Ising Model: Local Spin Correlations and Conformal Invariance

We study the 2-dimensional Ising model at critical temperature on a simply connected subset Ωδ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega_{\delta}}$$\end{document} of the square grid δZ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\delta\mathbb{Z}^{2}}$$\end{document}. The scaling limit of the critical Ising model is conjectured to be described by Conformal Field Theory; in particular, there is expected to be a precise correspondence between local lattice fields of the Ising model and the local fields of Conformal Field Theory. Towards the proof of this correspondence, we analyze arbitrary spin pattern probabilities (probabilities of finite spin configurations occurring at the origin), explicitly obtain their infinite-volume limits, and prove their conformal covariance at the first (non-trivial) order. We formulate these probabilities in terms of discrete fermionic observables, enabling the study of their scaling limits. This generalizes results of Hongler (Conformal invariance of Ising model correlations. Ph.D. thesis, [Hon10]), Hongler and Smirnov (Acta Math 211(2):191–225, [HoSm13]), Chelkak, Hongler, and Izyurov (Ann. Math. 181(3), 1087–1138, [CHI15]) to one-point functions of any local spin correlations. We introduce a collection of tools which allow one to exactly and explicitly translate any spin pattern probability (and hence any lattice local field correlation) in terms of discrete complex analysis quantities. The proof requires working with multipoint lattice spinors with monodromy (including construction of explicit formulae in the full plane), and refined analysis near their source points to prove convergence to the appropriate continuous conformally covariant functions.

that for β < β c the system is disordered at large scales while for β > β c a long-range ferromagnetic order arises.
In classical discrete probability, the phase transition can be described in terms of the infinite-volume limits: in the disordered phase β < β c only one Gibbs probability measure exists, while for β > β c , measures are convex combinations of two pure measures.It has a continuous phase transition since only one Gibbs measure exists at β = β c .Critical lattice models at continuous phase transitions are expected to have universal scaling limits (independent of the choice of lattice and other details), as (non-rigorously) suggested by the Renormalization Group.The scaling limits of (a large class of) critical 2D models are expected to exhibit conformal symmetry.This can loosely be formulated as follows: for a conformal mapping ϕ acting on the domain Ω ⊂ C, we have ϕ (scaling limit of M on Ω) = scaling limit of M on ϕ (Ω) .
There are two points of view describing the scaling limits of planar lattice models: curves and fields.The curves that arise in conformally invariant setups are Schramm-Loewner Evolution (SLE) curves: they describe the scaling limit of the interfaces between opposite spins.The fields on a discrete level, such as the ±1-valued spin field formed by the spin values, can be described by Conformal Field Theory (CFT): their correlations in principle can be computed (non-rigorously), using representation-theoretic methods.For the 2D Ising model, most of the above program can be implemented rigorously: the interfaces of ±1 spins can be shown to converge to SLE curves, and the correlations of the most natural fields can be shown to converge to the formulae predicted by CFT.
What makes it possible to mathematically analyze the model with great precision is its exactly solvable structure, revealed by Onsager [Ons44].The exact solvability can be formulated in many different ways; in the recent years, the formulation in terms of discrete complex analysis has emerged as one the most powerful ways to understand rigorously the scaling limit of the model.Specifically the model's conformal symmetry becomes much more transparent in this context.
The results of [Hon10] and [CHI13] concerning the (asymptotic) conformal invariance of lattice fields can be formulated, in their simplest instance, as follows: consider the critical Ising model on discretization Ω δ of a simplyconnected domain Ω by a square grid of mesh size δ > 0; put +1 spins on the boundary, take a point a ∈ Ω, identify it to the closest face of Ω δ and let a + δ be the face adjacent to that face.Then, as δ → 0 where C σ , C > 0 are explicit nonuniversal constants and ϕ is a conformal map from Ω to the unit disk D, mapping a to the origin.The 0 and √ 2 2 on the right hand side are the infinite-volume limits (i.e.values on graphs approaching the full square grid Z 2 ) of the quantities of the left hand side.The above results illustrate the relation between the infinite-volume limit description and the field-theoretic description: for a given local field, its correlations are described at first order by the infinite-volume limit, and the corrections are described by Conformal Field Theory quantities.The purpose of this paper is to study more general fields, to compute the infinite-volume limits, and to describe the CFT corrections.
More precisely, we look at local pattern probabilities (e.g. the chance that three adjacent spins are the same, the chance that a given spin is + and that its neighbor is −, etc).In the case of the dimer model, similar results, connecting pattern probabilities with conformal invariance, were obtained by Boutillier in [Bo07].We give a way to compute the infinite-volume limit of such probabilities for the planar Ising model, and we describe the conformally covariant corrections induced by the geometry of the domain up to order δ: where α is 1 for spin-symmetric patterns and 1 8 for spin-sensitive patterns.Another way to formulate our result is the following.The Ising CFT, conjectured to describe the scaling limit of the critical Ising model, contains three primary (conformally covariant) fields: the identity (dimension 0 -a constant field), the spin (dimension 1 8 ) and the energy (dimension 1).We formulate pattern probabilities up to order δ in terms of these three operators (corresponding to the three terms in the above formula).
Our proof relies mainly on discrete complex analysis methods: we use lattice observables, in the form introduced in [Hon10], to connect pattern probabilities with solutions to discrete boundary value problems.We then study the scaling limits of such solutions using discrete complex analysis techniques.The new techniques introduced for this purpose, are: refined discrete analysis of multipoint observables, constructions of lattice spinor observables on the full plane, and refined analysis of convergence of observables.
Applications and Perspectives.Besides giving a general connection between pattern probabilities, Gibbs measures and conformal covariance, our results can be useful in shedding light on specific questions.One such question of Benjamini that served as one of the motivations for this work arises in the context of Ising Glauber dynamics.These dynamics are Markov chains on the space of ±1 spin assignements, their equilibrium measures are the Ising measure.They pick a spin (uniformly at random), flip it with a certain probability, according to the state of the four neighbors, and then repeat this procedure.
A natural field, associated with these dynamics is the flip rate of a spin at a given location.It is tempting to relate it to the energy, as it is intuitively related to the thermal disorder of the system (the more the spin flips, the higher the disorder).Hence, at critical temperature, one would like to relate this flip rate (after correction) to the conformal invariance results of the energy density [CHI13].This paper makes this precise: the flip rate at x ∈ Ω δ depends on the frequency of occurence of the various configurations (patterns) at the five spins x, x ± δ, x ± iδ (in a spin-symmetric fashion).Hence the flip rate (at equilibrium) can be described by Theorem 1.1.
The methods introduced in this article pave the way for additional development, to appear in subsequent papers.Those will include probabilities of occurrence of multiple patterns at macroscopic distance from each other, eventually leading to a full connection between the scaling Iimits of the critical Ising model fields and the content of the corresponding minimal model of Conformal Field Theory [DMS97].The methods of [ChSm11] can also be used to generalize these results to two-dimensional isoradial lattices.
Acknowledgements.Most of this research was carried out during the Research Experience for Undergraduates program at Mathematics Department of Columbia University, funded by the NSF under grant DMS-0739392.We would like to thank the T.A., Krzysztof Putyra, for his help during this program and the program coordinator Robert Lipshitz, as well as all the participants, in particular Adrien Brochard and Woo Chang Chung.
Clement Hongler would like to thank Dmitry Chelkak and Stanislav Smirnov, for sharing many ideas and insights about the Ising model and conformal invariance; Itai Benjamini and Curtis McMullen for asking questions that suggested we look at this problem; Stéphane Benoist, John Cardy, Julien Dubédat, Hugo Duminil-Copin, Konstantin Izyurov, Kalle Kytölä and Wendelin Werner for interesting discussions, the NSF under grant DMS-1106588, and the Minerva Foundation for financial support.
1.1.Notation.Our graph notation is largely consistent with that in [CHI13]; caution is needed since our function notation is distinct from the notation there, mixing in features from [Hon10].
1.1.1.Graph Notation.The Ising model is a model of spin behavior which assigns spins of value ±1 to sites of given arrangement.In this paper, we consider spins on the faces of the discretizations Ω δ (or C δ ) of a given bounded simply connected domain with smooth boundary Ω (or C) via a rotated square graph of mesh size √ 2δ.More specifically (the notations for Ω δ are also used for C δ by putting C in place of Ω in the definitions): • Frequently we represent edges and faces by their midpoints.In particular, the set of medial vertices V m Ω δ is defined as the set of edge midpoints; given an edge e ∈ E Ω δ , m(e) ∈ V m Ω δ is its midpoint, and vice versa for m ∈ V m Ω δ and e(m) ∈ E Ω δ .• Furthermore, we collect the corners, which are points δ/2 off from the vertices in each of the four cardinal directions.
Note that for a τ ∈ {1, i, λ, λ} V τ Ω δ is a rotated square lattice, with nearest pairs of corners √ 2δ apart; in this lattice those corners are adjacent.• Given an edge e = {a, b} ∈ E Ω δ ⊂ V Ω δ , an orientation on the edge is a choice of a unit vector o in the edge direction between a−b |a−b| and b−a |a−b| .In most cases, we also arbitrarily choose one of its two square roots to get a double orientation, denoted ( √ o) 2 in case of ambiguity.We denote an oriented edge (midpoint), meaning a pair of an edge midpoint m = m(e) and a double orientation o thereon, by m o = m ( √ o) 2 .We collect oriented edge midpoints in the set V o Ω δ .Similarly we define (double) orientations of corners, which is fixed by their type, as the unit vector in the direction to the nearest vertex and its square root.
• The domain of choice for the discrete functions in the following sections is the set of both corners and medial vertices, or Ω δ , and E Ω δ respectively.Given a boundary edge, we define the unit normal outward vector v out as the unit vector in the direction of the vertex in C \ Ω viewed from the vertex inside Ω.
For the discrete functions with monodromy which will be introduced in Section 3, we work with graphs lifted to the double cover [Ω, a] of Ω \ {a} (or [C, a] of C \ {a}) for a given fixed point a ∈ Ω (or C).We assume a is a midpoint of a face; if not we can move the grid by less than √ 2δ so that it is the midpoint of a face.The convergence results we quote require Hausdorff convergence of the discrete domains to the continuous one-thus they hold regardless since as δ → 0 the shifted grid still converges to the continuous domain in the Hausdorff sense.
• To identify the branches of the double cover [C, a] we use the function √ z − a which is naturally defined on it; the left-slit plane X := C \ (a + R − ) lifts to two branches, X + and X − , the former where Re √ z − a is positive and the latter where it is negative.Similarly, the right-slit plane Y := C \ (a + R + ) lifts to Y + and Y − , the superscripts noting the sign of Im √ z − a.On the discrete level, define the lift of V 1 Ω δ to X ± as X ± δ , and the lift of • Our functions of interest on the double cover will be spinors, or functions with monodromy −1 around a; that is, we want functions that switch sign when one goes from a point on the double cover to the other point on the double cover that maps to the same point under the covering map.1.1.2.Ising Model.We consider the Ising model on the faces of Ω δ : a configuration σ assigns ±1 spins to each face in F Ω δ .
• σ i ∈ {±1} is the spin assigned to the face i ∈ F Ω δ .
• Each configuration σ has energy associated with it, defined as E(σ) = − i∼j σ i σ j , with i ∼ j denotes that i and j are adjacent.The Ising model declares the probability of σ, P(σ), as being proportional to e −βE(σ) at the inverse temperature β > 0. • We only consider the critical model at inverse temperature β c = 1 2 ln( √ 2 + 1).• If an edge e is incident to the faces i and j, define the renormalized energy density field at e as (e) := µ − σ i σ j , where µ = √ 2 2 , the infinite-volume limit, is defined so that the expectation of the full-plane energy density goes to 0 as δ → 0 ( [HoSm10]).
• Given a set of edges B (in particular, a spin-symmetric pattern defined in 1.13), we define the energy density of B as (B) := e∈B (e).A spin-sensitive pattern is an assignment of spins ±1 to the faces F (B); in other words, the pattern is an element of the set {−1, 1} F (B) .A corresponding spin-symmetric pattern is the union of the spin-sensitive pattern The spin-sensitive pattern associated with and its exact negative (i.e. the spin at every face is flipped).The simplest example of a spin-sensitive pattern is the event {σ a = +} while the simplest example of a spin-symmetric pattern is the event {σ a = σ a+δ }.
We consider the probability of such patterns occuring at a.In order to exploit the existing tools that compute the expectation of various energy densities, defined on edges, we prefer a notation based on the presence of edges.Using the low-temperature expansion, we identify each spin-symmetric pattern with an edge subset B ⊂ B by letting e ∈ B if and only if e separates faces whose spins are different; this will be our notation for a spinsymmetric pattern from now on, with P Ω δ (B) denoting the probability that the spin-symmetric pattern B appears on Since F (B) is connected, taking a spin-symmetric pattern given by B and fixing the spin at a to be ±1 specifies a spin-sensitive pattern.As such we define a spin-sensitive pattern by [B, σ a ] with B defined as before and σ a the spin at a. 1.1.4.Convergence.A family of functions {F δ : Ω m δ → C} δ>0 converges on compact subsets to the continuous function f : Ω → C if given any compact subset K ⊂ Ω in the continuous domain for all > 0, there exists a δ > 0 1.2.Main Results.In this section we present our main results regarding conformal invariance results of spinsymmetric and spin-sensitive patterns.We first present the conformal invariance result on spin-symmetric spin pattern probabilities, a result that follows from the generalization of [Hon10] to edges O(δ) apart from each other.
Theorem 1.1 (Conformal Invariance of Spin-Symmetric Pattern Probabilities).Given a base diagram B δ in Ω δ centered about a, and a spin-symmetric pattern B ⊂ B δ , where the convergence is uniform for B away from ∂Ω. a, B Ω is an explicitly defined function such that given a conformal map ϕ : Here P Z 2 [B] is the infinite-volume limit defined as Remark.Since we are at critical temperature the limit exists and is unique.
We have a similar formulation of our main result for spin-sensitive pattern probabilities.
Theorem 1.2 (Conformal Invariance of Spin-Sensitive Pattern Probabilities).Given B, a spin-symmetric pattern on the base diagram B δ centered about point a in Ω, and a designated spin σ a = ±1, where the convergence is uniform away from ∂Ω. a, [B, σ a ] Ω is a function such that given a conformal map ϕ : Ω → ϕ(Ω) we have is defined as before as the infinite volume limit, Remark 1.3.By taking ϕ : D → Ω to be the conformal map from the unit disk to our domain with ϕ(0) = a, we have the following formula for the renormalized spin-sensitive pattern probability: where C [B,σa] is an explicit lattice and pattern dependent constant, and rad(a, Ω) is the conformal radius of Ω as seen from a.That is, rad(a, Ω) = |ϕ (0)|.
1.3.Proof Strategy.In this section we outline the strategy for proving our main results, Theorems 1.1 and 1.2.We follow the proof structure of [Hon10].That is, we introduce functions (discrete fermionic observables) defined on the discretized domain that take specific values related to the probability of the presence or absence of an edge in the graph.We then associate the presence or absence of edges with spin patterns on the graph and examine the continuous limits of the discrete observables.The limits, defined on the continuum obey conformal covariance properties and thus yield conformal invariance results on spin-pattern probabilities.Specifically, we begin, in Section 2, with an overview of important definitions and results of discrete complex analysis: we construct discrete observables that satisfy certain Riemann boundary conditions with continuous counterparts.In this section, we also define the full plane discrete observables that are discrete holomorphic with a singularity and vanishing at ∞.We do so for both the complex plane, C, and its double cover with ramification at a point a ∈ C. In particular the construction of the full plane observable on the double cover with ramification at a and singularity at b is crucial to our result and relies on the explicit construction of the discrete harmonic measure on the slit plane, done in Appendix A. In constructing the full plane spinor we first argue there exists an infinite-volume limit of the fermionic spinors defined on Ω δ , then that they converge to a continuous function in the scaling limit.The full plane fermionic observables allow us to cancel out the singularities of the functions we construct in Section 3 and prove their convergence to continuous conformally covariant functions.
In Section 3, we first define the fermionic observable on Ω δ as first introduced in [Hon10].Specifically we show that the results of [Hon10] extend to adjacent source points and thus the multipoint discrete fermionic observable F Ω δ can be expressed in terms of the energy density E[e 1 ...e m ] for adjacent edges.This allows us to relate the discrete fermionic observable to the spin-symmetric pattern probability P Ω δ [B].In order to extend this to spinsensitive patterns we construct a multipoint version of the fermionic spinor of [CHI13], F [Ω δ ,a] living on the double cover of Ω δ with monodromy around a. Unlike the function of [CHI13] the function we construct has points of singularity away from a point of monodromy so that we can consider a collection of edges as our pattern.We follow the same procedure as for F Ω δ to relate this discrete spinor to the spin-sensitive pattern probability P Ω δ [B, σ a ].This section concludes with a Pfaffian relation that defines the multipoint fermionic observables in terms of their two point counterparts.
Section 4 is the core of the paper in which we prove convergence, as the mesh size δ goes to zero, of ) to a conformally covariant function.In [CHI13] such convergence was proven for a spinor with singularity located at the point of monodromy a.However, the multipoint spinors we introduce have points of singularity at all their source points, possibly all away from the point of monodromy.After expressing them using the Pfaffian relation from Section 3 in terms of two-point functions, the points of singularity approach the point of mondromy as δ → 0. We then prove convergence of these two-point functions to conformally covariant continuous functions up to higher orders of δ.
Finally in Section 5, we combine the results from Section 3 and Section 4 to prove the main results of the paper, Theorems 1.1 and 1.2.Using the Pfaffian relations from Section 3 and the relation between the discrete observables and edge pattern energy densities, we express the energy density of an arbitrary spin-symmetric pattern in terms of two-point fermionic observables.Then using the convergence results of [Hon10] and Section 4 to conformally covariant functions, we deduce that the energy densities of spin patterns converge to conformally covariant quantities.We use Appendix C to deduce conformal invariance of spin pattern probabilities in Theorems 1.1 and 1.2.

Discrete Complex Analysis and Full-Plane Observables
2.1.Discrete Complex Analysis.We introduce here basic notions of discrete complex analysis on the square lattice and introduce useful full-plane auxiliary functions H C δ , H [C δ ,a] .Note that we denote discrete functions by upper-case alphabet letters; the continuous counterparts will be denoted by the same lower-case alphabet letters.
Definition 2.1 (s-holomorphicity, [Smi07]).A function K : where P τ R denotes projection onto the line τ R. K is s-holomorphic on Ω δ if it is s-holomorphic at each x ∈ V c Ω δ .For a function K defined on the double cover [Ω δ , a], choose an open ball V ⊂ Ω \ {a} around x, whose preimage by the covering map will give two disjoint copies is s-holomorphic for any such x, V , and i.This way of defining complex analysis notions on double covers will be frequently implied, with a point in the planar domain being identified with one of the corresponding points in the double cover.
Remark 2.2.A more obvious candidate for the discrete notion of holomorphicity, called discrete holomorphicity, is implied by the stronger notion of s-holomorphicity.Specifically, it is easily seen that an s-holomorphic function Ω δ where the expression makes sense.The values on V 1,i Ω δ in fact determine an s-holomorphic function uniquely: given a function K defined on V 1,i Ω δ such that the above discrete holomorphicity relations hold.
We note that a family of s-holomorphic function converges on compact subsets to the continuous function f if and only if the restrictions converge on compact subsets respectively to Re f and Im f , where the notion of convergence on (a fixed single type of) corners is defined analogously to that on medial points.Definition 2.3 (Discrete Harmonicity).For a fixed τ ∈ {1, i} a function L : The function L is discrete harmonic if it is discrete harmonic at every point.
Remark 2.4.We only introduce the harmonicity notion on C δ , but it easily generalizes to Ω δ with some modification of the definition in case of boundary points; see [ChSm11] for details.Discrete holomorphicity implies discrete harmonicity of restrictions-the following can be checked straightforwardly: Proposition 2.5 (Proposition 3.6, [ChSm12]).The restriction of an s-holomorphic function on a] for some τ ∈ {1, i} is discrete harmonic at every point except at a ± δ 2 .Remark 2.6.As seen in Remark 2.2, an s-holomorphic function is defined by its discrete holomorphic restriction to V 1,i Ω δ .In fact, the full restriction to V 1,i Ω δ can be recovered by the restriction on just one type of the corner up to a constant.If one has a harmonic function on V 1 Ω δ and a fixed value on any one point in V i Ω δ (or vice versa), there is a unique harmonic function on , called the harmonic conjugate, which has the specified value at the point and forms a discrete holomorphic function together with the prescribed values on V 1 Ω δ (or V i Ω δ ).Harmonicity of the prescribed function ensures that the values defined by sums along different lines are well-defined (see Lemma 2.15 of [ChSm11]).
Discrete harmonic functions on a domain, like their continuous counterparts, are unique given a certain set of boundary conditions.Here, we present a discrete complex analysis version of such a result concerning s-holomorphic functions.
Proposition 2.7 (Corollary 29, [Hon10]).A solution K δ of the discrete Riemann boundary value problem on Ω δ with boundary data L : ∂V m Ω δ → C, which is a discrete function defined on V cm Ω δ satisfying the conditions Proof.In applying the proof of Proposition 28 in [Hon10] (which estimates the "area integral" of K 2 δ , which is the sum of all medial values, by the "line integral" of L 2 , or the sum thereof along the boundary), the only tool needed is the superharmonic antiderivative of K 2 δ ; its existence is shown in Remark 3.8 of [ChSm12].2.2.Full-plane Observables.We now present full-plane versions of various observables constructed in Section 3.These functions encode the infinite-volume limits that the model converges to when the domain approaches C δ ; from a functional point of view, they have discrete singularities, which are used to cancel out the same singularity types in the domain-dependent observables when applying uniqueness and convergence results (examples of which we saw above in Propositions 2.7, 2.8) concerning everywhere s-holomorphic functions.
To use procedures outlined in Remarks 2.4 and 2.6, we define an important harmonic function which can be defined on most discrete domains: Definition 2.9.Given medial vertices a 1 , a 2 and an orientation o ∈ O 2 on a 1 , we define the full-plane fermionic observable where G(a 1, a 2 ) = G(0, a 2 − a 1 ) and G(0, w(1 + i)) := 2C 0 (0, 2w), and where C 0 is the coupling function defined in Section 5 of [Ken00].
a2−a1 on compact subsets of Ω × Ω away from the diagonal.
Remark 2.10.By Thoerem 87 of [Hon10], the full-plane fermionic observable is the unique s-holomorphic function on Note that an otherwise s-holomorphic function that is possibly singular at an edge is in fact s-holomorphic there if and only if the residue at that edge is zero.
On [Ω δ , a], [CHI13] gives a formulation of an analogous function on the double cover based on harmonic measure and its various estimates.For a discrete domain Λ and A ⊂ ∂Λ, the harmonic measure of Λ as seen from z hm Λ A (z) is the probability that a random walk starting at z hits A before hitting ∂Λ\A, which, across all z ∈ Λ, happens to be the unique harmonic function on Λ with boundary value 1 on A and 0 elsewhere.We explicitly compute the version we will use (our needed harmonic measures are on graphs isomorphic to C δ \ R + ): Theorem 2.11.The discrete harmonic measure on the slit complex plane C δ \R + is given by hm where C(θ) := cos θ 1 + | sin θ| and the square root takes the principal value.In particular, hm (2sδ), which asymptotically gives hm Theorem 2.12 (Theorem 2.15, [CHI13]).There is a unique s-holomorphic spinor on Y ± δ , and zero on corners missed by either X ± δ or Y ± δ .The two functions respectively converge to Re 1 √ z−a and i Im 1 √ z−a uniformly on compact sets, and extend s-holomorphically to V cm [C δ ,a] per our Remark 2.2.Theorem 2.11 provides the asymptotic estimate.Note it is impossible to extend the function s-holomorphically to a + δ 2 ; the projections from the two adjacent midpoints have opposite signs.Now we define the two-point versions of the spinors; they are spinors with discrete singularities which can be moved to points other than a + δ 2 .Analogous to H C δ , our goal is to provide functions with singularities at edge midpoints which have the same behavior as H [Ω δ ,a] , to be defined in the next section.We overload the notation H [C δ ,a] in order to refer to the above defined spinor and the various two-point functions below; the definitions are clearly distinguishiable once one identifies whether the argument(s) are corners or edge midpoints.Write a] }, where the function is not defined: Theorem 2.13.There is a unique function a] a distance O (δ) from a as δ → 0, we have, uniformly over z on compact subsets away from a, for some The proof proceeds in two steps done in Appendix A. We first prove the existence and uniqueness of such a] with the above properties in A.4.Then we prove convergence to Theorem 2.14.There are s-holomorphic spinors a]   converge on compact subsets respectively to spinors g Proof.Theorem 2.16 of [CHI13] proves this for ) on X ± δ , zero elsewhere; [CHI13]'s objective is just to get the real corner harmonic restriction of the full s-holomorphic function.We refer to its proofs regarding convergence and harmonicity of the series; but for the limiting function we note that the sum is close to the Riemann sum of Re 1 2 √ z−a that approaches Re √ z − a, identifying the sum.
For the imaginary part, we define 2 ) = ∓i respectively), zero elsewhere.The proofs for convergence to Im √ z − a and harmonicity are exactly analogous; to get an s-holomorphic extension to V cm [C δ ,a] , we need to check discrete holomorphicity.We use the explicit formula for the harmonic measure for this, and we do it in the appendix, Corollary A.3.
For G, we analogously define a] ; the properties are checked exactly analogously.

Discrete Holomorphic Observables
In this section, we define discrete holomorphic observables on Ω δ and [Ω δ , a] and prove the relations linking them to various energy densities which form the pattern expectation vector.For the rest of the paper, as hinted at in the introduction we will refer to the discrete observable F Ω δ defined on Ω δ as the discrete fermionic observable.We will refer to the discrete observable F [Ω δ ,a] defined on [Ω δ , a], the double cover of Ω δ with monodromy at point a, as the discrete fermionic spinor.
3.1.Discrete Fermionic Observable with no Monodromy.We first reintroduce the discrete fermionic observable with no point of monodromy that was presented in [Hon10].We generalize the observable and the most relevant results from [Hon10] to the case when source points are adjacent.This allows us to compute expectations of energy densities of edges even when the edges are O(δ) apart from each other.Then the observable can be related to probabilities of arbitrary spin-symmetric patterns.Remark 3.1.Here we introduce some definitions that are borrowed from [Hon10] and [CHI13] useful for defining the discrete fermionic observables on Ω δ and [Ω δ , a].A point b i on V Ω δ are medial vertices on the graph Ω δ , each assigned a double orientation in O 2 , o ∈ {±1 ± i, ∓1 ± i} with a fixed square root branch.A walk ω between two points b i and b j is a collection of edges and half edges such that there is exactly one edge at each of b i and b j and every other vertex in Ω δ has an even number of incident edges.For well-definedness, we follow the conventions of [Hon10] and force the walk to go right any time it intersects itself.The winding number W (ω) represents the number of loops around a point the path ω makes, as is typical of CFT partition functions.c(γ 1 , ..., γ n ) is the crossing signature induced by the walks γ i : if we link 1, ..., 2n on R by simple paths in the upper half plane that connect the endpoints of each of the n γ i , the number is well defined modulo 2 and is what we define to be c(γ 1 , ..., γ n ) .For more rigorous definitions of the winding number and crossing signature see 5. where 2n ) = {edges and half-edges that form walks between b oi i 's and loops} It has been proven that n i=1 φ(γ i ) is well-defined for various admissible choices of walks and thus Hon10]).Given a collection of doubly oriented points (• • • ), a collection of signed edges e 1 , ..., e m disjoint from (• • • ), define the restricted real fermionic observable Further, given a collection of doubly oriented points (...), a collection of signed edges {...} disjoint from (...) and edges e 1 , ..., e m , define the fused real fermionic observable F for any branch choice of o 2n .From here we can define the complex fermionic observable Proposition 3.6.Let e 1 , ..., e m be a set of possibly adjacent interior edges.Then we have Proof.This is proven for general boundary conditions, not just plus boundary conditions in [Hon10] 5.3.
) has a discrete simple pole at b j , with front and rear values given by: where e(b j ) denotes the edge whose midpoint is b j .Further, the function ) can be extended to an s-holomorphic function Proof.This proof follows immediately from its proof where the edges and points are not allowed to be adjacent in Prop.76 of [Hon10] and Prop 3.7 which generalizes the projection relations to the case when they are adjacent.Proposition 3.9.For distinct edges e 1 , ..., e m and distinct doubly oriented medial vertices b o1 1 , ..., b o2n 2n , for each choice of orientations q i ∈ O(e i ) : 1 ≤ i ≤ m we have that where we associate with e i the medial vertex on the edge and where the 2p × 2p matrix, p = m + n, is defined for (not necessarily distinct) doubly oriented vertices x ξ1 1 , ..., x ξ2p 2p by Proof.Since everything else has been generalized from [Hon10] to the case when points are adjacent and the proof of the Pfaffian relation for fused fermionic observables only relies on previously generalized Lemmas and Propositions due to the corner formulation of the projection relations, we have the same proof as in section 6.6 of [Hon10].Thus we have reduced all problems for the real multipoint fermionic observable with points of possibly adjacent singularity to computations of two point fermionic observables.
Remark 3.10.From the above Pfaffian relation on the fused fermionic observable we can relate the expectation of energy density of edges to Pfaff(A Ω δ ) as ).
3.2.Discrete Fermionic Spinor with Monodromy.We now construct a multipoint fermionic spinor with a point of monodromy around a ∈ Ω building off the single point spinor introduced in [CHI13] on the double cover of the complex plane.Our extension of the single-point spinor to its multipoint equivalent is based on the extension of [HoSm10] to [Hon10].We move the source points away from the point of monodromy and allow for multiple source points and multiple end points.This allows us to compute expectations of energy densities of arbitrary edge collections while preserving information about σ a obtained by setting the point of monodromy at a. We will eventually use convergence results on this fermionic spinor with monodromy to get expectations of spin-sensitive patterns.
Remark 3.11.Since we are now working on the double cover of the complex plane, [C δ , a] there is some additional notation to introduce, borrowed from [CHI13].We define the double cover of a domain Ω δ ⊂ C δ with point of monodromy around a ∈ Ω δ as in the introduction.We then define the loop number #L(w, a) as the number of loops around a (i.e. the number of loops such that a is a face contained in the interior of the loop in Ω δ where the loops are viewed not on the double cover but on the principal domain).The sheet number is a multiplicative factor of ±1 indicating whether a walk ends on the same sheet as it began or on a different sheet.It is defined such that one full loop around a ends on the lift of the starting point to a different sheet and is thus responsible for the spinor property of the fermionic observable with monodromy.For a more rigorous definition of the loop number and the sheet number Sheet(w) refer to [CHI13].
Definition 3.12.Given a domain Ω δ and its double cover : The loop number of the above collection of walks and loops is 1 since there is precisely one loop with a in its interior.
Proof.The proof of this theorem is a lengthy set-theoretic proof that we relegate to the Appendix.See Appendix B.2 for the proof of well-definedness of the multi-point fermionic spinor.
Proof.We leave the proof of this to Appendix B.2 as it is simply a generalization of [Hon10] to the multipoint spinor.
for any branch choice of o 2n .From here we can define the complex fermionic observable where v out (b 2n ) is defined as the outer normal to the boundary at b 2n .
Proof.We leave the proof to the Appendix B.2.
Further, for each j ∈ {1, ..., 2n − 1} such that a j ∈ ∂V m Ω δ , the function ) can be extended to an s-holomorphic function at b j by setting the value at b j to Proof.We leave the proof to Appendix B.2.
Proposition 3.20.For distinct edges e 1 , ..., e m and distinct doubly oriented medial vertices b o1 1 , ..., b o2n 2n , for each choice of orientations q i ∈ O(e i ) 1 ≤ i ≤ m we have that where we associate with e i the medial vertex on the edge and where the 2p × 2p matrix, p = m + n, is defined for (not necessarily distinct) doubly oriented vertices x ξ1 1 , ..., x ξ2p 2p by Proof.We leave the detailed proof of several propositions that lead to this and this proposition to Appendix B.2.This reduces all values of the multipoint fermionic spinor to a Pfaffian of values of the two-point fermionic spinor for which we have a full plane counterpart F [C δ ,a] (x ξi i , x ξj j ) with the same singularity as F [Ω δ ,a] (x ξi i , x ξj j ).Remark 3.21.From the above Pfaffian relation on the fused fermionic spinor we can relate the expectation of energy density of edges to Pfaff(A [Ω δ ,a] ) as ).

Convergence of Observables
We apply discrete complex analysis results mainly cited from [HoSm10, Hon10, CHI13] in this section in order to prove convergence on the discrete fermionic spinors.

Riemann Boundary Value Problems.
Proposition 4.1 ([Hon10], Proposition 48; [CHI13], Remark 2.9).Given Ω ([Ω, a]), there is at most one solution k Ω , (k [Ω,a] ) for each of our continuous Riemann boundary value problems, which look for functions respectively satisfying given boundary data l : ∂Ω → C (l [Ω,a] : [∂Ω, a] → C with monodromy -1): These sets of boundary conditions, being analogous to the discrete boundary value problems, specify a unique limit for our observables, which solve the discrete problems.This is also where the conformal invariance arises; if φ is an injective conformal map on Ω, it is easy to see that h 1 , a 2 ) on compact subsets of Ω × Ω. 4.2.Analysis near the singularity.In this section we analyze the convergence of our s-holomorphic spinors at points order δ away from the point of monodromy a.Much of the notation used in this Section is from Section 2 where we constructed the full plane fermionic spinor δ , zero everywhere else, and similarly on Y ± δ .For details of the definitions of these full plane spinors, refer to Section 2. The s-holomorphic fermionic spinor H [Ω δ ,a] is defined as in Section 3 of the paper.
uniformly on compact subsets away from a. Proof.We note that the proof of Proposition 3.9 from [CHI13] can be applied without a problem, since a − δ + vδ scales with δ and thus for any > 0 can be included in the -ball around a if δ is small enough; a tool we possibly not have is the integral I δ := I δ (H δ ) of the square of H δ := H [Ω δ ,a] ((a − δ + vδ) o , •) defined as in Proposition 3.6 in the same paper.I δ is well-defined locally, but global well-definedness is a priori a problem because of the simple pole at a − δ + vδ and we have to define I δ away from it, thus creating a possible monodromy around the pole.However, we note that the monodromy of I δ along ∂Ω δ is zero because the increment along the boundary involves precisely adding over Im(H 2 δ v out ) = 0. Since Ω is simply connected, the only nontrivial loops when going around the pole are those homotopic to ∂Ω δ , and since those do not contribute any monodromy, I δ is globally well-defined on any compact subset away from a.
The other potential problem in applying the technique in the proof is the following: as noted in Remark 3.8 in the same paper, subharmonicity of the integral of the square of •), crucial in uniformly bounding the integral near a, fails at a because the spinor branches at the point.The subharmonicity at a is used twice in the proof: in extending the uniform boundedness apart from a to a, and in the proof of Lemma 3.10 in the same paper.Lemma 3.10 is easy; they already prove the Lemma for H δ near a branch point without singularity.We can utilize the same technique for H † δ and deduce the result for H δ (for small enough δ, z max (for converges as δ → 0 (Theorem 2.13), and the argument of the proof goes through).
To show H † δ is bounded near a, we use a simple generalization of Theorem 2.17 in [CHI13]; the proof is exactly the same as our Theorem 4.6, but it only depends on the convergence result for the one-point H [C δ ,a] from the same paper.The result we need states that where H [Ω δ ,a] (•) is a one-point spinor, defined the same to our two-point spinor but with a different choice of normalization and only with paths originating from a + δ 2 and culminating at another, A (to be defined in Remark 4.5) is a [Ω, a]-dependent constant, and the second equality comes from the fact that by definition.It is obvious from the defintion of the one-and two-point spinors that 2 ) vanishes at a + δ 2 and thus I δ (H † † δ ) is subharmonic everywhere including at a by Remark 3.8 in [CHI13].In addition, its uniform boundedness near a (which now can be proved with the technique in [CHI13], Proposition 3.9) is equivalent to uniform boundedness of depends on the domain [Ω, a].However, for ease of notation, we drop the [Ω, a] and just note that the constant depends on the domain.
where A is defined as in Remark 4.5 and Proof.We closely follow the strategy in the Section 3.5 of [CHI13].Note, since the spinor a] is s-holomorphic, by Remark 2.2 it suffices to show the asymptotic behavior not directly at a − δ + v 2 δ on 1, i-corners a − δ + vδ for v ∈ V c C1 .Write R for reflection across a + R; there is a small neighborhood Λ around a in Ω ∩ R(Ω).When a is the midpoint of a face, R(C δ ) = C δ , and thus Λ δ = Λ ∩ C δ is naturally a subgraph of both Ω δ and R(Ω δ ).Since for small enough δ we can assume Λ δ , we restrict our attention to Λ δ , where . Now define functions s-holomorphic everywhere on [Λ δ , a]: a] .
By construction, we have, on , and on a + R + , H δ = H (R) δ (the conjugation comes from the winding angles being negated, while the additional negative sign across the monodromy comes from the sheet factor).This, as well as the construction of functions near a (combined with spinor property, estimating those restrictions will estimate the magnitude of the function on the double cover): without loss of generality we show the o(δ) estimate for ϑ(δ)S 1 δ , since all the restricted harmonic functions are defined on a slitted planar square lattice which are isomorphic locally around a, and all the estimates go through in the other lattices exactly the same way.
Define the discrete circle w(r) := {z ∈ Domain(S 1 δ ) : r < |z − a| < r + 5δ}.A similar twist of the discrete Beurling estimate (Theorem 1, [LaLi04]) as Lemma 3.3, [CHI13] gives hm reversing time on which gives hm , an estimate at a − δ + vδ of a harmonic function identically 1 on w(r).Comparing with S 1 δ on w(r) and applying maximum principle in the interior gives where we use the fact that h (R) [Ω,a] = h [Ω,a] • R since the right hand side solves the boundary value problem.So we have (knowing ϑ(δ) ∼ δ 1/2 when δ is small) 1 and A is defined as before.
Proof.Note by definition of F [Ω δ ,a] for a two point function it is only a multiple of . We simply absorb this constant into C v o 1 1 so that Corollary 4.7 follows immediately from Theorem 4.6.This will be the convergence result we use in Section 5 to prove Theorems 1.1 and 1.2. 5. Proof of Theorems 1.1 and 1.2 5.1.Proof of Theorem 1.1: Spin-Symmetric Pattern Probabilities.In this section we complete the proof of our main results beginning with Theorem 1.1 on conformal invariance of spin-symmetric pattern probabilities.We use Pfaffian relations on our discrete holomorphic obsevables from Section 3 and the convergence results from [Hon10].
In order to do this we first introduce some simple notation for the Pfaffian of a 2m × 2m matrix.
Remark 5.1 (Pfaffian Notation).Call the partition of {1, ..., 2m} into pairs {i k , j k } k , π ∈ Π where Π is the set of all (2m − 1)!! partitions.sgn(π) will be the sign of the associated permutation mapping 1 to i 1 , 2 to j 1 , 3 to i 2 , 4 to j 2 and so on.Then the definition of the Pfaffian is given as Each of the a i k j k is a function of the pair (x Call Π 0 ⊂ Π the subset of permutations where for all k, x i k = x j k and Π 1 ⊂ Π the subset of permutations with exactly one k such that x i k = x j k .Proposition 5.2.Given a collection of edges e 1 , ..., e m with double orientations q 1 , ..., q m , )) with the matrix A Ω δ defined as before Proof.This follows immediately from Remark 3.10.
Proposition 5.3.For a given set of adjacent edges e 1 , ..., e m centered about the face a, as before call ). Then where ] mirroring earlier notation on our fermionic observables.Proof.We prove these relations by expanding the Pfaffian of the matrix A Ω δ .
The first convergence is immediate by taking Ω δ → C δ so terms of F C δ Ω δ vanish.Now to get the second statement, we notice that and thus We then subtract this term, the infinite-volume limit, from both sides and renormalize by dividing by δ.Taking δ → 0, gives The sum over all partitions times the sign of the partition is independent of the choice of orientation at a.
where given a conformal map ϕ : Ω → ϕ(Ω), a o , B Ω defined as , Proof.The first convergence result was shown in the proof of the previous proposition.
For the second convergence result, we note that F C δ (x i k , x j k ) is a full plane fermionic observable and thus only depends on the relative positions of the edges, or the base diagram B we are examining.Hence a, B is solely a function of the location of a and the base diagram B centered at a.
The conformal invariance follows directly from the fact that which is a result of the conformal covariance relation on f C Ω , Proposition 92 in [Hon10].
Theorem (Restatement of Theorem 1.1).Given B, a connected spin-symmetric pattern on the base diagram B centered about point a in Ω, where a, B is a linear combination For the second convergence result we multiply M by the difference of the domain expectation matrix and the full plane expectation matrix.Denote this matrix by Since all the subdiagrams of B are also centered about a as δ → 0, we have that for some k, l This completes the proof of the first part of our main result, the conformal invariance of probabilities of spinsymmetric patterns in the planar Ising model.5.2.Proof of Theorem 1.2: Spin-Sensitive Pattern Probabilities.We now proceed to the proof of the general case of spin-sensitive patterns in the planar Ising model.The proof of Theorem 1.2 will be similar to the proof of Theorem 1.1 except it will use results on the discrete fermionic spinor with monodromy at a and new convergence results from Section 4. Proposition 5.5.Given a collection of edges e 1 , ..., e m and double orientations q 1 , ..., q m ∈ O 2 and a face a in Ω δ , we have ), where the matrix A [Ω δ ,a] is defined as before as Proof.This follows immediately from Remark 3.21.
Proposition 5.6.For a given set of adjacent edges e 1 , ..., e m centered about the face a in Ω δ , as before call ). Then where . Furthermore, we have the higher order convergence result Proof.Just as in the spin-symmetric case, we expand the Pfaffian to find the delta dependence of the expectation of the energy density.
From [CHI13] we have that This implies For the higher order convergence result, first note from Corrolary 4.7 that asδ → 0, where the indices i, j indicate that the constant depends on the location of x i and x j .We have a similar formulation for the scaling limit of a] .We absorbed the full plane observables from a] into the constants since we focus on how they scale as δ → 0. Then subtracting the equation for the δ 1 8 -dependence of the expectation from both sides and dividing by δ yields the higher order convergence result.
Proposition 5.7.Consider the base diagram B centered about a face a in Ω.Given a conformal map ϕ : Ω → ϕ(Ω), where Proof.The first statement follows immediately from the previous proposition.
The next order conformal covariance result follows from the conformal covariance relation on σ a Ω from [CHI13]: Theorem (Restatement of Theorem 1.2).Given B, a spin-symmetric pattern on the base diagram B centered about point a in Ω and a spin σ a ∈ {±1} at a, where a, [B, σ a ] is a linear combination of spin-symmetric and spin-sensitive expectations of energy densities on subdiagrams of B such that given a conformal map ϕ : Proof.We have a matrix M that acts on a matrix of 2 n−1 spin-sensitive energy densities and 2 n−1 spin-symmetric energy densities of subdiagrams of B, B 1 , ..., B 2 n−1 denoted here by For the first convergence result multiply the matrix by the first convergence result of the combination of the previous proposition and the first convergence result of Theorem 1.1.Since the spin-sensitive probability is a linear combination of spin-symmetric and spin-sensitive expectations both of which converge to their full plane counterparts, the spin-sensitive probability also converges to its full plane counterpart, For the second convergence result we multiply it by the difference of the domain expectation matrix and the full plane expectation matrix (infinite-volume limit).Denote new this matrix by Since all the subdiagrams of B are also centered about a as δ → 0, we have that for any spin-sensitive pattern [σ a = ±, B] for some k, l for any conformal map ϕ : Ω → ϕ(Ω).
The convergence on the second line follows from the fact that for any spin-symmetric pattern, and its lowest order convergence term is δ so even after normalization by δ 1 8 all higher order terms vanish.We are only left with contributions from spin-sensitive energy densities which have the same conformal covariance relation (multiplication by |ϕ (a)| 1 8 which yields conformal invariance of spin-sensitive pattern probabilities.
Appendix A. Full Plane Fermionic Spinor and Harmonic Measure We now explicitly construct the discrete harmonic measure in the slit plane, using Fourier analysis analysis techniques (see also [CHI14]): Proposition A.1.The harmonic measure H 0 , defined explicitly by where C(θ) := cos θ 1 + | sin θ| and the square root takes the principal value, is the unique harmonic function on the discrete diagonal slit plane C δ=1 \ Z + with boundary values 1 at the origin and 0 elsewhere on the cut and ∞.
Proof.Since the solution to the Dirichlet problem for the discrete Laplacian is unique, it suffices to check the boundary values and check that the given function is harmonic.On the real axis, we have k = 0, and by the generalized binomial theorem so the s-th Fourier coefficient, which is precisely H 0 (−s), vanishes for odd and positive even s, and H 0 (0) = 1.
For the boundary estimates at infinity, we want to show H 0 → 0 uniformly in s as |k| → ∞, and vice versa.For the former, we use dominated convergence: ↓ 0 pointwise a.e. and , which is integrable.For the s → ∞ estimate, without loss of generality we show that H0 (s + ik) = ´π 0 if g is any smooth function on [0, π], we can integrate by parts and get Since smooth functions are dense in L 1 , we can choose a g such that the first term becomes less than /2 for a given > 0. In controlling the remaining terms with s, the only dependence on k is in the last term; we show it is in fact uniformly bounded: where we make the substitution φ := π 4 − θ 2 .
Harmonicity is easy to prove at points with k = 0 once we notice that for those points m,n∈{1,−1} where the last term vanishes for positive s, in other words, the following function has no negative Fourier modes: Remark A.2.The harmonic measure on the slit plane with boundary value 1 at m ∈ Z + and 0 elsewhere, denoted H m , can be obtained by the recursion relation H m (z) = H m−1 (z − 1) − H m−1 (−1)H 0 (z), since the right hand side is again harmonic and satisfies the desired boundary conditions.
, as defined in Theorem 2.14, is discrete holomorphic.
Proof.Without loss of generality, show discrete holomorphicity at z = a where we use discrete holomorphicity of H [C δ ,a] at (A), and Riemann-Lebesgue Lemma at (B).
We proceed to construct and prove convergence results on the full plane spinor H x (z) := H [C δ ,a] (x ox , z) for x off the slit a + R + .Recall that we denote by R a the orthogonal reflection with respect to the axis {a + t : t ∈ R}.
First note using oriented corners a] we can define spinors 2 ) in the manner analogous to the case where the arguments are oriented edges.a] \ {x}, we have that if Ω (1) δ ,a (x ox , y oy ) exists and is independent of the sequence .
The function F [C δ ,a] (x ox , y oy ) satisfies the following antisymmetry properties: a] .Then H x (•) := H [C δ ,a] (x ox , •) satisfies the following: (1) H x (•) has monodromy −1 around a, is s-holomorphic on [C δ , a] except at x, where it has the following discrete singularity: a] chosen on the same sheet above C \ {a − t : t > 0} (or on different sheets if both are above {a − t : t > 0}), set H x := H [C δ ,a] (x ox , •).Then, we have the following cancellations on {a + t : t ∈ R} (except at x if x = x): • a] and x as in 3., we have the same cancellations as in 3, if we replace t > 0 by t < 0. Furthermore, F [C δ ,a] is uniquely determined by the antisymmetry properties and the fact that (x ox , z) → H [C δ ,a] (x ox , z) satisfies the properties 1,2,3 (or equivalently 1,2,4).Proof.To prove this proposition, we will follow the following strategy, centered around H [C δ ,a] (which is equivalent to constructing and studying F [C δ ,a] ) • We first prove the uniqueness statement (Lemma A.5).
• We then prove (Lemma A.6) that H Ω (n) δ ,a n is uniformly bounded and that lim z→∞ lim sup n→∞ H Ω (n) δ ,a (x ox , z) = 0 • By symmetry arguments we check 1,2,3 for H Ω (n) δ ,a (for x, z within Ω (n) δ ).We obtain that any subsequence as n → ∞ satisfies 1,2,3 ; by uniqueness this concludes the convergence.a] satisfies properties 1,2,3, except that it is s-holomorphic everywhere, including if z = x.Set H * x (z) := H * [C δ ,a] (x ox , z) (the choice of o x is irrelevant: we want to show H * x (z) = 0 for all x, z).First suppose x ∈ V i [C δ ,a] ∩ {a + t : t > 0}.We have that H * x = 0 on V i [C δ ,z] ∩ {a + t : t > 0}, by Property 3 (since x is its own symmetric).Hence by harmonicity of the imaginary part and assumption about the decay at infinity, we obtain that H * x = 0 (first the imaginary part is 0, but the real part also by discrete Cauchy-Riemann equations and monodromy).Now suppose x ∈ V i [C δ ,a] ∩ {a − t : t > 0} .By Property 3 again, we have that H * x vanishes on V i [C δ ,a] ∩ {a − t : t > 0}.Notice that we cannot conclude H * x = 0 using maximum principle directly because there could be a failure of harmonicity at a + δ.However, by the antisymmetry of F * (we can swap the role of x and z) and the result when x ∈ {a + t : t > 0}, we can deduce that a] we obtain that be the discrete analogue of the antiderivative Re ´(H n,x δ ) 2 , normalized to be 0 at x as in earlier in the paper; it is single-valued.On where we can bound it from below by 1 + |H n,x δ (y)| 2 where y is the corner opposite to x (e.g.y = x + δ with a] ).Moreover, its outer normal derivatives on ∂V m Summing the Laplacian of , we obtain the inequalities w∈V Hence, to prove the boundedness and decay at infinity, thanks to the monodromy, it is enough to bound For x ∈ a ± δ 2 , , H n,x δ (y) has a probabilistic interpretation as the ratio E Ω (n) ), which is uniformly bounded (it converges actually to 1) by finite-energy property of the Ising model.Hence the result holds for such x.a] .Again, by variable swap, we have boundedness for any . Hence we can apply the maximum principle to bound H n,x δ (z) with respect to all z, and the same argument applies if and finally (swapping again the variables) to any x, z ∈ V Ω (n) δ ,a .By the first paragraph, it is enough to show boundedness to obtain decay at infinity, so this concludes the proof.a] can be constructed as a linear combination of translation of harmonic measure of the tip of the slit plane  α #γ (−1) c(γ1,...,γn) n i=1 φ(γ i ).
Further, given a collection of doubly oriented points (...), a collection of signed edges {...} disjoint from (...) and edges e 1 , ..., e m , define the fused real fermionic observable F for any branch choice of o 2n .From here we can define the complex fermionic observable where the partial edge from the vertex to the corner counts as a half edge towards the length of α and the winding is calculated with a contribution half of a right or left turn coming at the turn into the corner.Suppose without loss of generality, b2n = b 2n + iδ such that b 2n + i 2 δ lies on the line l(e).We claim that Note that proving the first part of the identity proves that the corner complex weight takes the value of the projection of the medial edge complex weight onto the line connecting the two points in all orientations and is thus enough to prove the second part and complete the proof of the lemma.There are two cases to consder when proving the claim: the case when b 2n , b 2n + (1+i)δ 2 ∈ γ ⊕ c(e ) and the case when b 2n , b 2n + (1+i)δ 2 / ∈ γ ⊕ c(e ).For ease of notation, set which given the required orientation implies • Define the multi-point real function F [Ω δ ,a] : Here, we define the partition functions a) .
Proof.Since any admissible choices of walks have to either start or end at each of the b i 's, i ∈ {1, ..., 2n} we can begin by choosing γ 1 and γ1 to have an endpoint at b 1 .Call b k1 and b k1 the other endpoints of the two walks respectively.Then we are left with a collection of walks γ 2 , ..., γ n and likewise for the other choice of walks such that the lemma does not necessarily hold.Now there are two cases to consider.If k 1 = k1 then naturally we do the same with γ 2 and γ2 so they both share endpoint b m where m = min({2, ..., 2n} \ {k 1 }.Clearly we can continue in this manner until we reach two walks for which k i = ki .So suppose k i = ki .Then choose b m to be the other endpoint of the walk γ i with b ki as one of its endpoints.Choose the permutation such that both γ 2 and γ2 share endpoint b m . It is obvious that we are again in a situation where the other endpoints of γ 2 and γ2 are different.Call the indices of the new endpoints k i+1 = ki and ki+1 and continue in this manner.If at any point, ki = k j for some j ≤ i then we restart the process with the remaining walks and choose m = min({i + 1, ..., 2n} \ j≤i k j ).Then it is easy to continue as before and we will never run into the possibility of not having a possible choice of shared endpoint for γ or γ.
Proof of Proposition B.6.We begin by noting that the value of our function F [Ω δ ,a] (...) is completely independent of the order of the walks so we can choose any permutation of the choices of walks γ i and γi that we desire.Hence we choose the one we proved to exist in Lemma B.8 where γ i shares an endpoint with γi for all i.
It follows that γ i ⊕ γi is a collection of loops and possibly a path from an endpoint of γ i to an endpoint of γi .Call the path γ i and call the collection of loops γ i0 ⊕ γi0 \ γ i0 .Note that if both endpoints are shared then γ i = ∅.
Lemma B.9.Given two admissible choice of walks γ i and γi such that γ i and γi share an endpoint for each i, where γ i is defined as before.
where the sheet number is defined by choosing one endpoint and choosing one of the sheets to fix it on.
Thus Sheet(λ j ), which is by definition 1 if λ j doesn't contain a and -1 if it does, is the product of the sheet number of the λ ji 's: = (−1) #L( γ i ) , because sheet numbers of the intersections all occur even number of times.The choice of sheet that we fix the chosen endpoint on does not matter because every endpoint is shared by an even number of constituent loop components.
This completes the proof that the multipoint fermionic spinor is well-defined.Proof.For ease of notation, set W [Ω δ ,a] (•) = W [Ω δ ,a] (•, o 1 , ..., o 2n−1 ).The lemma has already been shown for the case in which n = 1 with source point taken at the monodromy.We show this holds in general.As before, there are two cases to consider: either b 2n , b 2n + 1+i 2 δ ∈ γ or it isn't.First we show that in both cases, (−1) #L(γ\∪γi,a) (−1) c(γ1,...,γn) i Sheet(γ i ) = (−1) #L(γ\∪γi,a) (−1) c(γ1,...,γn) i Sheet(γ i ) but only the γ n is affected by the XOR operation so the crossing signature is the same on both sides.If c(e) destroys a loop in γ that changed the sheet of [Ω δ , a] then the loop becomes a part of γn so the loop number changes on the right hand side but the sheet factor also changes on the right hand side.Thus we only need to be concerned about changes to the winding factor and the number of edges in the configuration.As shown before, • In the former case, we have [Ω δ ,a] (...) in terms of the complex weights.The monodromy -1 around ramification point a is a direct result of the multiplicative factor i Sheet(γ i ).
To show that the boundary condition holds, it suffices to recognize that any point on the boundary of the domain can only be reached from one direction and thus i √ o2n e − iW (γn ) 2 = v out (b 2n ) mod 2π.

Figure 1
Figure 1.1.A possible Ising model configuration at critical temperature along with the interfaces between plus and minus spins.

Figure 1
Figure 1.2.V Ω δ are denoted by • with the solid lines connecting them being the edges in E Ω δ ; V m Ω δ are denoted by •; V c Ω δ are denoted by .

Figure 1. 3 .
Figure 1.3.The double cover of the complex plane is a Riemann surface with ramification at a point of monodromy a.Here the ramification is at the origin.
is the double cover of Ω δ with ramification at point a; in other words, the vertices, medial vertices, and corners get lifted to give the lifted vertex, edge and corner sets.• We use similar notations for the lifted vertex, edge, and corner sets as above by replacing Ω δ with [Ω δ , a]. • [Ω δ , a] can be naturally viewed as a subgraph of [C δ , a] in view of the natural inclusion [Ω, a] ⊂ [C, a].
1.1.3.Pattern Notation.A base diagram B is a collection of edges in the graph on the square grid (1 + i)Z 2 .Denote by F (B) the set of faces incident to edges in B and let a 0 be a marked midpoint of a face in F (B).Given a graph Ω δ and a face at a ∈ Ω δ we associate with any b ∈ B ∪ F (B) the corresponding a + δ(b − a 0 ) ∈ E Ω δ ∪ F Ω δ : call this embedded diagram centered at a, B δ ⊂ Ω δ (omit the δ subscript when clear from context).

Figure 1. 4 .
Figure 1.4.Pattern Notation.The set of four edges surrounding σ a are the elements of the base diagram B centered at a.The spin-symmetric pattern is defined by the presence of the edge e a,z so B = {e a,z } separating opposite spins.The spin-sensitive pattern associated with σ x = σ y = σ v = σ a = +, σ z = − is defined by B|σ a = +.
simple pole at b (in other words, every projection-based s-holomorphicity relation holds except for those involving H C δ (b o , b)) with residue i √ o1 at b: for b + 1±i 2 √ 2 oδ and b − 1±i 2 √ 2 oδ corners adjacent to b, and h

Figure 3
Figure 3.1.An example of a two walks between b 1 and b 3 and b 2 and b 4 .The winding of γ : b 1 → b 3 is 2π while the winding of γ : b 2 → b 4 is − π 2 .

Definition 4. 2 .
The continuous fermionic observable and spinor on Ω, respectively denoted h Ω and h [Ω,a] , are functions with the property that• h C Ω := h Ω − h Csolves the above continuous Riemann boundary value problem on Ω, with the boundary data given as −h C | ∂Ω .• h [C,a] [Ω,a] := h [Ω,a] − h [C,a] solves the above continuous Riemann boundary value problem on [Ω, a] with the boundary data given as −h [C,a] [∂Ω,a] .

Proposition 5. 4 .
Let B be a base diagram centered about point a in Ω.Here we use the notation (B) to denote the product of energy densities of edges in the base diagram B centered about a. Then

2 n− 1 i=1D
i a, B i Ω where B 1 , ..., B 2 n−1 are 2 n−1 subdiagrams of B (n := |F(B)|).Given a conformal map ϕ : Ω → ϕ(Ω), a, B satisfies a, B Ω = |ϕ (a)| ϕ(a), B ϕ(Ω) .Proof.We have a matrix M that when multiplied by a matrix of normalized energy densities of subdiagrams of B, B 1 , ..., B 2 n−1 denoted here by {E Ω δ [a, (B i )]}, gives probabilites of all 2 n−1 possible spin-symmetric diagrams B on base diagram B. The choice of what subdiagrams, of the 2 m possible subdiagrams to use is defined along with an explicit construction of the matrix M in Appendix C.For the statement, multiply the matrix M by both sides of the first convergence result of the previous proposition.
and gives probabilites of all 2 n possible spin-sensitive diagrams B on base diagram B. Note again that n := |F(B)|.A proof of this and an explicit construction of the matrix can be found in Appendix C.

•
The properties for F [C δ ,a] are immediate for F [Ω δ ,a] and hence are obtained by passing to the limit.Lemma A.5.With the notation of Proposition A.4, the function F [C δ ,a] is uniquely determined by the antisymmetry properties and the fact that(x ox , y) → H x (z) := H [C δ ,a] (x ox , z) satisfies Properties 1,2,3.Proof.By linearity, it is enough to show that if F * [C δ ,a]is the difference of two functionsF [C δ ,a] satisfying the above properties, then F * [C δ ,a] is zero.Denote by H * [C δ ,a]the difference of the two corresponding functions H [C δ ,a] : we have that H * [C δ , {a + t : t ∈ R}, by antisymmetry of F * and the discussion above.Hence, by harmonicity (remember that H *x does not have singularities) and decay at infinity, H *x = 0 and we get the result.