RSW on the quantum model

(C) normalization: F_δ(e^δ_b) = Proj[F_δ(b_δ^w), ν] =^√ν

δ

, whereF_δis defined via Equation (6.3).

Existence of such a solution has been shown already. In effect, the observable we in-troduced earlier satisfies these three conditions, as shown in Section6.2.2. When it comes to uniqueness, we will use the primitiveH_δ we constructed in Section6.2.3along with the boundary modification trick.

Proposition 6.12(Existence of solution). The observableF_δ defined in(6.2)is a solution to the above boundary value problem.

Proof. It is direct from the properties of the observable as shown in Propositions6.4and6.5.

Proposition 6.13(Uniqueness of solution). For each semi-discrete Dobrushin domain(Ω_δ, a_δ, b_δ), the semi-discrete boundary value problem has a unique solution.

Proof. Assume that there are two solutions F_δ,1 and F_δ,2 to the boundary value problem mentioned above. LetF_δ :=F_δ,1−F_δ,2. Note thatF_δ is still s-holomorphic being difference of two such functions. DefineF_δ as in Equation (6.3). Consider H_δ := ImR

(F_δ(z))²dz the primitive defined in Section6.2.3. The functionH_δ is constant on the arcs(a_δb_δ)and(b_δa_δ) respectively due toe the property (B). Moreover, the identityF_δ(e^δ_b) = 0says that these two constants should be the same. Apply the boundary modification trick to extendΩ_δ into the primal domainΩe_δ.

Extend the primitiveH_δ to the new boundary ofΩe_δ as in Lemma6.11. The lemma also says thatH_δstays subharmonic inL_δ∩Ωe_δand superharmonic inL^?

δ∩Ωe_δ. This gives us that 0>(H_δ)_|L?

δ∩Ωe_δ>(H_δ)_|L?

δ∩Ωe^?_δ >0. By uniqueness of the Dirichlet problem (Proposition5.17), we know thatH_δ is constant everywhere.

The fact thatH_δis constant everywhere tells us thatF_δis zero everywhere onΩ^[_δ. Thus, these two solutions are equal.

6.4 RSW on the quantum model

In Chapter4, we showed the RSW property on the critical quantum random-cluster model for16q64, thus in particular, on the random-cluster representation of the quantum Ising model. Here, we provide an alternative using the fermionic observable (6.2) and the second--moment method.

6.4.1 RSW property: second-moment method

In the previous section, we established the conformal invariance of the limit of our semi-dis-crete observables. To show that the interface can be identified with an SLE curve in the limit and to determine its parameter, we need the so-called Russo-Seymour-Welsh (RSW) property.

This provides the hypothesis needed in [KS12] which, along with Theorem6.22, shows the main Theorem6.1.

The goal of this section is to show the following property.

6. Scaling limit of the qantum Ising model

Proposition 6.14(RSW property). Fixα >0. Consider the rectangular domainR_n,α= [−n, n]× [−αn, αn]and writeR^δ_n,αfor its semi-discretized counterpart (primal domain). Letξbe a bound-ary condition onR^δ_n,α. Then, there existsc(α)>0which is independent ofnandδsuch that

c6P^ξ(C_h(R^δ_n,α))61−c, c6P^ξ(C_v(R^δ_n,α))61−c,

whereC_handC_vdenote the events “having a horizontal / vertical crossing”.

The proof of Proposition6.14is based on the use of the same fermionic observable intro-duced in Section6.2and the second moment method to estimate the crossing probabilities.

This is inspired from [DCHN11] where the classical Ising case is treated and here we adapt the proof to the case of quantum Ising.

To show the RSW property in Proposition6.14, we only need to show the lower bound for free boundary condition by duality [DCHN11]. In this section, we will just show the property for the horizontal crossing, since the proof to estimate the probability of the vertical crossing is similar.

We recall that(Ω_δ, a_δ, b_δ)is a Dobrushin domain, meaning that the arc(a_δb_δ) is wired and the arc(b_δa_δ)is free. In Section6.3.1, we introduced the notion of modified primal and dual domains of a Dobrushin domain, which are denoted by Ωe_δ andΩe^?_δ respectively. Let us write HM_• and HM_◦ the harmonic functions on modified domainsΩe_δ andΩe^?_δ having boundary conditions 1 on the (extended) wired arc (∂_abforΩe_δ and∂^?_abforΩe^?_δ) and 0 on the (extended) free arc (∂_baforΩe_δ and∂^?_ba forΩe^?_δ).

We start by noticing that the connection probability of a vertex next to the free arc(b_δa_δ) to the wired arc(a_δb_δ)can be written in a simple way by using the parafermionic observable.

Proposition 6.15. Letu∈Ω_δsuch that{u⁺, u⁻} ∩(b_δa_δ),∅(equivalently,u is next to the free arc). Writeefor the mid-edge betweenuand the free arc(b_δa_δ). Then, we have

P_(Ω

δ,a_δ,b_δ)(u↔(a_δb_δ))²=δ|F(e)|². Proof. We take the definition ofF,

δ|F(e)|²=

E_(Ω

δ,a_δ,b_δ)

exp

i

2W(e, b_δ) 1e∈γ_δ

2

=|E_(Ω

δ,a_δ,b_δ)[1e∈γ_δ]|²=P_(Ω

δ,a_δ,b_δ)(e∈γ_δ)²

where the windingW(e, b_δ)is always a constant ifeis adjacent to the boundary. We also notice thate∈γ_δ is equivalent tou connected to the wired arc(a_δb_δ).

By using harmonic functions HM_•and HM_◦, we can get easily the following proposition.

Proposition 6.16. Letu∈Ω_δnext to the free arc. Writew∈ {u⁺, u⁻}which is not on the free arc. We have

p

HM_◦(w)6P_(Ω

δ,a_δ,b_δ)(u↔(a_δb_δ))6p

HM_•(u) Proof. Writew_∂ the neighbor ofuwhich is on the free arc. We have

H(u) =H(u)−H(w_∂) =δ|F(e)|²=P_(Ω

δ,a_δ,b_δ)(u↔(a_δb_δ))².

6.4. RSW on the quantum model

Now, we are ready to show the RSW property. We keep the same notation as in the statement of Proposition6.14. We write∂−and∂₊for the left and right (primal) borders of R^δ_n,α. We define the random variableN given by the 2D Lebesgue measure of the subset of

∂−×∂₊⊂R²consisting of points which are connected inR^δ_n,α. More precisely,

To show Proposition6.14, we use the second moment method. In other words, by using Cauchy-Schwarz, we need to show that the lower bound of

P⁰(N >0) =E⁰[1²_{N >0}]>E⁰[N]² E⁰[N²]

(6.12) is uniform innandδ. First, we get a lower bound forE⁰[N].

Lemma 6.17. There exists a uniform constantcindependent ofnandδsuch that E⁰[N]>cn.

Proof. We decompose the right boundary intom=bn/δcparts,

∂₊= a neighbor ofxwhich is not on the free arc, and the harmonic measure is with respect to the modified domainΩe^?_δ where the Dobrushin domain(Ω_δ, a_δ, b_δ)is given byΩ_δ=R^δ_n,α,a_δ wherec⁰ is a uniform constant.

6. Scaling limit of the qantum Ising model

To estimate E⁰[N²], we need Proposition6.18, a consequence of Lemma 6.19. Both of them make use of the so-calledexploration path, which is the interface between the primal wired cluster and the dual free cluster. The proof in the discrete case [DCHN11] can be easily adapted to the semi-discrete case, since the interface is well-defined and we have similar estimates on harmonic functions, by means of semi-discrete Brownian motion, local central limit theorem and gambler’s ruin-type estimates. Therefore, we will just give the proof of Lemma6.19.

For any givenα, nandδ, let us considerR_n,α as before and(R^δ_n,α, a_δ, b_δ)the semi-dis-cretized Dobrushin domain obtained fromR_n,α with the right boundary∂₊= (a_δb_δ)which is wired.

Proposition 6.18([DCHN11, Proposition 14]). There exists a constantc >0which is uniform inαandnsuch that for any rectangleR^δ_n,αand any two pointsx, z∈∂−, we have

P(R^δ_n,α,a_δ,b_δ)(x, z↔∂₊) =P⁰

R^δ_n,α(x, z↔∂₊)6 c

√

|x−z|n

Lemma 6.19([DCHN11, Lemma 15]). There exists a constantc >0which is uniform inα,n, δandx∈∂−such that for any rectangleR^δ_n,αand allk>0,

P(R^δn,α,a_δ,b_δ)(B_δ(x, k)↔(a_δb_δ))6c rk

n.

Proof. Letn, k, δ, α >0, the rectangular domainR^δ_n,α and its semi-discrete counterpart, the Dobrushin domain(R^δ_n,α, a_δ, b_δ)where∂₊= (a_δb_δ) is the wired arc. Considerx∈∂−. For k>n, the inequality is trivial, so we can assumek < n.

Since the probabilityP

(R^δn,α,a_δ,b_δ)(B_δ(x, k)↔(a_δb_δ))is non-decreasing inα, we can bound it by above by replacingαbyα+ 1, which we bound by above by a longer wired arc(c_δd_δ), wherec_δ andd_δ are respectively the left-bottom and the left-top points of the rectangular domainR^δ_n,α. See Figure6.9for notations.

P(R^δ_n,α,a_δ,b_δ)(B_δ(x, k)↔(a_δb_δ))6P

(R^δ_n,α+1,a_δ,b_δ)(B_δ(x, k)↔(a_δb_δ))

6P

(R^δ_n,α+1,c_δ,d_δ)(B_δ(x, k)↔(c_δd_δ))

x

2αn

2n p

x

b^w_δ b^b_δ bδ

aδ

a^w_δ a^b_δ

x d^w_δ d^b_δ

dδ

cδ

c^w_δ c^b_δ

x d^w_δ d^b_δ

dδ

cδ

c^w_δ c^b_δ p γδ(T)

Figure 6.9 – First: The primal domainR^δ_n,α. Second: The Dobrushin domain (R^δ_n,α, a_δ, b_δ). Third: The Dobrushin domain (R^δ_n,α, c_δ, d_δ). Third: The Dobrushin domain (R^δ_n,α, c_δ, d_δ). Fourth:The exploration path starting atc_δ and touchingB_δ(x, k)at timeT.

6.4. RSW on the quantum model Letγ_δ be the interface as defined in Section1.2.5and used in Section6.2to define the fermionic observable. The definition ofγ_δ tells us that the ballB_δ(x, k)is connected to the wired arc if and only ifγ_δgoes through a mid-edge which is adjacent to the ball. We param-eterizeγ_δ by its length and denotesT the hitting time of the set of the mid-edges adjacent to the ballB_δ(x, k). Therefore,B_δ(x, k)is connected to the wired arc if and only ifT <∞.

Writepfor the top-most point ofB_δ(x, k). We can rewrite the probability of{p↔(c_δd_δ)} by conditioning onγ_δ[0, T]and using Markov domain property to obtain

P(R^δ_n,α+1,c_δ,d_δ)(p↔(c_δd_δ)) =E

We estimate this quantity in two different ways to get the desired inequality. Firstly, since pis at distance at leastnfrom the wired arc, we have

P(R^δ_n,α+1,c_δ,d_δ)(p↔(c_δd_δ))6 c₁

√ n, which follows from Proposition6.16 and the fact that HM_• 6 ^c

n. Secondly, we can write γ_δ(T)asz+ (s,−r, s)where06s6kand06r62k. Thus, the linez+Z× {−r}disconnects afrom the free arc, we estimate the harmonic function and we obtain a.s.

P(R^δ_n,α+1\γ_δ[0,T],γ_δ(T),d_δ)(p↔(c_δd_δ))> c₂ which again comes from Proposition6.16 and the estimate HM_◦ > ^c

r. This final estimate

Now, we can complete the proof of the RSW property.

Proof of Proposition6.14. From Equation (6.12) and Lemma6.17, we just need to show that E⁰[N²] =

because the left-hand side ofland the right-hand side oflare independent. Therefore, the integral in Equation (6.13) can be cut into two independent parts, each of whom gives the same contribution,

6. Scaling limit of the qantum Ising model where in the last line we use Proposition6.18.

The proof is thus complete.

Dans le document Universality of the random-cluster model and applications to quantum systems (Page 172-177)