mula, scaling limits. C. R. Acad. Sci. P aris Sér . I Math. , 333(3):239–244, 2001.

The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality

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