Memoire de Master 2

Percolation on Isoradial Graphs and the qKZ

Equations

Mendes Oulamarasous la direction de Hugo Duminil-Copin

(IHES, Universite de Geneve)

Septembre 2018

M2 Probabilites et Modeles Aleatoires, Sorbonne Universite (ex UPMC)

Image : Trois grilles se deformant, Francois Morellet, 1965.

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Contents

1 Percolation on Isoradial Graphs 41.1 Isoradial Graphs and their Diamond Graph . . . . . . . . . . 41.2 Self-dual Percolation . . . . . . . . . . . . . . . . . . . . . . . 51.3 The Star-Triangle Transformation . . . . . . . . . . . . . . . 61.4 Tracks, Loops and Spectral Parameters . . . . . . . . . . . . 91.5 Yang-Baxter Equation and Other Symmetries . . . . . . . . . 15

2 Boundary Passage on a Strip 172.1 Percolation on a Strip . . . . . . . . . . . . . . . . . . . . . . 172.2 The transfer matrix on link patterns . . . . . . . . . . . . . . 192.3 A classical analysis . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Spectral Analysis of Link Patterns Distribution 233.1 The Temperley-Lieb Algebra . . . . . . . . . . . . . . . . . . 233.2 Symmetries of the Transfer Matrix . . . . . . . . . . . . . . . 243.3 The q-deformed Knizhnik–Zamolodchikov Equations . . . . . 25

4 Path representation of the qKZ equation 304.1 Dyck Paths Representation . . . . . . . . . . . . . . . . . . . 304.2 Inductive Construction of Ψ . . . . . . . . . . . . . . . . . . . 33

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Figure 1: Examples of a percolation configuration, a six-vertex configurationand a dimer confiugration on the same portion of the square lattice.

Introduction

The mathematical study of planar statistical physics on lattices deals witha variety of models that can be defined as a collection of configurations onsome two dimensional lattice. For example, a configuration of the six vertexmodel on the square lattice (i.e. the usual square grid Z2) consists in anorientation of every edge such that every vertex has exactly two incomingand two outgoing edges (or arrows). A configuration of the dimer model isa matching of the vertices of a graph such that all vertices are matched withexactly one of their neighbors. A percolation configuration on any lattice canbe seen as a choice of one of its subgraphs (see Figure 1). Every configurationc gets some nonnegative (Gibbs) weight wc. The sum of all these weights iscalled the partition function Z and one can define a probability law, calledthe Gibbs measure, over configurations by giving the probability wc/Z tothe configuration c. When the graph is not finite, the partition functioncannot be directly defined and one needs some limiting procedure to definea (non necessarily unique) Gibbs measure.

We can roughly distinguish two types of approaches to study these mod-els. One has its roots in the traditional approach from Physics based onexact integrability and conformal field theory: some systems have enoughsymmetries so that it is possible to compute explicit formulas for quan-tities of interest (called observables). For example, Kasteleyn’s theorem[TF61, Kas61] exhibits an exact formula for the partition function of thedimer model. These methods are mostly algebraic using for instance trans-fer matrices and commutation identities. The other approach is the moreclassical probabilistic one, based on inequalities, stochastic dominations andestimation of events. An example of a tool related to this approach is theRusso-Seymour-Welsh theory [DCT15] to bound the probability of crossinga box in percolation. This approach needs less symmetry to apply but is farfrom giving results as precise as the algebraic approach in general.

It is important to note that even though it is common to talk about“integrable models”, a given model doesn’t belong to one or the other cat-egory. Instead, depending on the geometry of the model, both approaches

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might be applied. This is the case of the model we deal with in this text,namely Bernouilli percolation on isoradial graphs. The model will be de-fined at length in Section 1, but we can say for now that isoradial graphswere first introduced [Duf68, Mer01] as a convenient class of lattices to de-fine what could be a discrete holomorphic function and to study the criticalIsing model [CS11] and the dimer model [Ken02b]. Through the use ofanalyticity and complex functional operators, these works might be morerelated to the “exact integrability approach”. A percolation process on iso-radial graphs was defined in [Ken02a], and was proved to be critical and toshow some universality in [GM14], using extensively methods of the prob-abilistic approach: the Russo-Seymour-Welsh theory, couplings, increasingevents, large deviations. . . One can refer to [BdT12] for a review on statisti-cal physics models defined on isoradial graphs.

The work we present here is mainly based on [IP12] and deals with aparticular geometry, the strip, where methods from exact integrability canbe used to get explicit formulas for the probability of some connectivityevents (a boundary passage probability) in a percolation model on an iso-radial embedding of the strip. In Section 1, we define isoradial graphs ingeneral, and the notions and tools at the basis of exact integrability ap-proaches, namely tracks, rapidity lines and the Yang-Baxter equation. InSection 2, we present an attempt to get as much information as possibleon the quantity of interest, the boundary passage probability, using onlyclassical probability methods. This will expose the use of the star-triangletransformation and might serve as a more down-to-earth illustration of thespectral analysis that follows. In Section 3 we define a Markov process ona space of link patterns whose invariant distribution is sufficient to com-pute the boundary passage probability. We explain how the computation ofthis distribution can be reduced to the resolution of a set of complex val-ued polynomial equations called the q-deformed Knizhnik–Zamolodchikov(qKZ) Equations.

Section 4 is devoted to describe a path representation of these equa-tions that could lead to their rigorous resolution. Unfortunately, in thismaster thesis, we did not manage to make rigorous the resolutiondescribed in [IP12] (there is one symmetry lacking in the equations). Theanalysis of the qKZ equations is essentially based on [DGP+10, ZJ07, DF05,IP12].

1 Percolation on Isoradial Graphs

1.1 Isoradial Graphs and their Diamond Graph

Let G = (V,E) be a planar graph with edges embedded as straight line seg-ments. It is said to be isoradial if every bounded face of G is circumscribedin a circle of radius 1 with its center in the interior of the face. Isoradiality

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Figure 2: Example of an isoradial graph G in blue. Its diamond graph G

is represented in dotted black line. One of the faces of G is shown with itscircumscribing circle with center c (which is a vertex of G∗). An edge e isidentified with its adjacent vertices v1, v2 and dual vertices f1, f2, and theangle θe is represented.

is a property of the embedding of G, but we will nonetheless speak of an iso-radial graph. The usual embeddings of the square lattice Z2, the triangularlattice T or the hexagonal lattice H are examples of isoradial graphs.

When G is isoradial, there exists a natural embedding of its dual graphG∗ = (V ∗, E∗). Every f ∈ V ∗ corresponds to a bounded face F of G, andtake for the embedding of f the center of the circle on which vertices of Flie. Then join the vertices of V ∗ by straight line segment. Since the facesof G are convex, two edges of E∗ cannot cross except at vertices of V ∗. Sothis construction gives a planar embedding. The definition of the diamondgraph will make it clear that G∗ is also isoradial and G can be obtained as(G∗)∗.

The diamond graph G has vertex set V = V ∪ V ∗ and edge set E =v, f where v ∈ V, f ∈ V ∗, v and f are adjacent to a common edge. (seeFigure 2). Let e ∈ E, v1, v2 ∈ V its two ends, f1, f2 ∈ V ∗ its two adjacentdual vertices, e∗ = f1, f2. Denote by | · | the euclidean norm in theembedding. Then by isoradiality, |f1− v1| = |f1− v2| = |f2− v1| = |f2− v2|,so the faces of G are rhombi with side length 1, and G∗ is isoradial. Thisconstruction can be seen as a motivation to study the diamond graph, withrespect to which G and G∗ play symmetric roles.

1.2 Self-dual Percolation

Let G = (V,E) be a planar lattice with dual graph G∗ = (V ∗, E∗). Whene ∈ E, e∗ denotes the corresponding dual edge in E∗. Let p = (pe)e∈E ∈[0, 1]E a collection of probability parameters, and (Xe)e∈E a collection ofindependent Bernouilli random variables with P(Xe = 1) = pe for all e ∈ E.The Bernouilli percolation process on G with parameter p is defined as the

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measure on Ω = 0, 1E , denoted Pp, given by the law of (Xe)e∈E . An edge eis said to be open if Xe = 1 and closed otherwise. The dual of a configurationω ∈ Ω is the configuration ω∗ ∈ 0, 1E∗ defined by ω∗e∗ = 1 − ωe forevery e ∈ E. As usual in percolation, one is interested by the connectivityproperties of the model. For v1, v2 ∈ V we define the event v1 ↔ v2 as theset of configurations ω such that there exists a path of open edges from v1

to v2. If A is a subgraph of G, and ω a configuration, we write v1A←→ v2 for

the event that v1 and v2 are connected through a path in A, and v1A,ω←−→ v2

for the statement that ω ∈ v1A←→ v2.

Now suppose that G is isoradial with diamond graph G. Then everyedge e = v1, v2 ∈ E is the diagonal of some rhombic face F of G. Defineθe to be the value of the angle of F at v1 (or v2, see Figure 2). Notice thatif we invert the role of G and G∗ with respect to G, we get θe∗ = π − θe.We define p = (pe)e∈E by

∀e ∈ E, pe1− pe

=sin(13θe)

sin(13(π − θe)

) . (1)

The isoradial percolation process is defined on G by PG = Pp. It is easy tocheck that if p∗ is defined by the previous equation for G∗, then p∗e∗ = 1−pe.Hence this process is self dual in the sense that the dual process of theisoradial process on G has the same law as the isoradial process on G∗.

Note that in Equation (1), we could have defined pe via any increasingfunction f : [0, π]→ R+ with f(0) = 0 and still have self-duality. The choiceof f : x 7→ sin(x/3) will only appear to be necessary when we will studythe star-triangle transformation in Section 1.3. It is shown in [GM14] thatthis percolation process is critical: it has no infinite cluster, the radius ofthe cluster at the origin has polynomial decay, if we slightly increase theprobability for edges to be open an infinite cluster appears, on the contraryif we slightly lower this probability the radius of the cluster at the originhas exponential decay. In particular, the isoradial percolation process isequal to the critical percolation process on the square, the triangular andthe hexagonal lattices.

1.3 The Star-Triangle Transformation

Definition

One of the major questions of planar statistical physics is the universalityconjecture: the large scale behavior of a lattice model should not depend onthe particular local shape of the lattice. When restricted to isoradial graphs,this means that we would want to compare two percolations defined on twodifferent isoradial graphs. Fortunately, the star-triangle transformation givesa hint that this might be possible to achieve (see [GM+13] for details).

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Figure 3: The star-triangle transformation with corresponding probabilities.

Figure 4: Definition of the random coupling maps T and S. P = (1−p1)(1−p2)(1− p3)

We first define the transformation on a general graph G = (V,E) with(pe)e∈E a percolation parameter. Suppose that there are three vertices of G,v1, v2, v3 ∈ V that form a triangle as shown in Figure 3. Let G′ = (V ′, E′) bea copy of G where the triangle v1v2v3 has been replaced by a star with thecorresponding probabilities p′ shown in Figure 3, in particular V ′ = V ∪v∗.We claim that the percolation processes on G and G′ can be coupled so thatthe connectivity relations between vertices of V remain unchanged. This

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means that except at the vertex v∗ that we add or remove, all the percolationproperties are the same in G and in G′.

Proposition 1. Suppose that p1 + p2 + p3 − p1p2p3 = 1. There exists tworandom maps T : 0, 1E → 0, 1E′ and S : 0, 1E′ → 0, 1E such that ifω is distributed as a percolation on G with parameter (pe)e∈E and ω′ as apercolation on G′ with parameter (p′e)e∈E′, then T (ω) has the same law asω′ and S(ω′) has the same law as ω. Moreover, for any x, y ∈ V , we havealmost surely that

xG,ω←−→ y iff x

G′,T (ω)←−−−→ y

xG′,ω′←−−→ y iff x

G,S(ω′)←−−−→ y

Proof. Let T (resp. S) such that it changes the configuration ω (resp. ω′)only on the triangle (resp. the star) as defined by Figure 4. It is clear fromthe definition that no connectivity between any pair x, y ∈ V is changed. Soit only remains to show that when S and T have different possible outcomesin Figure 4, the probability of the outcomes sum to 1. And under ourassumption on p1, p2, p3,

p1p2p3 + (1− p1)p2p3 + p1(1− p2)p3 + p1p2(1− p3)= (1− p1)(1− p2)(1− p3)− 1 + p1 + p2 + p3 − p1p2p3 = P + 0

Star-triangle transformation on isoradial graphs

Even though this transformation can be defined on any graph, it is particu-larly adapted to percolation on isoradial graphs. As shown in Figure 5, botha triangle and a star correspond to three rhombi in the diamond graph thatform the shape of a 3D-cube (or a hexagon). With the choice of v∗ to bethe point at which the rhombi meet, the star-triangle transformation justamounts to a flip of the 3D-cube (or a rotation by π of the hexagon formed

Figure 5: The star-triangle transformation on an isoradial graph.

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by the three rhombi). Since for any angle θ, pπ−θ = 1 − pθ, the isoradialprobabilities of the new edges correspond to the probability given by thestar-triangle transformation (in Figure 3).

Our goal is to show that the condition of Proposition 1 is always verifiedwhen dealing with isoradial probabilities. Doing so, we will justify the choiceof the coefficient 1

3 in Equation (1). Let α 6= 0 be some real number anddefine zi = eiαθi , i = 1, 2, 3 and w = eiαπ where θ1, θ2, θ3 are defined in Figure5. Note that θ1+θ2+θ3 = π, hence z1z2z3 = w. Define ri = zi−1/zi

zi−1/zi+w/zi−zi/wand remark that if α = 1

3 , then pθi = ri, i = 1, 2, 3. If we check the conditionof Proposition 1 for general ri’s, we find

r1 + r2 + r3 − r1r2r3 − 1 = 0

⇐⇒ (1− w)

(w +

1

w2

)+

(w + 1− 1

w

)( 3∑i=1

w

z2i−

3∑i=1

z2iw

)= 0.

The only solutions for this to be true for all angles θ1, θ2, θ3 are α = ±13 .

In particular setting α = 13 shows that the star-triangle transformation can

always be applied on isoradial graphs.

1.4 Tracks, Loops and Spectral Parameters

Track system representation

The representation as (train) tracks of a rhombic tiling was introduced byde Bruijn [dB81] in its generalization of Penrose’s tiling [Pen79]. Let G =(V , E) be the diamond graph of an infinite isoradial graph such that G

is a tiling of the plane. Let e0 be one of its edges, it belongs to two rhombicfaces of G, and in each of these faces, there is one edge opposite to e0. Callthese edges e−1 and e1. Again, e1 belongs to another rhombic face. Callthe opposite edge of this face e2. Continuing this process, we end up witha doubly infinite sequence t = . . . e−2, e−1, e0, e1, e2 . . . that we call a track.The sequence of edges gives a choice of orientation of the track. It is easyto see that since the faces are rhombi, all the edges of t, seen as straightline segments oriented positively with respect to the orientation of the track,have the same angle α relative to any oriented axis ~D. We call this angle thetransverse angle relative to ~D (see Figure 8). If we look at the projectionof the edges of t on an axis with angle α + π/2 relative to ~D, we can seethat the sequence t is strictly monotonic. Hence, t does not cycle and all itsedges are distinct. When the graph is finite, tracks can be defined the sameway except that they are finite sequences of edges. The collection of all thetracks of G is called its track system representation.

The way we constructed tracks shows directly that every edge of G

belongs to exactly one track. Oriented tracks can also be seen as orientedarcs joining the midpoints of their edges (see Figure 6). Two tracks t, t′ are

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Figure 6: A percolation configuration on an isoradial graph. The diamondgraph is shown as thin black lines, open edges as thick black lines, duallyopen edges (i.e. duals of closed edges) as thick and dashed black lines, tracksas dashed blue curves and loops as red curves.

said to intersect at some rhombic face F if there are two edges e ∈ t, e′ ∈ t′with e, e′ ∈ F . This means that the corresponding arcs intersect in theinterior of F . Two tracks have at most one intersection: if they intersectonce, then the potential first next intersection (say along t seen as an arc)cannot happen as the transverse angles are not compatible. This result relieson the fact that an arc separates the plane in two disjoint domains. This isnot true for rhombic tilings of the torus for instance.

Loops and cluster interfaces

Given a percolation configuration on an isoradial graph, we can considerits clusters (i.e. connected components) and dual clusters (i.e. dually con-nected components). It is possible to define the interfaces of these clustersas non-intersecting loops and infinite paths separating every cluster fromthe adjacent dual clusters. We call this collection of interfaces a loop con-figuration. There is a bijection between percolation configurations and theirloop representations. Usually the loop representation is formally defined onthe medial lattice which is the dual of the diamond graph (e.g. [DC17]).However it will be more convenient for us to define it on the track system.

Let G be an isoradial graph, G = (V , E) its diamond graph and Tits track system. A loop l on T is a finite or doubly infinite sequence ofedges of G such that every two consecutive edges e, e′ belong to the samerhombic face of G but are not opposite edges. In the case of a finite loopl = e0, . . . , en, we consider that e0 and en are consecutive, except if G isfinite and e0 and en belong to its boundary, in which case we impose no

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Figure 7: Link pattern corresponding to the configuration in Figure 6. Pri-mal clusters are v1, v2, v4, v5, v6, v3, dual clusters are w1, w2, w3,w4, w5, w6.

other condition. Finally we impose that every edge appears at most once ina loop. A loop can be seen as a curve living on the arc representation of thetracks that makes a left or right turn whenever it reaches an intersectionbetween two tracks (as represented on Figure 6). We insist on the fact thata loop can either be a proper loop or a path.

Take ω a percolation configuration on G. The loop representation of ωis a collection ` of loops such that every edge e ∈ G belongs to exactly onel ∈ `, and such that loops, seen as curves, make their left or right turn soas to avoid open and dually open edges of ω (see Figure 6). More formally,for every two consecutive edges e, e′ of some l ∈ `, let F be the commonrhombus to e and e′. Then in ω, F is split by either a primal or a dual edgeg. In this case e and e′ must be on the same side of g in F .

Link patterns

Suppose that G is finite, then the union of the rhombic faces of G formsa bounded simply connected domain whose boundary is a collection ∂G =e1, . . . , en of edges of G. As before, ω is a percolation configurationon G and ` its loop representation. Then ` induces a perfect matching of∂G by the fact that every loop l ∈ ` contains either 0 or 2 edges of theboundary. This matching is called the link pattern λ(`) on ∂G induced by`. We denote by LP(∂G) the set of all possible link patterns on ∂G. Werepresent in Figure 7 the link pattern corresponding to the configuration ofFigure 6. The loops being interfaces between primal and dual clusters, it iseasy to see that the link pattern contains exactly the information necessaryto recover the connectivity relations between primal and dual vertices of∂G: two (dual) vertices are (dually) connected if and only if they are notseparated by a link.

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Figure 8: The tracks t1 and t2, with transverse angle θ1 and θ2 and ra-pidities z1 and z2 intersect on a rhombus. The two possible outcomes forthe loop configuration at this intersection are shown on the right, with thecorresponding probabilities. q = exp

(2iπ3

)and [x] = x− 1/x.

From isoradial graphs to rapidity lines

As we have seen, a percolation configuration on G is equivalent to a loopconfiguration on T and induces a link pattern of LP(∂G). To generate aloop configuration, one just needs to know how the tracks of T intersect,and what is the probability of a left or right turn of the loops at everysuch intersection. Moreover this probability only depends on the transverseangles of the tracks. Take two oriented tracks t1, t2 intersecting on a rhombusF , with transverse angles θ1, θ2 relative to some axis ~D as in Figure 8.We define the spectral parameter, or rapidity, of t1 (resp. t2) to be z1 =exp(iθ1/3) (resp. z2 = exp(iθ2/3)). An oriented track together with itsspectral parameter is called in the Physics literature a rapidity line. We willus the same terminology.

In a percolation configuration, the rhombus F can either be crossed bya (primal or dual) edge transverse to the direction of the tracks (this isthe first case of Figure 8). This happens with probability pθ2−θ1 . Or it iscrossed by a (primal or dual) edge in the direction of the tracks (this is thesecond case of Figure 8). This happens with probability pπ−(θ2−θ1). Fromnow on we use the notation [x] = x − 1/x for any scalar x, and we defineq = exp

(2iπ3

). Remark that

2i sin

(θ2 − θ1

3

)=z2z1− z1z2

= [z2/z1].

Likewise

2i sin

(π − (θ2 − θ1)

3

)= 2i sin

(π − π − (θ2 − θ1)

3

)= 2i sin

(2π + θ2 − θ1

3

)=qz2z1− z1qz2

= [qz2/z1].

Moreover one can check that for any w ∈ C, [w] + [qw] = [q/w].

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Figure 9: For the oriented tracks on the left to be realized as rapidity lines,we need a determination in R of the transverse angles verifying θ3 > θ2,θ2 > θ1, θ1 > θ3. This is impossible. To realize the oriented tracks on theright, we just need θ3 > θ2 > θ1. Take for instance θ1 = 0, θ2 = π/2,θ3 = 3π/4.

Hence according to Equation (1), we have

pθ2−θ1 =sin(θ2−θ1

3

)sin(θ2−θ1

3

)+ sin

(π−(θ2−θ1)

3

) =[z2/z1]

[qz1/z2](2)

pπ−(θ2−θ1) =sin(π−(θ2−θ1)

3

)sin(θ2−θ1

3

)+ sin

(π−(θ2−θ1)

3

) =[qz2/z1]

[qz1/z2]. (3)

Given an isoradial graph with oriented tracks t1, . . . , tn, it is tempting totry to reduce its study to its rapidity line configuration and to forget aboutthe isoradial embedding. To do so, we need to choose a reference axis ~Dand a determination in R of the transverse angles θ1, . . . , θn to compute thespectral parameters. Indeed, because of the division by 3 in the computationof the rapidities, it is not enough to know the angles modulo 2π. As aresult, it is not always possible to represent an isoradial graphwith its oriented tracks as a rapidity line configuration. Supposethat ti and tj cross in this order as in Figure 8, then we need the followingcondition for Equation (2) to hold:

θ2 − θ1 ∈ (0, π) (4)

Otherwise they do not give true probabilities (see Table 1). Figure 9provides an example of a very simple graph with two orientations of itstracks. The first one cannot be realized as a rapidity line configuration. Itis possible for the second one.

Abstract rapidity lines

Let us now consider arbitrary configurations of rapidity lines. In par-ticular configurations where, unlike tracks, two rapidity lines can inter-sect more than once. The formulas of Figure 8 still give weights to local

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z2z1

= eiα, α ∈ [0, π/3] (π/3, 2π/3) (2π/3, π)

[z2/z1][qz1/z2]

∈ [0, 1] (1,+∞) (−∞, 0)

[qz2/z1][qz1/z2]

∈ [0, 1] (−∞, 0) (1,+∞)

Table 1: Domain of the local loop weights as functions of the spectral ratioz2/z1. The weights are π-periodic as functions of α.

loop configurations at every intersection but we must be careful as theseweights are real but not necessarily probabilities in [0, 1]. We always have[z2/z1][qz1/z2]

+ [qz2/z1][qz1/z2]

= 1, and Table 1 gives the domain of the weights as afunctions of the rapidities.

If the rapidity line configuration is finite then we can give a weight toeach loop configuration by multiplying its local weights. This produces asigned measure on loop configurations. We believe that this level of formal-ism should be sufficient, even more so since for the sake of readability, thefollowing proofs on rapidity lines in this text will distinguish cases graphi-cally.

However we finish this section with a formal definition for what we in-tuitively see as a family of ‘nice’ curves in general position. If we look atthe union of these curves as the trace of a graph in the plane, then the dualof this graph has quadrangular faces, like the diamond graph. Hence thefollowing: let H = (V,E) be a finite planar graph such that all its boundedfaces are quadrangles (only the combinatorial structure matters here, thegeometry is not relevant) and such that its outer boundary is a simple se-quence of edges, or equivalently every edge e ∈ E is adjacent to two distinctfaces. A rapidity line r = (z, e0, . . . , en) is defined by a complex spectralparameters z ∈ U and a sequence of edges of E such that if e, e′ are twoconsecutive edges, there exists a bounded face F of H such that e and e′

belong to F but are not adjacent with each other (they are to opposite edgesof F ). Either e0 and en both belong to the boundary ∂H, or we considerthat they are consecutive. Every edge appears at most once in r. Let R bea collection rapidity lines on G, such that every edge appears in R exactlyonce. If r, r′ ∈ R, we write r∩r′ for the set of faces F of H having both edgesbelonging to r and edges belonging to r′. Examples are given in Figure 10.

A loop configuration ` on E is defined as in the beginning of the cur-rent section (1.4), changing rhombic faces to quadrangular faces. Let Lbe the set of all loop configurations. If F is a bounded face of H withboundary edges f1, f2, f3, f4 in cyclic order, these edges are paired in ` byappearing and being consecutive in the same loop. Denote by `F the pairingdescribing the configuration of ` on F , that is either f1, f2, f3, f4 or

f1, f4, f2, f3. Define wR(`F ) = [z2/z1][qz1/z2]

or [qz2/z1][qz1/z2]

the correct weight of

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Figure 10: Example of rapidity line configuration in blue with their quad-rangular construction graph H in black.

the pairing according to Figure 8, where z1, z2 are the spectral parametersof the two rapidity lines intersecting at F with the correct orientation. Thenthe weight of a loop configuration is defined by

wR(`) =∏

F face of H

wR(`F ). (5)

We use the same notation for the induced weight of link patterns on ∂G.For any link α ∈ LP(∂H),

wR(α) =∑

`∈L,λ(`)=α

wR(`). (6)

1.5 Yang-Baxter Equation and Other Symmetries

Let R and R′ be two rapidity line configurations defined on quadrangularconstruction graphs H and H ′ such that ∂H = ∂H ′. These two configu-rations induce two signed measures wR and wR′ on the same space of linkpatterns LP(∂H). We define the equivalence relation 'LP on rapidity lineconfigurations by

R 'LP R′ ⇐⇒ wR = wR′ . (7)

This section is devoted to show the 'LP-equivalence of some configurations.These equivalences will be useful to transform bigger rapidity line configura-tion: if R,R′,R′′ are three configurations with R 'LP R′ and R appears assome sub-pattern of R′′, then the link pattern distribution on R′′ dependsonly on the link pattern on its version of R and not on the particular loopconfiguration realizing it. This allows us to replace R by R′ in R′′ and stillget an 'LP-equivalent configuration giving the same weights to the samelink patterns.

In the following results, the configurations will be defined pictorially,with a specification of the edges of ∂H. The first equivalence result justshows what happens when the orientation of a rapidity line is reversed.

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Theorem 1 (Crossing relation).

(8)

Where a, b, c, d are labels of the edges joined by the rapidity lines, z1, z2 ∈ Uare rapidities.

Proof. Define R (resp. R′) to be the configuration on the left hand side(resp. right hand side) of (8). Since LP(a, b, c, d) has only two (non-intersecting) link patterns and since the total weight is always 1, it is enoughto check the equality of the weight for one of them, say α = a, b, c, d.We have

wR(α) =[z1/z2]

[qz2/z1]=

[z2/z1]

[z1/(qz2)]=

[qz2/(qz1)]

[q(qz1)/z2]= wR′(α).

Theorem 2 (Unitary relation).

(9)

Where a, b, c, d are labels of the edges joined by the rapidity lines, z1, z2 ∈ Uare rapidities.

Proof. As before let R,R′ be the two configurations, it is enough to checkthat weights coincides on one of the two possible link patterns. The patternα = a, d, b, c is realized by only one loop configuration. Hence

wR(α) =[qz1/z2]

[qz2/z1]× [qz2/z1]

[qz1/z2]= 1 = wR′(α).

Theorem 3 (Yang-Baxter equation). We have

(10)

16

where a, b, c, d, e, f are labels of the edges joined by the rapidity lines, z1, z2, z3 ∈U are rapidities.

The term ‘Yang-Baxter equation’ was introduced by Faddeev to describea principle of invariance appearing in a wide variety of physical systems, thatcan be seen as a generalization of the star-triangle transformation. The firstappearance of such transformation goes back to the work of Kennelly in1899 on electrical networks. One can refer to [PAY06] for an account on thehistory and the various versions of the Yang-Baxter equation.

In our case, this equation is really a direct generalization of the star-triangle transformation defined in Section 1.3. On isoradial graphs, Figure5 shows that the star-triangle transformation is just a permutation of rhombiof the diamond graph, which is exactly equivalent to sliding one of the threetracks (or rapidity lines) corresponding to these rhombi over the other two.This track transformation is exactly the one appearing in Equation (10).

Building on our work on the star-triangle transformation on isoradialgraphs, we can give the following proof of Theorem 3.

Proof. Let R,R′ be the two rapidity line configurations appearing in Equa-tion (10). Let α be a link pattern with weights wR(α) and wR′(α). Fig-ure 9 shows that R can be obtained from an actual isoradial graph as inFigure 5. In this case the weights are probabilities and we showed thatwR(α) = wR′(α). It is easy to see that if θ1, θ2, θ3 are the angles of theedges of the rhombi of the the isoradial graph, then we can slightly moveone of these angles while keeping the other fixed and still having an isora-dial representation (there are three degrees of freedom on these angles). Thismeans that there exists an open set of U3 such that (z1, z2, z3) correspondsto an isoradial graph. Moreover wR(α) and wR′(α) are rational functions ofthe rapidities z1, z2, z3. So wR(α) = wR′(α) in general.

A more straightforward proof is to check that the weights of link patternscorrespond. Up to symmetries, we can reduce this verification to two linkpatterns, namely α = a, f, b, e, c, d and β = a, f, b, c, d, e.α is realized by only one loop configuration in R and R′, β is realized by 4loop configurations in R and 1 in R′. One can do the calculation and checkwR(α) = wR′(α) and wR(β) = wR′(β).

2 Boundary Passage on a Strip

2.1 Percolation on a Strip

From now on, we will focus on a particular isoradial graph, the strip, rep-resented in Figure 11. The strip can be constructed from a square isoradialgraph (i.e. a graph isomorphic to Z2) by selecting two (vertical) paralleltracks that we call the left and the right boundaries. We fix all edges of

17

Figure 11: (a): A section of the infinite isoradial strip with L = 5. Theprimal graph is drawn in straight blue line and the diamond graph in blackdotted line. The right boundary (in thick blue line) is fixed to be open andthe left boundary (in dashed thick blue line) is set to be dually open (i.e.closed). (b): The rapidity line configuration corresponding to the strip. Theleft and right boundary conditions translate by the fact that extremities ofloops ending on the boundaries are linked two by two. This is representedby small link with a square (to distinguish from the continuation of a ra-pidity line). All the rapidity lines from left to right have the same spectralparameter w, vertical lines have rapidities z1, . . . , z5. (c): A percolation con-figuration on two lines of the strip and the corresponding loop configurationon the rapidity line configuration.

the left (resp. right) boundary to be closed (resp. open). The strip is thegraph between these boundaries. Let L be the number of tracks of the stripparallel to the boundaries. We consider only the case when L is odd.

It will be convenient to take the x-axis oriented in the negative direc-tion as a reference ~D for transverse angles. We assume that all horizontaltracks have the same transverse angle −π2 , corresponding to the rapidityw = exp

(−iπ6

)on Figure 11. We take the determination of the transverse

angles of the vertical tracks in (−π/2, π/2). Since the crossing always in-volve a vertical and an horizontal track, it is easy to see that the consistencycondition (4) is verified. The percolation on a strip can hence be describedby the rapidity line configuration shown in Figure 11.

Denote by z1, . . . , zL the rapidities of the vertical lines as in Figure 11,and define the corresponding probabilities (or weights when the rapiditiesare out of the scope of isoradial graphs):

pi =[qzi/w]

[qw/zi], i = 1, . . . , L.

In every column of the strip, primal or dual edges in the south-west to north-east direction are open with probability pi and closed with probability 1−pi.

18

Figure 12: Two configurations realizing respectively v0 ↔ B and v1 ↔B.

Define two vertices v0 and v1 as in Figure 11, and B the set of verticesof the right boundary. We are interested by the computation of

PL = P (v0 ↔ B) and PL = P (v1 ↔ B) . (11)

We call them the boundary passage probabilities. They depend on therapidities of the system, or equivalently on the probabilities pi, i = 1 . . . L.We will abuse notation and see them as function, writing PL(z1, . . . , zL)(since w is fixed) or PL(p1, . . . pL), and likewise for PL.

2.2 The transfer matrix on link patterns

Define LPL = LP(1, . . . , L,∞) to be the link patterns where 1, . . . , L,∞are thought of as L + 1 points lying on a line in this order, and such thatthe links when drawn in one of the half-plane do not intersect. We representthese patterns as links between 1, . . . , L together with one path starting ata point of 1, . . . , L and thought of as going to ∞ (see Figure 12).

Since the width of the strip L is odd, the loop configuration induced bya random percolation configuration is almost surely composed of finite loopsand one single infinite path, that we call the interface, stretching from −∞to +∞. The uniqueness of the infinite path can be deduced from the factthat almost surely, two-line configurations letting only one path to traversethem (like the one in Figure 11.c) happen infinitely often. Hence, if we fixone line of the strip and look at the cut loop configuration, it induces aprobability law on the upward (resp. downward) link patterns on the halfplane below (resp. above) the line by linking the edges joined by a loop (and

linking the remaining edge to ∞). We call π↑L (resp. π↓L) this probabilitylaw on LPL.

Let L be the set of loop configurations on two horizontal lines of thestrip and ` ∈ L a random configuration induced by the percolation process.It is easy to see that v0 ↔ B and v1 ↔ B are equivalent to the factthe interface passes to the left of either v0 or v1 (see Figure 12). And this

19

Figure 13: The action ↑ of a loop configuration on two lines of the strip onan upward link pattern.

can be deduced from the upward link pattern, the downward link pattern,and the state of the two lines around v0: there exist two indicator functionsI : LP2

L → 0, 1 and I : LPL×L× LPL → 0, 1 such that

PL =∑

α∈LPL

∑β∈LPL

π↑L(α)π↓L(β)I(α, β) (12)

and PL =∑

α∈LPL

∑l∈L

∑β∈LPL

π↑L(α)P(` = l)π↓L(β)I(α, l, β) (13)

We can define the action of l ∈ L on α ∈ LPL by l ↑ α (resp. l ↓ α) thatconsists in the link resulting from the concatenation of the loop configurationl with the link pattern α seen as an upward (resp. downward) link patternon the strip. An example of the action l ↑ α is shown in Figure 13. The lawsπ↑L and π↓L are invariant by a translation of two line (we still get the same half

infinite strip). This means that π↑L (resp. π↓L) is the invariant distributionof the Markov chain on LPL defined by the action of ` ↑ · (resp. ` ↓ ·).Indeed this Markov chain is easily shown to be irreducible and aperiodic (onits support which might not be all LPL).

With this Markov chain interpretation, the link between the upward andthe downward distribution is clearer. If we make explicit the dependency ofthe distribution on the pi’s, we can see that

∀α ∈ LPL,∀p1, . . . , pL ∈ [0, 1], π↓L,p1...,pL(α) = π↑L,1−p1...,1−pL(α). (14)

To simplify notations, we will only consider upward link patterns. Wedefine the transfer matrix t to be the transposed of the transition matrixof the Markov chain. When necessary we will make the dependency inp1, . . . , pL or z1, . . . zL explicit.

∀α, β ∈ LPL, tβ,α = P(` ↑ α = β) (15)

Our goal for the rest of this text is to identify π↑L and hence the boundarypassage probabilities. Here is a first result in this direction:

20

Theorem 4. The coefficients(π↑L(α)

)α∈LPL

are rational functions in the

parameters p1, . . . , pL. Hence so are PL and PL.

Proof. By definition, the coefficients of t are polynomials in p1, . . . , pL (andhence rational functions in z1, . . . , zL). They belong to the field Q(p1, . . . , pL).

Moreover π↑L can be seen as a right eigenvector of t for the eigenvalue 1.Hence, its coefficients are also defined in the field Q(p1, . . . , pL) of rationalfractions in p1, . . . , pL.

2.3 A classical analysis

This section is independent of the rest of this text and can be skipped. Itcan be thought of as a classical illustration of the following sections, wherewe use algebraic methods, spectral parameters and the loop representationto identify the invariant probability on link patterns. Here we translatesome of these ideas to the primal percolation representation depending onp1, . . . , pL, and we get some symmetries on the boundary passage probabilityPL(p1, . . . , pL). PL can be dealt with similarly.

Theorem 5. For any p1, . . . , pL ∈ R, the rational fraction PL verifies

(i) PL(p1, . . . , pi, pi+1, . . . , pL) = PL(p1, . . . , pi+1, pi, . . . , pL) for any 1 ≤i ≤ L− 1.

(ii) PL(p1, . . . , pi−1, 1− pi, pi+1, . . . , pL) = PL(p1, . . . , pi−1, pi, pi+1, . . . , pL)for any 1 ≤ i ≤ L.

(iii) PL(p1, . . . , pi,pi−1pi, pi+2, . . . , pL) = PL−2(p1, . . . , pi−1, pi+2, . . . , pL) for

any 1 ≤ i ≤ L− 1.

Proof. Define SH to be the subgraph of the strip where we take only H lines

above and below v0. Then define PHL = P(v0SH←→ B). PHL is a polynomial

in the pi’s and for p1, . . . , pL ∈ [0, 1], PHL is increasingly converging to PLwhen H → +∞, hence uniformly. It is sufficient to prove (i) and (ii) forPHL .

Let H be such that there is a vertex in the highest line between columni and i + 1. We can apply the following transformations for the value x =1−pi+pipi+1

1−pi+1+pipi+1.

21

We first add an edge on the very top with percolation probability x, thenwe apply a serie of star-triangle transformation that exchange the columnsand transport the vertex x to the bottom of SH , where we finally remove it.None of these transformation modifies the probability of connectivity fromv0 to the right boundary (in particular because v0 never happens to be inthe middle of a star-triangle transformation). This proves the invariance ofPHL under any permutation.

We can group the edges of the last columns in pairs of consecutive edgesthat totally determine whether their left extremity is linked directly to theright boundary. This proves (ii) for i = L, and the invariance by permuta-tion allows to conclude for every 1 ≤ i ≤ L.

To prove (iii) we need to exit the domain of probabilities and really seePHL as a polynomial. The star-triangle transformation can still be applied,we just have to think in terms of weight rather than of probabilities. Takepi = p, pi+1 = p−1

p and as before, H so that we can apply these connectivitypreserving transformations:

This shows that

PHL (p1, . . . , pi,pi − 1

pi, pi+2, . . . , pL) = PHL−2(p1, . . . , pi−1, pi+2, . . . , pL)

for infinitely many H. We do not have probabilities anymore so we must becareful to extend this property to PL. However, notice that if we call

qHk = P(v0 is connected to k vertices of the last column

through columns 1, . . . , L− 1 in SH)

we can decompose PHL in

PHL =∑k≥1

qHk (pL(1− pL))k .

This decomposition shows that the series with coefficients qHk converges ab-solutely if pL(1− pL) is close enough to zero. The value of p can be chosenclose to 1 so that 1

p ×p−1p is close to zero. Hence the series with coefficients

qk = limH→∞ qHk converges absolutely in a neighborhood of zero and we

have

PL

(p1, . . . , pL−2, p,

p− 1

p

)= PL−2(p1, . . . , pL−2).

This equality holds for p in a complex neighborhood of 1, and both sideare rational fractions. So it holds as an equality between rational fractions.Thus we can use the invariance under permutations (ii) to get (iii).

22

Figure 14: (a) example of an action of ei on a link pattern. (b) eiei+1ei = ei.

3 Spectral Analysis of Link Patterns Distribution

We saw with Theorem 4 that the values of π↑L are rational fractions in thevariables z1, . . . , zl. We can multiply them by the same factor and definea vector (Ψα)α∈LPL

proportional to π↑L whose coefficients are Laurent poly-nomial and have no factor in common. The goal of this section is to finda set of equation, called the q-deformed Knizhnik–Zamolodchikov, that aresufficient to identify a particular such Ψ.

3.1 The Temperley-Lieb Algebra

The Temperley-Lieb algebra is generated by the collection (ei)1≤i≤L−1 ofoperators acting on LPL (seen as upward links) by applying a downwardlink between i and i + 1 and then an upward link between i and i + 1 (seeFigure 14.a). These operators verify some relations:

e2i = ei

eiei±1ei = ei (see Figure 14)

eiej = ejei if |i− j| ≥ 2.

We define the canonical base of CLPL by (|α〉)α∈LPLwhere |α〉 is the

vector with coefficient 1 at coordinate α and 0 otherwise. We abuse notationby also considering ei as an operator on CLPL defined on the base by ei|α〉 =|eiα〉. Define the operator corresponding to the crossing of rapidity linesexiting from sites i and i+ 1:

Ri(w) =[q/w]

[qw]1− [w]

[qw]ei (16)

The operator Ri(zi/zi+1) corresponds to the transformation on upward linkpattern induced by

.

23

3.2 Symmetries of the Transfer Matrix

Lemma 1. The transfer matrix satisfies the interlacing relation

Ri(zi/zi+1)t(z1, . . . , zi, zi+1, . . .) = t(z1, . . . , zi+1, zi, . . .)Ri(zi/zi+1), (17a)

and the boundary relations

t(z1, z2, . . .) = t(1/z1, z2, . . .) (17b)

t(z1, . . . , zL−1, zL) = t(z1, . . . , zL−1, 1/zL) (17c)

Proof. The left hand side and the right hand side of (17a) correspond to thefollowing rapidity line configurations which are 'LP-equivalent thanks to arepeated application of the Yang-Baxter equation (10).

The boundary relations are easily deduced by considering all the possibletiles on the crossings of the first and the last vertical rapidity line. Forinstance for the first boundary relation, this loop configuration is the onlyone giving its link pattern on the boundary:

Using the fact that w = exp(−iπ/6), it has weight

[qz1/w][z1/w]

[qw/z1][qw/z1]=

[z1/w5][z1/w]

[1/(w3z1)][1/(w3z1)]=

[1/(z1w)][z1/w]

[z1/w3][1/(z1w3)]

which invariant under z1 ↔ 1/z1.

For 1 ≤ i ≤ L − 1, define the function ϕi : LPL−2 −→ LPL by in-serting two points at position i and a link between them. Formally, if

α =x1, y1 , . . . ,

xL−1

2, yL−1

2

∈ LPL−2,

ϕ(α) = i, i+ 1(L−1)/2⋃j=1

xj + 2× 1xj≥i, yj + 2× 1yj≥i.

24

Lemma 2. When zi+1 = qzi, the transfer matrix verifies

tL(z1, . . . , zi, qzi, zi+2, . . .)ϕi = ϕitL−2(z1, . . . , zi−1, zi+2, . . .) (18)

where we made explicit the dependency of t in L.

Proof. If we look at the rapidity line configuration induced by ϕi on thevertical lines i and i + 1, we can apply twice the following transformationsbased on the crossing relation (8) and the unitary relation (9).

By doing so, we prove the 'LP-equivalence of the configurations correspond-ing to the two sides of the relation.

We finish this section with a remark on the horizontal rapidity lines.We made the choice of giving the same rapidity w to the two lines of thetransfer matrix and of fixing the value of w = exp

(−iπ6

)from the beginning.

Suppose that we gave two different rapidities w and w′ to the two horizontallines of the transfer matrix. Then using the relations (8, 9, 10) and Lemma5, it is possible to show that two transfer matrices with the same value ofthe product ww′ commute. Hence they have the same Perron-Froebeniuseigenvector. This means that if we squeeze the strip like an accordion, thelink pattern distribution, and hence the boundary passage probabilities, donot change.

3.3 The q-deformed Knizhnik–Zamolodchikov Equations

When we take the rapidities z1, . . . , zL to be induced from transverse anglesin (−π/2, π/2), t(z1, . . . , zL) is the transposed of the transition matrix of anirreducible aperiodic Markov chain. Hence it has a Perron-Froebenius eigen-vector for the eigenvalue 1, say v. Following Theorem 4, the coordinates of vare rational fractions in z1, . . . , zL and we can normalize this eigenvector toget another proportional Perron-Froebenius eigenvector whose coordinatesare Laurent polynomials with no factor in common. The following theoremexhibits a set of relations that we will show to be sufficient to identify onesuch eigenvector.

Theorem 6. Let Ψ(z1, . . . , zL) ∈ CLPL be a vector valued function in thevariables z1, . . . , zL. Define the partition function

Z(z1, . . . , zL) =∑

α∈LPL

Ψα(z1, . . . , zL).

Denote for any function f , πjf(. . . , zj , zj+1, . . .) = f(. . . , zj+1, zj , . . .). Thenthe following are equivalent:

25

(i) Ψ verifies the following relations called the qKZ equations

Rj(zj/zj+1)Ψ = πjΨ for 1 ≤ j ≤ L− 1 (19a)

Ψ(z1, . . . , zL) = Ψ(1/z1, z2, . . . , zL) (19b)

Ψ(z1, . . . , zL) = Ψ(z1, . . . , zL−1, 1/zL). (19c)

(ii) Z(z1, . . . , zL) is invariant under any πi, i = 1, . . . , L−1 and the trans-formations z1 ↔ 1/z1 and zL ↔ 1/zL, and

t(z1, . . . , zL)Ψ(z1, . . . , zL) = Ψ(z1, . . . , zL).

In particular if Ψ is non-zero, it is a Perron-Froebenius eigenvector oft.

Proof. We first show (ii) =⇒ (i) using Lemma 1. If we apply both sidesof Equation (17a) to Ψ, we find

Ri(zi/zi+1)Ψ = t(z1, . . . , zi+1, zi, . . .)Ri(zi/zi+1)Ψ.

The Perron-Froebenius eigenvector being unique, this tells us that Ri(zi/zi+1)Ψand πiΨ are proportional (they can be zero a priori). Moreover R is a mea-sure preserving transformation, hence∑

α∈LPL

(Ri(zi/zi+1)Ψ

)α

=∑

α∈LPL

Ψα = Z = πiZ =∑

α∈LPL

(πiΨ)α .

So Ri(zi/zi+1)Ψ = πiΨ. The same strategy works to equally deduce (19b)and (19c) from (17b) and (17c).

Now we show (i) =⇒ (ii). The strategy of the proof is to define anothermatrix acting on CLPL called the scattering matrix S(z1, . . . , zL), which isthe transposed of an irreducible stochastic matrix. We will show that S andt commute, hence they have the same Perron-Froebenius eigenvectors. Toconclude, we show that if Ψ is a vector verifying the qKZ equations, it is aneigenvector of S for the eigenvalue 1 (or zero).

The scattering matrix S is defined as the matrix representation of theaction on upward link patterns LPL of the rapidity line configuration repre-sented in Figure 15. Formally, it is defined by

S(z1, . . . , zL) =L∏i=2

i−1∏j=1

Ri−j(zizj)×L∏i=2

i−1∏j=1

Rj+L−i(zjzj) (20)

where the order of the products is very important since these operators donot commute. If θ1, . . . , θL are the transverse angles corresponding to thevertical rapidities, we can take them to be positive and close to 0. Then thecoefficients of the operators R in (20) are positive and S is the transposed

26

Figure 15: Rapidity line representation of the scattering matrix S for L =5. The squares represent junctions between rapidity lines with differentrapidities.

of an irreducible stochastic matrix (the irreducibility is easy to check). LetΨ be a vector verifying the qKZ equations (19a, 19b, 19c), then by Lemma3 we have SΨ = Ψ. By Lemma 4, t and S commute, so they have the samePerron-Froebenius eigenvectors, hence tΨ = Ψ.

Since R is measure preserving, it is clear from (19a) that Z = πiZ fori = 1, . . . , L − 1, and from (19b,19c) that Z is invariant under z1 ↔ 1/z1and zL ↔ 1/zL.

Lemma 3. Let Ψ be a vector verifying the qKZ equations (19a, 19b, 19c)then SΨ = Ψ.

Proof. A graphical way to see this is to say that on the diagram in Figure15, the arguments of Ψ after a sequence of upward operations are exactlythe rapidities read in order on the rapidity lines just above the crossingscorresponding to the sequence of operations. When two line cross, the indexof their rapidities is exchanged and (19a) shows that it is exactly whathappens to the argument of Ψ. When a rapidity line ends on the boundaryand another one with inversed rapidity starts, it appears in Ψ at first or lastposition, and (19b,19c) show that we can apply the same inversion to thearguments of Ψ.

27

Figure 16: Graphical proof that tS = St for L = 3. This shows a sequenceof 'LP-preserving transformation. We first change the rapidities on somelines using (8) and the fact that the weights are invariant if we replace arapidity by its opposite. The next step is done via a repetition of the Yang-Baxter equation. The next one relies on the boundary result in the proofof Lemma 4, and it is repeated to get the final rapidity line configurationcorresponding to tS, where we replace the rapidities by the original ones.

We show the formal computation for L = 3.

S(z1, z3, z3)Ψ(z1, z2, z3)

= R1(z1z2)R2(z1z3)R1(z2z3)R2(z1z2)R1(z1z3)R2(z2z3)Ψ(z1, z2, 1/z3)

= R1(z1z2)R2(z1z3)R1(z2z3)R2(z1z2)R1(z1z3)Ψ(z1, 1/z3, z2)

= R1(z1z2)R2(z1z3)R1(z2z3)R2(z1z2)Ψ(1/z3, z1, 1/z2)

= R1(z1z2)R2(z1z3)R1(z2z3)Ψ(z3, 1/z2, z1)

= R1(z1z2)R2(z1z3)Ψ(1/z2, z3, 1/z1)

= R1(z1z2)Ψ(z2, 1/z1, z3) = Ψ(1/z1, z2, z3) = Ψ(z1, z2, z3)

Lemma 4. The transfer matrix and the scattering matrix commute:

t(z1, . . . , zL)S(z1, . . . , zL) = S(z1, . . . , zL)t(z1, . . . , zL).

28

Proof. If we concatenate the rapidity line representation of t and S, weneed to make the two lines with rapidity w of t cross the representation ofS. Figure 16 shows the sequence of operations for L = 3. The Yang-Baxterequation (10) shows that all the internal crossing of S can be crossed easilyby t. It remains to show that t can cross the boundary points of S, that isto show the following 'LP-equivalences:

The first 'LP-equivalence on the right comes from two applications of (8),first to vertical lines and then to the horizontal lines. On the left, −z ×qz−1 = −q = w × w. On the right, −qz × q2z−1 = −1 = qw × qw. Hencewe can conclude using Lemma 5.

Lemma 5 (Boundary decoupling). If x, x′, y, y′ are rapidities such thatxx′ = yy′, then we have the following 'LP-equivalence:

Proof. The loop configurations not equivalent to the right hand sides arethe following:

The two configurations have weight

[x/y′]

[qy′/x]+

[qx/y′][qx′/y′][qx/y][x′/y]

[qy′/x][qy′/x′][qy/x][qy/x′]

=[x/y′][qy′/x′][qy/x][qy/x′] + [qx/y′][qx′/y′][qx/y][x′/y]

[qy′/x][qy′/x′][qy/x][qy/x′]

=[y/x′][qx/y][qx′/y′][qx/y′] + [qx/y′][qx′/y′][qx/y][x′/y]

[qy′/x][qy′/x′][qy/x][qy/x′]= 0.

29

4 Path representation of the qKZ equation

The goal of this section is to exhibit a path representation that could leadto a construction of an explicit solution to the qKZ equations. We first givean equivalent statement of (19a, 19b, 19c).

Proposition 2. A vector valued function Ψ(z1, . . . , zL) satisfies the qKZequations (19a, 19b, 19c) if and only if ∀z1, . . . , zL ∈ U, ∀α ∈ LPL, ∀1 ≤ j ≤L− 1,

Ψα(z1, . . . , zL) = Ψα(1/z1, z2, . . . , zL) (21a)

Ψα(z1, . . . , zL) = Ψα(z1, . . . , zL−1, 1/zL) (21b)

j, j + 1 /∈ α =⇒ Ψα

[qzj/zj+1]= πj

Ψα

[qzj/zj+1](21c)

j, j + 1 ∈ α =⇒ [qzj/zj+1]

[zj/zj+1](1− πj)Ψα =

∑β∈LPL \αejβ=α

Ψβ. (21d)

Proof. It suffices to project (19a, 19b, 19c) on |α〉 and distinguish cases.

4.1 Dyck Paths Representation

The Dyck path representation is given by a bijection between LPL and theset of paths D defined as the set of function f : 0, . . . , L −→ Z such that

f ≥ 0

f(0) = 0, f(L) = 1

∀i ∈ 0, . . . , L− 1, |f(i+ 1)− f(i)| = 1

(22)

We define the bijection d : D −→ LPL on any α ∈ LPL by d(α)(0) = 0 and∀i ∈ 0, . . . , L−1 take j such that i, j ∈ α, then d(α)(i+1)−d(α)(i) = 1if j > i and −1 otherwise. It is easy to check that d is indeed a bijection.If α is seen as a well parenthesized expression, d(α)(i) can be interpreted asthe number of open pairs of parentheses just after i (see Figure 17).

We will abuse notation by writing α for d(α) and indexing Ψ by Dyckpaths or link patterns, it will be clear from the context when we do so. For2 ≤ i ≤ j ≤ L− 1, we use the notation α+ xi, jy (or α+ xiy if i = j) for thefunction d(α) + 2× 1i,...,j. For this function to still be in D we must haved(α)(i − 1) > d(α)(i) and d(α)(j + 1) > d(α)(j), it will always be the casewhen we apply this operation. Similarly we define α − xi, jy and α − xiy.Moreover we will consider that α+ xj, iy = α+ xi, jy.

The Dyck path representation induces a partial order on link patternsdefined by

∀α, β ∈ LPL, α ≤ β ⇐⇒ ∀i ∈ 0, . . . , L, d(α)(i) ≤ d(β)(i). (23)

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Figure 17: Example of link patterns with their Dyck path representation.

This partial order will enable us to define the coordinates of Ψ induc-tively. Another reason to use Dyck paths is that the operators ei can beinterpreted graphically and this will allow us to manipulate Equation (21d).We first prove some lemmas to work with Dyck path representations.

Lemma 6. For α ∈ LPL and i, j ∈ 1, . . . , L,∞, i < j, we have

i, j ∈ α⇐⇒ d(α)(i) > d(α)(i− 1) and j = infk > i, d(α)(k) = d(α)(i− 1).

Proof. There must be a unique j matched to i in α, so it is enough to showthat the j defined by the right hand side is matched to i. We can show thisby induction on j − i. If j − i = 1, then the definition of d tells that i ismatched to its right and j to its left. For the pattern to be non-crossing,they must be matched together. The induction hypothesis and the fact thatDyck paths increase or decrease by 1 allows us to deduce that any k with1 < k < j is matched to another k′ also strictly between i and j. Thuswe can reapply the argument that the pattern must be non-crossing andi, j ∈ α.

Lemma 7. Let i ∈ 2, L and α ∈ LPL be such that i, i + 1 ∈ α andd(α)(i) ≥ 2. Then (α−xiy) is a well defined link pattern and ei(α−xiy) = α.

Proof. By definition of d, there exists j < i and j′ > i+1 such that j, i, i+1, j′ ∈ (α − xiy). Let k, k′ ∈ 1, . . . , L,∞ \ j, i, i + 1, j′ be such thatk, k′ ∈ (α−xiy). Lemma 6 tells us that if k < x ≤ k′, d(α−xiy)(x) > d(α−xiy)(k), hence d(α)(x) > d(α)(k). Moreover the value of d on k, k−1, k′, k′−1does not change. So the characterization of k, k′ by Lemma 6 is the same forα and (α−xiy) and k, k′ ∈ α. Obviously we also have k, k′ ∈ ei(α−xiy).Since i, i + 1 appears in both α and ei(α − xiy), then so does the linkbetween the last unmatched indices j, j′.

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Figure 18: The first Dyck path is the representation of some pattern α. If weadd the blue square to the path we get α+xiy. The set Oi(α) = Oi(α+xiy)is represented with red squares. The second Dyck path represents α+xk, iywhere k corresponds to the leftmost red square.

Figure 19: The first two paths represent a pattern α with d(α)(i) < d(α)(i−1) = d(α)(i+ 1) and α+ xiy = eiα. The last two represent α+ xi, ky for thetwo possible values of k ∈ Oi(α).

Lemma 8. Let i ∈ 1, . . . , L− 1 and α ∈ LPL be such that i, i+ 1 ∈ α.Define

Oi(α) = k ∈ 2, L−1 | d(α)(i−1) = d(α)(k) < d(α)(k−1) = d(α)(k+1)

and if i < j < k or k < j < i, d(α)(i− 1) ≤ d(α)(j).

Then

α− xiy + xi, ky | k ∈ Oi(α) = β ∈ LPL | α = eiβ, β 6= α, β 6= (α− xiy)

where by convention β 6= (α−xiy) is verified if d(α)(i) < 2. If d(α)(i) ≥2, define Oi(α− xiy) = Oi(α).

The crucial consequence of this lemma is that all β such that eiβ = αverify β ≥ α except β = α − xiy if it exists. Thus Equation (21d) can beused as a recurrence relation to define Ψ on β = α− xiy using only greaterlink patterns.

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Proof. Let i ∈ 2, . . . , L − 1, α be as in the statement of the lemma andβ ∈ LPL such that eiβ = α (see Figure 19). Suppose that i, i+1 ∈ β, thenβ = eiβ = α. Otherwise, there exist distinct j, j′ ∈ 1, . . . , L,∞ such thati, j, i + 1, j′ ∈ β. If j < i and i + 1 < j′, then d(β)(i) < d(β)(i − 1) =d(β)(i+ 1). Hence by Lemma 7, eiβ = β+ xiy = α, so β = α− xiy. Since βmust be non-crossing, we have otherwise that j′ < j < i or i+ 1 < j′ < j.

By symmetry we assume j′ < j < i. Clearly if k, k′ /∈ i, i + 1, j, j′,k, k′ ∈ β ⇐⇒ k, k′ ∈ eiβ = α. If k < j or k ≥ i then it is precededby the same number of beginnings and endings of links in α and in β, sod(α)(k) = d(β)(k). If j ≤ k < i, then k is preceded by one more beginningand one less ending in β compared to α. Hence d(β)(k) = d(α)(k)+1−(−1).That is, β = α+xj, i−1y. Moreover, since j, i ∈ β, d(α)(j)+2 = d(β)(j) =1 + d(β)(i) = 2 + d(β)(i+ 1) = 2 + d(α)(i+ 1). So j ∈ Oi(α).

Conversely, take j ∈ Oi(α), say j < i, and β = α + xj, i − 1y. There isj′ such that j, j′ ∈ α. By definition of Oi and d, d(α)(j − 1) > d(α)(j) soj′ < j. Using the infimum characterization of Lemma 6 and the definitionof Oi(α), it is easy to see that we must have j′, i + 1, j, i ∈ β and thatall the others links of β are also in α. So eiβ = α.

4.2 Inductive Construction of Ψ

Since all Dyck paths must have L+12 upward steps and L−1

2 downward steps,there exists a greatest link pattern with respect to the order induced byDyck paths, that is build by doing all the upward steps first. We call it τ(for L = 7, it is represented as the last path of Figure 19).

Does there exist an algorithm to inductively build Ψ, going downfrom the value of Ψτ , such that Ψ is a solution Ψ to the qKZ equations(19a,19b,19c) and

Ψτ =∏

1≤i<j≤L+12

k(zj , zi)∏

L+32≤i<j≤L

k(1/zi, zj) (24)

where k : a, b 7→ [qb/a][q/(ab)] ?

References

[BdT12] Cedric Boutillier and Beatrice de Tiliere. Statistical mechanicson isoradial graphs. In Probability in Complex Physical Systems,pages 491–512. Springer, 2012.

[CS11] Dmitry Chelkak and Stanislav Smirnov. Discrete complexanalysis on isoradial graphs. Advances in Mathematics,228(3):1590–1630, 2011.

33

[dB81] Nicolaas Govert de Bruijn. Algebraic theory of penrose’s non-periodic tilings of the plane. Kon. Nederl. Akad. Wetensch. Proc.Ser. A, 43(84):1–7, 1981.

[DC17] Hugo Duminil-Copin. Lectures on the ising and potts models onthe hypercubic lattice. arXiv preprint arXiv:1707.00520, 2017.

[DCT15] H Duminil-Copin and V Tassion. Rsw and box-crossing propertyfor planar percolation. IAMP proceedings, 2015.

[DF05] Philippe Di Francesco. Inhomogeneous loop models with openboundaries. Journal of Physics A: Mathematical and General,38(27):6091, 2005.

[DGP+10] Jan De Gier, Pavel Pyatov, et al. Factorised solutions oftemperley-lieb qkz equations on a segment. Advances in The-oretical and Mathematical Physics, 14(3):795–878, 2010.

[Duf68] Richard James Duffin. Potential theory on a rhombic lattice.Journal of Combinatorial Theory, 5(3):258–272, 1968.

[GM+13] Geoffrey R Grimmett, Ioan Manolescu, et al. Inhomogeneousbond percolation on square, triangular and hexagonal lattices.The Annals of Probability, 41(4):2990–3025, 2013.

[GM14] Geoffrey R Grimmett and Ioan Manolescu. Bond percolation onisoradial graphs: criticality and universality. Probability Theoryand Related Fields, 159(1-2):273–327, 2014.

[IP12] Yacine Ikhlef and Anita K Ponsaing. Finite-size left-passageprobability in percolation. Journal of Statistical Physics,149(1):10–36, 2012.

[Kas61] Pieter W Kasteleyn. The statistics of dimers on a lattice: I. thenumber of dimer arrangements on a quadratic lattice. Physica,27(12):1209–1225, 1961.

[Ken02a] Richard Kenyon. An introduction to the dimer model. LectureNotes ICTP, 2002.

[Ken02b] Richard Kenyon. The laplacian and dirac operators on criticalplanar graphs. Inventiones mathematicae, 150(2):409–439, 2002.

[Mer01] Christian Mercat. Discrete riemann surfaces and the ising model.Communications in Mathematical Physics, 218(1):177–216, 2001.

[PAY06] Jacques HH Perk and Helen Au-Yang. Yang-baxter equations.arXiv preprint math-ph/0606053, 2006.

34

[Pen79] Roger Penrose. Pentaplexity a class of non-periodic tilings of theplane. The mathematical intelligencer, 2(1):32–37, 1979.

[Pon11] Anita Kristine Ponsaing. Finite size lattice results forthe two-boundary temperley–lieb loop model. PhD Thesis,arXiv:1109.0374, 2011.

[TF61] Harold NV Temperley and Michael E Fisher. Dimer problemin statistical mechanics-an exact result. Philosophical Magazine,6(68):1061–1063, 1961.

[ZJ07] Paul Zinn-Justin. Loop model with mixed boundary conditions,qkz equation and alternating sign matrices. Journal of StatisticalMechanics: Theory and Experiment, 2007(01):P01007, 2007.

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