title: The two-dimensional continuum random field Ising model
authors: Adam Bowditch Rongfeng Sun
text: 
  
  
  
  
  The two-dimensional continuum random field Ising model
  
  
  A]Adam Bowditch[label=e1,mark]a.m.bowditch@gmail.com
  A]Rongfeng Sun[label=e2,mark]matsr@nus.edu.sg
  
  [A]Department of Mathematics,
  		National University of Singapore,
   e1,e2
  
  
  
  In this paper we construct the two-dimensional continuum random field Ising model via scaling limits of a random field perturbation of the critical two-dimensional Ising model with diminishing disorder strength. Furthermore, we show that almost surely with respect to the continuum random field given by a white noise, the law of the magnetisation field is singular with respect to that of the two-dimensional continuum pure Ising model constructed by Camia, Garban and Newman in <cit.>.
  
  
  [class=MSC2020]
  [Primary ]82B44
  [; secondary ]82B20
  82B27
  60G60
  60K35
  
  
  
  Random Field Ising Model
  Continuum Scaling Limit
  Magnetisation Field
  
  
  
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  
    
  *
  
  Since its introduction by Lenz <cit.> as a model for ferromagnetism, the Ising model has become one of the most fundamental models in statistical mechanics, maintaining an important role in the theory of critical phenomena since Peierls <cit.> proved that it undergoes a phase transition in dimensions two and above. It is natural to consider disorder perturbations of the model by i.i.d. random external fields, known as the random field Ising model (RFIM), and ask whether the critical behaviour changes or not.
  Imry and Ma <cit.> gave a physical argument which suggested that in low dimension, the phase transition is rounded off under the influence of arbitrarily weak random field disorder.
  
  This was confirmed in dimension d=2 by Aizenman and Wehr <cit.> who showed the absence of a first order phase transition at any temperature for any non-zero disorder strength. For dimension d≥ 3, the question was settled by Bricmont and Kupiainen <cit.> who showed that the first order phase transition persists at low temperatures.
  
  In this paper we consider the Ising model in dimension d=2 with the aim of further understanding the issue of disorder relevance; that is, how the addition of arbitrarily weak disorder changes the nature of the phase transition of the underlying pure model.
  We show that disorder relevance manifests itself via the convergence of the disordered model to a disordered continuum limit when the disorder strength and lattice mesh are suitably rescaled. In the absence of disorder, such a continuum limit for the critical two-dimensional Ising magnetisation field has been constructed by Camia, Garban and Newman in <cit.>. We prove that, for almost every instance of disorder, the pure and disordered continuum limits are singular.
  
  In the particular case of Gaussian disorder, the decay rate of the spin correlations for the RFIM has been of much recent interest.
  It has been shown in <cit.> that, at any temperature and any disorder strength, the correlations between spins of distance N are at most 1/ loglog N. The decay rate has then been improved to polynomial order in <cit.>, and then further improved to exponential decay in <cit.>  (see also <cit.>), which resolves a long-standing conjecture.
  
  
  
  Let us first recall the basic ingredients before stating our results.
  
  
  
    
  *The pure Ising model Let Ω⊂^2 be a simply connected bounded open domain with a piecewise C^1 boundary. For >0, define Ω_:=Ω∩^2 and write x ∼ y if x,y ∈Ω_ are neighbouring vertices. Denote by ∂Ω_:={y∈^2 \Ω_: y∼ x  x∈Ω_} the external boundary of Ω_. Given boundary condition ξ∈{±1}^∂Ω_, we then define the pure Ising model as the law over spins σ∈{± 1}^Ω_∪∂Ω_ with σ|_∂Ω_=ξ|_∂Ω_ by
  
      _Ω^, ξ(σ) = 1/Z_Ω^, ξexp(β∑_x∼ yx ∈Ω_, y∈Ω_∪∂Ω_σ_xσ_y)
  
  where the sum is over unordered pairs x∼ y and Z_Ω^, ξ 
  is the partition function. When ξ≡+1, it is known as the + boundary condition and we simply write + in place of ξ.
  It is well known that there is a critical inverse temperature β_c=log(1+√(2))/2 such that the boundary effect is negligible in the infinite volume limit for β<β_c, leading to a unique infinite volume Gibbs state; and non-negligible for β>β_c, leading to multiple infinite volume Gibbs states, see <cit.> for a more detailed introduction. Henceforth, we set β=β_c and let _Ω^ denote the two dimensional critical Ising model with + boundary condition, and we denote by _Ω^ expectation with respect to this law. We will assume + boundary condition throughout the rest of the paper and omit + from the superscripts.
  
  
  
  
    
  *The random field Ising model
  The random field Ising model is a disorder perturbation of the Ising model by introducing i.i.d. random external field for each spin.
  Denote by  a law over a family ω=(ω_x)_x∈^2 of i.i.d. centred random variables with unit variance and finite exponential moments. Write  for expectation with respect to this law. Given λ, h:Ω→, for each >0 and x ∈Ω_, write
  
      λ_x^ :=^7/8λ(x),       h_x^ :=^15/8h(x),      ω^_x:=ω_x/.
  
  For ω fixed, we define the two dimensional critical random field Ising model (with + boundary condition) as the law
  
      _Ω; λ,h^ω,(σ)
      := 1/Z_Ω;λ, h^ω,exp(∑_x∈Ω_(λ_x^ω^_x+h_x^)σ_x)_Ω^(σ)
  
  where _Ω^ is the pure Ising model at critical inverse temperature β_c and
  
      Z_Ω;λ, h^ω,
      = _Ω^ [exp (∑_x∈Ω_(λ_x^ω^_x+h_x^)σ_x ) ]
  
  is the random partition function depending on the random field ω.
  
  In <cit.>, it is shown that the rescaled partition function
  
      Z_Ω;λ, h^ω,:=θ_ Z_Ω;λ, h^ω,,       θ_:=e^-1/2^-1/4‖λ‖ _L^2^2,
  
  converges in -distribution to a non-trivial limit _Ω;λ, h^W, which admits a Wiener chaos expansion with respect to a spatial white noise W. This is the first step toward the construction of the two-dimensional continuum random field Ising model and the starting point of our paper, which will be explained further in Section <ref>.
  
  
  
  
    
  *Magnetisation field
  We will study the convergence of the RFIM through its magnetisation field.
  For >0 and x∈Ω_, let S_(x):={y∈Ω:‖ x-y‖ _∞</2} be the box centred at x with side length . We then define the rescaled piecewise constant magnetisation field as the distribution
  
      Φ_Ω^ := ^-1/8∑_x∈Ω_σ_x_S_(x).
  
  Denote by μ_Ω^:=_Ω^∘(Φ_Ω^)^-1 the law of Φ_Ω^ without disorder. It has been shown by Camia, Garban and Newman in <cit.> that, as → 0, μ^_Ω converges weakly to a limiting probability measure μ_Ω, which can be regarded as the law of the magnetisation field Φ_Ω for the continuum two-dimensional critical Ising model. The magnetisation field Φ_ω^ was regarded as an element of the Sobolev space H^-3, which was subsequently improved to the optimal Besov-Hölder space ^α_loc(Ω) for α<-1/8 by Furlan and Mourrat <cit.>.
  
  Similarly, for each fixed realisation of the random field ω, define
  
      μ^ω,_Ω;λ,h:=_Ω;λ,h^ω,∘(Φ_Ω^)^-1
  
  to be the quenched law of the magnetisation field with disorder ω. The main focus of this article is to show that μ^ω,_Ω;λ,h converges weakly in -distribution to a disordered continuum limit μ^W_Ω;λ,h (with + boundary conditions) where the disorder is given by a white noise W that arises as
  the limit of the random process
  
      W^ω,:=^-1∑_x∈Ω_ω_x^_S_(x).
  
  
  
  
  
    
  *Statement
  We will regard the magnetisation field Φ_Ω^ as an element of the Besov-Hölder space ^α_loc(Ω) for α<-1/8 defined in <cit.> (see Section <ref> for more detail). Let _1(^α_loc(Ω)) denote the space of probability measures on ^α_loc(Ω) equipped with the topology of weak convergence, so that for every ω, μ^ω,_Ω;λ,h∈_1(^α_loc(Ω)). Denote by C^1(Ω) the space of bounded, continuously differentiable functions with bounded first derivatives.
  
  Our first result shows that the disordered continuum limit μ^W_Ω;λ,h exists and, for almost every realisation of the white noise W,  is a probability measure on ^α_loc(Ω) and can be interpreted as the law of the continuum RFIM magnetisation field with external field W.
  
  
  Let λ,h∈ C^1(Ω) with λ_min:=inf_x∈Ωλ(x)>0 and let α<-1/8, α'<-1.
  
  As → 0,
  
      (W^ω,,Z^ω,_Ω;λ,h,μ^ω,_Ω;λ,h)⇒(W,^W_Ω;λ,h,μ^W_Ω;λ,h)
  
  weakly as random variables in ××_1(^α_loc(Ω)), where W is white noise, and -a.s., ^W_Ω;λ,h and μ^W_Ω;λ,h are determined uniquely by W.
  
  
  With a slight abuse of notation, we will also use ^ω, _Ω; λ, h and ^ω, _Ω; λ, h to denote probability and expectation with respect to μ^ω, _Ω; λ, h, and similarly use ^W_Ω; λ, h and ^W_Ω; λ, h for μ^W_Ω; λ, h. We will omit ω (or W) and λ when λ≡ 0, and write ^_Ω and ^_Ω (or _Ω and _Ω) when λ=h≡ 0.
  
  
  A natural strategy for proving convergence of the RFIM measure μ^ω,_Ω;λ,h would be to look at the exponential weight in the disordered Gibbs measure (<ref>) and define the candidate continuum disordered model by
  
      μ̣^W_Ω;λ,h/μ̣_Ω(σ) “=" 1/^Wexp(∫(σ_x λ(x)W(x)x̣+σ_x h(x)x̣)).
  
  However, this formula is not well defined because the continuum magnetisation field σ is a generalised function and so is W, which makes σ W ill-defined. In fact, our next result proves that the disordered continuum limit μ^ω,_Ω;λ,h is almost surely singular with respect to the pure continuum limit μ_Ω, which shows that it is hopeless to define the continuum disordered model directly through a Radon-Nikodym density. However, when averaged over the disorder W, the limit is absolutely continuous with respect to μ_Ω.
  
  For -a.e. W, the probability measure μ^W_Ω;λ,h is singular with respect to μ_Ω. However, the averaged quenched measure μ^W_Ω;λ,h is absolutely continuous with respect to μ_Ω.
   
  
  
  
  
  
    
  *Discussion
  One of the first results that identifies a disordered continuum limit for a disordered system is the work by Alberts, Khanin and Quastel,  who showed that for the directed polymer model in dimension 1, if the disorder strength is sent to zero at a suitable rate as the lattice spacing tends to 0, then the partition functions converge to the solution of the one-dimensional KPZ equation <cit.>, while the polymer measure converges to a continuum limit called the continuum directed polymer <cit.>. Subsequently, Caravenna, Sun and Zygouras made the observation that such disordered continuum limits should exist for more general disorder relevant systems <cit.>, where arbitrarily weak disorder perturbation of an underlying pure model changes its behaviour on large scales, and hence tuning the disorder strength down to zero suitably as the lattice spacing tends to zero allows one to construct a continuum limit with non-trivial disorder dependence. They formulated general criteria for the partition functions of a disordered model to have non-trivial limits, which is the first step to construct the disordered continuum model. These criteria were verified for the disordered pinning model, with the continuum disordered pinning model subsequently constructed in <cit.>. They were also verified in <cit.> for the partition functions of the RFIM with + boundary condition, which provides the starting point of the present paper.
  
  
  
  Both the directed polymer model and the disordered pinning model have a time direction, which allows one to use the Markov property
  to construct the continuum models directly from the continuum limit of the partition functions. For the RFIM in two dimensions, such an approach is no longer feasible because it would require knowledge of the continuum limit of the partition functions for all domains with all boundary conditions, and it is not even clear how to define general boundary conditions for the continuum RFIM since the magnetisation field is only a distribution. Instead, we will use characteristic functions to characterise the law of the continuum magnetisation field and prove convergence in Theorem <ref>.
  
  
  
  Heuristically, the singularity in Theorem <ref> can be understood as follows: the disorder splits the system into sub-domains which behave as essentially independent components with a small random external field, tilting the law with respect to the measure with no external field. This occurs on arbitrarily small scales. If the effect of random tilting on each subdomain remains strong enough on smaller and smaller scales, then this gives singularity. In particular, such singularity should arise on any open subdomain, as we will show in the proof. When the disordered law is averaged, the spatial fluctuation is averaged out and the effect of the disorder is smoothed, removing the singularity. The same heuristic applies to the continuum directed polymer model and disordered pinning model. Indeed, the analogue of Theorem <ref> has been proved for both models, where the Markov property plays an essential role. The proof of Theorem <ref> for the continuum RFIM is much more subtle for the reasons described before and will constitute the bulk of the paper.
  
  Our results also extend the work of Camia, Garban and Newman <cit.>, where they constructed and analysed the near-critical scaling limit of the two-dimensional Ising model, which corresponds to the continuum RFIM with a deterministic external field, namely, μ^ω,_Ω;λ,h with λ=0. It was then shown in <cit.> that when h≡ c≠ 0, the continuum model has exponential decay of correlations.
  
  
  
    
  *Open problems
  Given recent breakthroughs on the exponential decay of correlations for the RFIM on ℤ^2 <cit.>, it would be very interesting to prove the same result for the continuum RFIM and to understand how the magnetisation field depends on the mean h and strength λ of the random field. The latter are interesting and challenging questions that are also open for the lattice RFIM.
  
  For the critical Ising model on ℤ^2, instead of considering the magnetisation field as in <cit.> and study its scaling limit, one can also study the interfaces between + and - spins, which was shown to converge to the conformal loop ensemble CLE(3) in <cit.>. When a deterministic external field is present in the continuum, the law of the magnetisation field is tilted and the same should hold for the law of the interfaces. Is it possible to characterise the law of interfaces in the continuum RFIM when disorder is present? By Theorem <ref>, we expect it to be singular with respect to the law of CLE(3). However, if we consider the law of a single interface, such as the interface that arises from imposing a Dobrushin boundary condition, then it is conjectured (communicated to us by Christophe Garban) that almost surely in the disorder, the law of the interface is absolutely continuous with respect to the law of an SLE(3).
  
  Another interesting question is to investigate whether one can make sense of (<ref>) in the same spirit as in the solution theory for singular SPDEs <cit.>, which also had to deal with products of distributions such as σ W in (<ref>). And is it possible to construct the measures μ_Ω and μ_Ω; λ, h^W as the equilibrium solutions of SPDEs, similar to the stochastic quantisation of Φ^4 theory <cit.>? Note that the Φ^4_3 measure can formally be seen as a Gibbs change of measure of the Gaussian free field (GFF), similar in spirit to (<ref>), although it is also singular with respect to the reference GFF <cit.>. We also remark that in the study of singular SPDEs, results such as Theorem <ref> are known as weak universality (see e.g. <cit.>).
  
  
  
  
    
  *Organisation
  The rest of the paper is organised as follows. In Section <ref> we provide some background and technical results that we will need for the rest of the paper. This includes details on Ising spin correlations, conformal invariance, Besov-Hölder spaces, white noise and Wiener chaos expansions.
  
  In Section <ref> we prove the uniqueness of the limit in Theorem <ref>. The characteristic function of the magnetisation field tested against smooth functions can be written as the ratio of partition functions.
  This reduces this part of the proof to establishing joint convergence of the white noise approximation and finite families of partition functions.
  For this, we use a Lindeberg principle similar to <cit.> and show that the limit is a family of Wiener chaos expansions.
  
  In Section <ref> we complete the proof of Theorem <ref> by showing tightness, which can be reduced to the case without a random field that was considered in <cit.>.
  
  In Section <ref> we prove Theorem <ref> by studying the Radon-Nikodym derivatives of the continuum magnetisation fields conditioned on their average as well as the average of the white noise on disjoint subdomains. These conditioned Radon-Nikodym derivatives are then controlled by discrete approximations and, via suitable fractional moment bounds, are shown to converge to zero as the conditioning is gradually refined.
  
  In Appendix <ref>, we give a sufficient condition for the weak limit of one sequence of probability measures to be absolutely continuous with respect to the weak limit of a second sequence.
  
  In Appendices <ref> and <ref>, we prove that the RFIM partition functions Z^ω, _Ω;λ, h have uniformly bounded positive and negative moments of all orders, and hence the same holds for their continuum limit ^W_Ω; λ, h.
  
  In Appendix <ref>, we show that for any Φ in a Besov-Hölder space ^α with α>-1, we can define ∫_B Φ
  for any subdomain B with a regular enough boundary, so that the total magnetisation of the continuum RFIM on B
  is well-defined. This is needed in Section <ref>.
  
  
  
  
  
  
  § PRELIMINARIES
  
  In this section, we provide some background and list some technical results needed in the proof.
  
  
    
  *Spin correlations
  A feature of many two-dimensional critical statistical mechanical models is that the continuum scaling limits are conformally invariant.
  In <cit.>, Smirnov established the conformal invariance of fermionic observables in the critical Ising model.
  This facilitated rigorously establishing conformally invariant scaling limits including convergence of Ising loops to a CLE <cit.>, convergence of the Ising interface to a chordal SLE <cit.> and, fundamental to this work, convergence of spin correlations <cit.>. More specifically, we will need the following result.
  
    Let Ω⊂^2 be a simply connected bounded open domain with a piecewise C^1 boundary.
  For the critical Ising model on Ω_:=Ω∩^2 with + boundary condtition, there is a symmetric function
  ϕ^+_Ω:⋃_k=1^∞Ω^k→ such that for all n∈ and distinct x_1,...,x_n∈Ω,
  
  ^-k/8_Ω^,+[∏_i=1^kσ_x_i] ⟶^kϕ^+_Ω(x_1,...,x_k)      ↓0,
  
  where  is a known constant, σ_x:=σ_x_ with x_:=a ⌊ a^-1x⌋, and the convergence holds both pointwise and in L^2(Ω^n).
  
  There exists a symmetric function f_Ω: ⋃_k=1^∞Ω^k →^+ continuous everywhere except on the diagonal, such that uniformly in I⊂Ω with |I|=k, k∈, and ∈(0,1], we have
  	
      0≤_Ω^, +[∏_x∈ Iσ_x] ≤^k/8 f_Ω(I),	 
      ‖ f_Ω‖_L^2(Ω^k)^2≤ C^k(k!)^1/4.
  
  
  
  The pointwise convergence in (<ref>) was established in <cit.>, which was then extended to L^2 convergence in <cit.> by dominated convergence, using (<ref>) (<cit.>) and (<ref>)
  (<cit.>). The assumption that Ω has a piecewise C^1 boundary is to ensure that ∫_Ω dx/d(x, ∂Ω)^1/4<∞, which is used to prove the L^2 convergence in (<ref>) (see <cit.>).
  
  
  
  
    
  *White noise and chaos expansions
  The partition functions Z^ω,_Ω;λ,h defined in (<ref>) encode much of the essential information of the system and will be vital for studying the law of the quenched magnetisation field μ^ω,_Ω;λ,h defined in (<ref>). The  partition functions and their scaling limits have been studied in <cit.> using polynomial and Wiener chaos expansions which we now review.
  
  Let  be a countable set (typically Ω_) and ():={I⊆:|I|<∞}. For I∈() and a vector u∈^ we write u^I:=∏_i∈ Iu_i. Any function ψ:()→ can be used to define a multi-linear polynomial
  
      Ψ(u)=∑_I∈()ψ(I)u^I
  
  and we call ψ the kernel of Ψ. Let ζ:=(ζ_i)_i∈ be a family of independent random variables. We say that a random variable  admits a polynomial chaos expansion with respect to ζ if it can be expressed as Ψ(ζ) for some multi-linear polynomial Ψ. For l∈, we write Ψ^≤ l for the chaos expansion with kernel ψ_≤ l defined by ψ_≤ l(I)=ψ(I)_{|I|≤ l}.
  
  Recall the definition of Z_Ω;λ, h^ω, from (<ref>). Write ξ^_x:=λ_x^ω^_x+h_x^ for the random external field; then, using a high temperature expansion,
  
  Z_Ω;λ, h^ω,
      =θ__Ω^[exp(∑_x∈Ω_ξ^_xσ_x)] 
  
      =θ__Ω^[∏_x∈Ω_(cosh(ξ^_x)+σ_xsinh(ξ^_x))] 
  
      = θ_cosh(ξ^_·)^Ω_∑_I⊆Ω__Ω^[σ_·^I] tanh(ξ^_·)^I. 
  
  where the prefactor θ_cosh(ξ_·^)^Ω_ converges to 1 in probability by a Taylor expansion (see Lemma <ref>). We want to understand the limiting behaviour of the remaining polynomial chaos expansion; for this we first introduce white noise and the Wiener chaos expansion.
  
  The limit of Z_Ω;λ, h^ω, will be expressed as a Wiener chaos expansion with respect to a white noise W on ℝ^2, which can be identified with a Gaussian process W=(W(f))_f∈ L^2(^2) with mean [W(f)]=0 and covariance [W(f)W(g)]=∫ f(x)g(x)x̣.
  
  If A_1, A_2,… are disjoint Borel sets with finite Lebesgue measure then the random variables W(A_i):=W(_A_i) are independent centred Gaussian random variables with variance the Lebesgue measure of A_i. Moreover, we have that the relation W(⋃_i≥ 1A_i)=∑_i≥ 1W(A_i) holds a.s. and we write ∫ f(x)W(x̣):=W(f) even though W(·) is a.s. not a signed measure.
  
  Recalling that S_(x) is the square of side length a centred around x we have that
  
  ϑ_x^:=^-1∫_S_(x)W(ỵ)
  
  is a standard Gaussian for any >0 and x∈^2. Furthermore, the sequence of distributions
  
   W^:=∑_x∈Ω_ϑ^_x δ_x=∑_x∈Ω_W(S_(x)) δ_x
  
  converges a.s. to the white noise W.
  
  
  
  
  
  
  
  
  
  For white noise W on ^2 we define the Wiener chaos expansion with kernel ϕ_Ω^+ from (<ref>) as
  
      _Ω;λ, h^W = 1+∑_n=1^∞^n/n!_Ω^nϕ_Ω^+(x_1,...,x_n)∏_i=1^n(λ(x_i)W(x̣_i)+h(x_i)x̣_i).
  
  By <cit.>, the rescaled partition function Z_Ω;λ, h^ω, converges in -distribution to _Ω;λ, h^W, and its Wiener chaos expansion is convergent in L^2. In this paper, we will need the following stronger result.
   
  Let Z_Ω;λ, h^ω, be as in Theorem <ref>, then  Z_Ω;λ, h^ω, converges in distribution to _Ω;λ, h^W as ↓ 0.
  Furthermore,
  for any p ≥ 0 (or p<0 with ω satisfying the concentration of measure inequality (<ref>)), [(_Ω;λ, h^W)^p]= lim_↓ 0[(Z_Ω;λ, h^ω,)^p] <∞. In particular, ^W_Ω; λ, h>0 -a.s.
  
  
  
  
  
  
  
  
  
  
  
  
  We only need to show that for any p∈, lim sup_↓ 0[(Z_Ω;λ, h^ω,)^p] <∞. We will distinguish between p>0 and p<0, which will be treated in Appendix <ref> and <ref> respectively.
  
  
  
  
    
  *Besov-Hölder spaces
  We now describe the Besov-Hölder spaces ^α in which the magnetisation field takes values, following <cit.>.
  First, for r∈, write C^r to denote the space of r-times continuously differentiable functions on ^2 and, for f ∈ C^r, define the norm
  
      ‖ f‖_C^r:=∑_|i|≤ r‖∂_i f‖_∞.
  
  For α<0 let r_α=-⌊α⌋ and
  
      ^r_α := {f∈ C^r_α : ‖ f‖_C^r≤ 1  and Supp(f) ⊂ B(0, 1)}.
  
  Then, for f∈ C_c^∞, denote
  
      ‖ f‖_^α:=sup_θ∈(0,1]sup_x∈^2sup_g∈^r_αθ^-α-2∫ f(y)g(y-x/θ)ỵ.
  
  The Besov-Hölder space ^α is the completion of C_c^∞ with respect to the norm ‖·‖_^α. For every open domain Ω⊂^2, the local Besov-Hölder space ^α_loc(Ω) is the completion of C_c^∞ with respect to  the family of seminorms (‖χ·‖_^α)_χ∈ C_c^∞(Ω).
  
  
  
  There is an equivalent characterisation of ^α through multi-resolution analysis. A multi-resolution analysis of L^2 is an increasing sequence (V_n)_n∈ of subspaces of L^2, together with a function ϕ∈ L^2(^2), such that
  
  
    * ⋃_n V_n is dense in L^2;
  
    * ⋂_n V_n={0};
  
    * f ∈ V_n if and only if f(2^-n(·)) ∈ V_0;
  
    * (ϕ(· - k))_k∈^2 is an orthonormal basis of V_0.
  
  Such a function ϕ is called the scaling function of (V_n)_n∈.
  Denote by W_n the orthogonal complement of V_n in V_n+1. Then for any r∈, there exist ϕ, (ψ^(i))_i=1,2,3 such that
  
  
    * ϕ, (ψ^(i))_i=1,2,3 all belong to C^r_c with support in B(0,1);
  
    * ϕ is the scaling function of a multi-resolution analysis (V_n)_n∈;
  
    * (ψ^(i)(·-k)_k∈^2, i=1,2,3) is an orthonormal basis of W_0;
  
    * for each i=1,2,3 and β_1, β_2∈_0 with β_1+β_2<r, we have ∫ x_1^β_1 x_2^β_2ψ^(i)(x)x̣=0.
  
  For any n ∈ and x∈^2, let ϕ_n,x(y) := 2^nϕ(2^n(y - x)) and ψ^(i)_n,x(y) := 2^nψ^(i)(2^n(y - x)), and denote
  Λ_n = ^2/2^n. Then (ϕ_n,x)_x∈Λ_n is an orthonormal basis of V_n, while (ψ^(i)_n,x)_x∈Λ_n, n∈, i=1,2,3 is an orthonormal basis of L^2.
  
  Denote by _n and _n the orthogonal projections on V_n, W_n respectively. For  f∈ L^2(^2), we have
  
      _nf=∑_x∈Λ_n⟨ f,ϕ_n,x⟩_L^2ϕ_n,x,   _nf=∑_x∈Λ_n, i=1,2,3⟨ f,ψ^(i)_n,x⟩_L^2ψ^(i)_n,x.
  
  We then have that, for any k∈,
  
      f=_kf+∑_n=k^∞_nf.
  
  By <cit.>, ^α can be equivalently defined as the completion of C_c^∞ with respect to the norm
  
      ‖ f‖_^α:=‖_0f‖_∞+
      sup_n∈2^α n‖_nf‖_∞.
  
  For an open domain Ω⊂^2, ^α_loc(Ω) is the completion of C_c^∞ with respect to  the family of seminorms (‖χ·‖_^α)_χ∈ C_c^∞(Ω). We note that the discrete piecewise constant magnetisation field Φ_Ω^ belongs to the space of distributions ^α_loc(Ω) for any α<0, and we write ⟨Φ,f⟩ to denote Φ acting on the function f.
  
  
  Theorem <ref> also holds with ^α_loc(Ω) replaced by the Besov spaces _p,q^α,loc(Ω) for any p,q∈[1,∞], also constructed in <cit.>. For simplicity, we restrict ourselves to ^α_loc(Ω) which is continuously embedded in _p,q^α,loc(Ω) for any p,q∈[1,∞].
  
  
  
  
  
    
  *Taylor expansions
  For ϕ∈ L^2(Ω), write
  
      ϕ(x) :=^-2∫_S_(x)ϕ(y)ỵ,     ϕ_x^  :=^15/8ϕ(x),     ϕ_x^ :=^-1/8∫_S_(x)ϕ(y)ỵ
  
  for the smoothed version, the scaled version, the smoothed and scaled version, respectively.
  To prove convergence of finite dimensional distributions of the magnetisation field, we will consider its Fourier transform, which leads to 
  polynomial chaos expansion of the random field partition functions as in (<ref>) with ξ_·^ replaced by ξ_·^+iϕ_·^. We then use a Lindeberg principle to replace tanh(ξ_·^+iϕ_·^) with Gaussian random variables. The following estimates give the required control over the moments of these random variables.
  Using a Taylor expansion we have that tanh(x)=x-x^3/6+O(x^5) for x small. Therefore using the exponential moments of ω and recalling that ξ^_x:=λ_x^ω^_x+h_x^ and their definitions from (<ref>),
  we have that
  
  [(tanh(ξ_x^+iϕ_x^))]     = h_x^+O(^21/8),     [(tanh(ξ_x^+iϕ_x^))^2]  = (λ_x^)^2+O(^7/2),  
  
  [(tanh(ξ_x^+iϕ_x^))]     = ϕ_x^+O(^29/8),       [(tanh(ξ_x^+iϕ_x^))^2]  = (ϕ_x^)^2+O(^11/2),   
  
      [(tanh(ξ_x^+iϕ_x^))(tanh(ξ_x^+iϕ_x^))]
  = h_x^ϕ_x^+O(^9/2),  
  
  
  
  
  
  
  
  
  
  
  where (·) and (·) denote the real and imaginary parts respectively.
  
  
  
  
  § UNIQUENESS OF THE LIMIT
  
  
  To prove Theorem <ref>, it suffices to show that the laws of (W^ω,,Z_Ω;λ, h^ω,,μ^ω,_Ω;λ,h)_∈(0,1] are tight and the limit is unique. In this section, we will assume tightness and prove uniqueness of the limit.
  
  
  
  
  
   §.§ Convergence of the finite dimensional distributions
  
  Note that we can find a countable set of functions Λ⊂ C_c^∞(Ω) (including the function 0) such that every μ∈_1(^α_loc(Ω)) is uniquely determined by its characteristic functions
  
      μ̂(ϕ) := ∫ e^i ⟨ϕ, Φ⟩μ( dΦ),     ϕ∈Λ.
  
  Given tightness, to show that (W^ω,,Z_Ω;λ, h^ω,,μ^ω,_Ω;λ,h) converges to a unique limit as a↓ 0, it then suffices to show that (W^ω,,Z_Ω;λ, h^ω,, (μ̂^ω,_Ω;λ,h(ϕ))_ϕ∈Λ) converges to a unique limit, since the limit of μ̂^ω,_Ω;λ,h(ϕ) must be the characteristic function of the limit of μ^ω,_Ω;λ,h by the continuity of μ̂(ϕ) in μ. To identify the limit of (μ̂^ω,_Ω;λ,h(ϕ))_ϕ∈Λ, note that
  
      μ̂^ω,_Ω;λ,h(ϕ)     =^ω,_Ω;λ,h [ exp (i <ϕ,Φ^_Ω > )]
         =_Ω^ [exp (∑_x∈Ω_ (λ_x^ω^_x+h_x^+iϕ_x^ )σ_x ) ]/Z_Ω;λ, h^ω, = Z_Ω;λ, h+iϕ^ω,/Z_Ω;λ, h^ω,,
  
  and convergence would follow if we show convergence of (Z_Ω;λ, h+iϕ^ω,)_ϕ∈Λ to a limit (^W_Ω; λ, h+iϕ)_ϕ∈Λ. The limiting measure μ^W_Ω; λ, h would then be uniquely determined by its characteristic functions
  
      μ̂^W_Ω; λ, h(ϕ) := ^W_Ω; λ, h+ iϕ/^W_Ω; λ, h,     ϕ∈Λ,
  
  which is well-defined since ^W_Ω; λ, h>0 -a.s. by Lemma <ref>.
  
  Also note that W is uniquely determined by W_φ:=⟨ W, φ⟩, φ∈Λ.
  Therefore uniqueness of the limit for (W^ω,,Z_Ω;λ, h^ω,,μ^ω,_Ω;λ,h) would follow from the convergence in distribution of
  
      ( (W^ω,_φ)_φ∈Λ, (Z_Ω;λ, h+iϕ^ω,)_ϕ∈Λ).
  
  This is the content of the following proposition.
  
  Let Λ⊂ C_c^∞(Ω) be as above, then ((W^ω,_φ)_φ∈Λ, (Z_Ω;λ, h+iϕ^ω,)_ϕ∈Λ) converges in finite-dimensional distribution to ((W_φ)_φ∈Λ, (_Ω;λ, h+iϕ^W)_ϕ∈Λ) as ↓ 0, where _Ω;λ, h+iϕ^W are defined as in (<ref>). Furthermore, -a.s., (_Ω;λ, h+iϕ^W)_ϕ∈Λ are uniquely determined by W.
  
  
  
  
  
  The proof of the convergence of (Z_Ω;λ, h+iϕ^ω,)_ϕ∈Λ is based on chaos expansions as outlined in Section <ref>. For ϕ=0, this result was proved in <cit.>. Here we adapt the proof to handle the case with a complex external field. It requires several approximations (Lemmas <ref>-<ref>), the proof of which will be deferred to the next subsection. Along the way, we will also formulate a multivariate version of the Lindeberg principle (Lemma <ref>]) for polynomial chaos expansions.
  
  
  Let F, G be any finite subsets of C_c^∞(Ω).
  By (<ref>) we have that
  
      Z_Ω;λ, h+iϕ^ω,=θ_cosh(ξ_·^+iϕ_·^)^Ω_∑_I⊆Ω__Ω^[σ^I_·]tanh(ξ_·^+iϕ_·^)^I
  
  where, by Lemma <ref> below, the prefactor θ_cosh(ξ_·^+iϕ_·^)^Ω_ converges to 1 in probability. It remains to prove the joint convergence of ((W^ω,_φ)_φ∈ F, (Υ_ϕ)_ϕ∈ G) where
  
      Υ_ϕ:=∑_I⊆Ω_ψ^(I)(tanh(ξ_·^+iϕ_·^)/(ξ_·^)^1/2)^I,       ψ^(I)=(ξ_·^)^|I|/2_Ω^[σ^I_·].
  
  Note that (ξ_·^)=(λ_·^)^2. When ϕ=0, the convergence of Υ_ϕ to _Ω;λ, h^W was proved in <cit.>
  by verifying conditions in <cit.>. We will adapt the proof to the case of complex external fields.
  
  
  First we show that Υ_ϕ can be truncated to order l∈ such that the error is uniform for small  and can be made arbitrarily small by choosing l large. More precisely, let Υ_ϕ^≤ l denote the restriction of the sum  in (<ref>) to I with |I|≤ l. Then we have
  
  For ϕ∈ C_c^∞(Ω),
  
      lim_l→∞lim_→ 0^+[|Υ_ϕ-Υ_ϕ^≤ l|^2]    =0.
  
  
  Next, we linearise tanh(ξ_·^+iϕ_·^) and approximate Υ_ϕ^≤ l  by
  
      Ξ_ϕ^≤ l  :=∑_I⊆Ω_, |I|≤ lψ^(I)(ξ_·^+iϕ_·^/λ_·^)^I
  
  and show that
  
  For ϕ∈ C_c^∞(Ω),
  
      lim_l→∞lim_→ 0^+[|Υ_ϕ^≤ l-Ξ_ϕ^≤ l|^2]    =0.
  
  
  We then further approximate Ξ_ϕ^≤ l by
  
      Θ_ϕ^≤ l  :=∑_I⊆Ω_, |I|≤ l a^-|I|ψ^(I)(ϑ_·^+·h_·^+iϕ_·^/λ_·^)^I,
  
  where we have replaced ω^a_x in ξ^a_x= λ^a_x ω^a_x+h^a_x by the normal random variable ϑ_x^:=^-1∫_S_(x)W(ỵ) defined from the white noise W, and S_(x) is the box centred around x with side length . Note that Θ_ϕ is a Wiener chaos expansion with respect to W, where by Lemma <ref>, ^-nψ^(x_1, …, x_n) converges pointwise and in L^2(Ω^n) to the kernel ^n ϕ^+_Ω(x_1, …, x_n)∏_i=1^n λ(x_i) appearing in the definition of ^W_Ω; λ, h  in (<ref>).
  Using Itô isometry and the assumption that λ, h ∈ C^1_b(Ω), infλ>0, and ϕ∈ C_c^∞(Ω), it is then straightforward to check that ((W^_φ)_φ∈ F, (Θ_ϕ^≤ l)_ϕ∈ G) converges in L^2 to ((W_φ)_φ∈ F, (_Ω;λ,h+iϕ^W,≤ l)_ϕ∈ G) for any l∈, where given a chaos expansion Ψ, Ψ^≤ l denotes its truncation to terms of order at most l.
  
  
  
  
  
  
  
  
  
  It was proved in <cit.> that the chaos expansion for _Ω;λ,h^W converges in L^2. It is easily seen that the proof also applies to _Ω;λ,h+iϕ^W with ϕ∈ C_c^∞(Ω) (we can dominate h+iϕ by |h+iϕ|).  Therefore _Ω;λ,h+iϕ^W,≤ l converges in L^2 to _Ω;λ,h+iϕ^W as l→∞. To conclude that ((W^ω,_φ)_φ∈ F, (Υ_ϕ)_ϕ∈ G) converges in distribution to ((W_φ)_φ∈ F, (_Ω;λ,h+iϕ^W)_ϕ∈ G) as ↓ 0, we can just truncate the chaos expansions to an arbitrarily large order l and then show that for any bounded f with bounded first derivatives, we have
  
      lim_↓ 0|[f((W^ω,_φ)_φ∈ F, (Ξ^≤ l_ϕ)_ϕ∈ G)]-[f((W^_φ)_φ∈ F, (Θ_ϕ^≤ l)_ϕ∈ G)]|=0.
  
  Separating the real and imaginary parts and writing Ψ_ϕ,^≤ l=(Ψ_ϕ^≤ l) and Ψ_ϕ,^≤ l=(Ψ_ϕ^≤ l) for Ψ∈{Υ, Ξ,Θ}, it suffices to show that
  
  For all g bounded and differentiable with bounded first derivatives, we have
  
      lim_↓ 0|[g((W^ω,_φ)_φ∈ F, (Ξ^≤ l_ϕ,)_ϕ∈ G,(Ξ^≤ l_ϕ,)_ϕ∈ G)]-[g((W^_φ)_φ∈ F, (Θ_ϕ,^≤ l)_ϕ∈ G,(Θ_ϕ,^≤ l)_ϕ∈ G)] |
  
  is equal to 0.
  
  We will prove Lemmas <ref>–<ref> in the next subsection. Since (_Ω;λ, h+iϕ^W)_ϕ∈Λ is a countable family defined via Wiener chaos expansions with respect to the white noise W, they are almost surely determined by W. This concludes the proof of Proposition <ref>.
  
  
  
  
   §.§ Proof of Lemmas <ref>–<ref>
  
  
  Before starting the proof of Lemma <ref>, we first show that the prefactor in (<ref>) tends to 1 in probability as ↓ 0. The proof is similar to that in the proof of Theorem 3.14 in <cit.>.
  
  
  Let θ_:=e^-1/2^-1/4‖λ‖ _L^2^2. Then uniformly in σ∈{±1}^Ω_a, we have that
  
      θ_[exp(∑_x∈Ω_λ_x^ω_x^σ_x)] ⟶ 1   ↓ 0.
  
  Moreover, θ_∏_x∈Ω_cosh(λ_x^ω_x^+h_x^+iϕ_x^) converges to 1 in -probability for any ϕ∈ C_c^∞(Ω); this convergence also holds in L^p for any p>0 when ϕ≡ 0.
  
  
  Recall that λ_x^ :=^7/8λ(x) and h_x^ :=^15/8h(x). Note that
  
      log[exp(∑_x∈Ω_λ_x^ω_x^σ_x)] = ∑_x∈Ω_log[e^λ_x^ω_x^σ_x] = ∑_x∈Ω_( (λ^_x)^2/2 + R(λ_x^ω_x^σ_x)),
  
  where we applied Taylor expansion to the log-moment generating function of ω^_x, with error term
  
      |R(λ_x^ω_x^σ_x)| ≤ C ‖λ‖_∞^21/8.
  
  Therefore
  
      |logθ_a + log[exp(∑_x∈Ω_λ_x^ω_x^σ_x)]|
      ≤    | -1/2^-1/4‖λ‖ _L^2^2 + 1/2∑_x∈Ω_ (λ^_x)^2 | + C‖λ‖_∞∑_x∈Ω_^21/8.
  
  Since λ∈ C^1(Ω), the first difference is of order O(^7/4) and tends to 0 as ↓ 0 by a Riemann sum approximation with control on the error. The second term also tends to 0 uniformly in σ, which completes the proof of the first statement.
  
  
  
  
  
  Recall that ξ_x^=λ_x^ω_x^+h_x^ and note that by a Taylor expansion, we have that
  
      [log(cosh(ξ_x^+iϕ_x^))]=(λ_x^)^2/2+O((h_x^)^2+(ϕ_x^)^2+(λ_x^)^4)=(λ_x^)^2/2+O(^7/2)
  
  where the error term O(^7/2) is uniform over x by continuity of λ, h and ϕ. In particular,
  
      ∑_x∈Ω_[log(cosh(ξ_x^+iϕ_x^))] = ∑_x∈Ω_(λ_x^)^2/2 + O(^3/2)
      = -logθ_ +O(^3/2).
  
  Therefore
  
      θ_∏_x∈Ω_cosh(ξ_x^+iϕ_x^)
      
      =    e^O(^3/2)exp(∑_x∈Ω_log(cosh(ξ_x^+iϕ_x^))-[log(cosh(ξ_x^+iϕ_x^))]).
  
  
  The sum is over |Ω_| independent centred random variables, each with variance at most
  
      [log(cosh(ξ_x^+iϕ_x^))^2]=O((λ_x^)^4)=O(^7/2).
  
  Therefore (<ref>) converges to 1 in probability by a weak law of large numbers.
  
  Lastly, to show that θ_∏_x∈Ω_cosh(λ_x^ω_x^+h_x^) converges to 1 in L^p for any p>0, it suffices to show that for any k∈, [θ_a^k ∏_x∈Ω_cosh(λ_x^ω_x^+h_x^)^k] is bounded as ↓ 0. This is straightforward to verify by expanding
  (cosh y)^k = (e^y +e^-y/2)^k and applying Taylor expansion to the exponential moment generating function of ω^_x. We omit the details. We remark that the L^p convergence statement should also hold for ϕ≠ 0, but the details will be more tedious and is not needed so we left it out.
  
  
  
  The proof follows by using a Taylor expansion of tanh(x) to bound moments of tanh(ξ_x+iϕ_) and L^2 estimates of the kernel ψ^. This is similar to <cit.>.
  We will bound the difference of the real parts |Υ_ϕ, -Υ_ϕ, ^≤ l|. The imaginary part follows similarly.
  
  Let
  
      _x  :=(tanh(ξ_x^+iϕ_x^))/λ_x^        _x  :=(tanh(ξ_x^+iϕ_x^))/λ_x^
  
  and _x :=_x-[_x], _x :=_x-[_x], so that
  
      tanh(ξ_x^+iϕ_x^))/λ_x^ = _x+ [_x] + i_x + i[_x].
  
  Let ψ^_>l(I)=ψ^(I)_|I|>l. Expanding (<ref>), we note that
  
      Υ_^>l  := Υ_ϕ, -Υ_ϕ, ^≤ l    = ∑_0pt3K_1, K_2, K_3, K_4⊂Ω_ |K_2|+|K_4|  (-1)^(|K_2|+|K_4|)/2
      (_·)^K_1 (_·)^K_2[_·]^K_3[_·]^K_4ψ^_>l(∪_i K_i) 
          = ∑_K_1, K_2 ⊂Ω_ (_·)^K_1 (_·)^K_2ψ^_>l(K_1,K_2),
  
  where
  
      ψ^_>l(K_1,K_2)
         := ∑_0pt3K_3, K_4 ⊂Ω_: K_1, K_2, K_3, K_4 |K_2|+|K_4|  (-1)^(|K_2|+|K_4|)/2[_·]^K_3[_·]^K_4ψ^_>l(∪_i K_i)
         ≤∑_0pt3K_3 ∩ K_4 =∅ (K_3∪ K_4)∩ (K_1∪ K_2)=∅ψ^_>l(∪_i K_i) (C)^|K_3∪ K_4|,
  
  and we used the fact that by (<ref>),  |[_x]|,|[_x]| are uniformly bounded by C for some C>0. Note that the above bound depends only on K_1∪ K_2.
  Using that [(_·)^K_1 (_·)^K_2(_·)^K'_1 (_·)^K'_2]=0 when K_1∪ K_2 ≠ K_1'∪ K_2', we can compute
  
      [(Υ_^>l)^2] = ∑_U⊂Ω_a∑_0pt3K_1∩ K_2=K'_1∩ K'_2=∅K_1∪ K_2=K_1'∪ K_2'=Uψ^_>l(K_1,K_2) ψ^_>l(K'_1,K'_2) [(_·)^K_1 (_·)^K_2(_·)^K'_1 (_·)^K'_2].
  
  The expectation can be bounded using Cauchy-Schwarz and independence as follows:
  
      [(_·)^K_1 (_·)^K_2(_·)^K'_1 (_·)^K'_2]
         ≤[(^2_·)^K_1 (^2_·)^K_2]^1/2[(^2_·)^K'_1 (^2_·)^K'_2]^1/2
          = ([^2_·]^K_1[^2_·]^K'_1[^2_·]^K_2[^2_·]^K_2')^1/2
         ≤ (1+C ^7/4)^|K_1|+|K'_1| (C a)^|K_2|+|K'_2|,
  
  where we used that by (<ref>),  [_x^2]≤ 1+C a^7/4 and [_x^2]≤ Ca^2 for some C>0 uniformly in x∈Ω_a. On the other hand, denoting U:=K_1∪ K_2=K_1'∪ K_2', by (<ref>) we have
  
      ψ^_>l(K_1,K_2) ψ^_>l(K'_1,K'_2)    ≤(∑_0pt3K_3 ∩ K_4 =∅(K_3∪ K_4)∩ U=∅ψ^_>l(∪_i K_i) (Ca)^|K_3∪ K_4|)^2  
         ≤∑_0pt3K_3 ∩ K_4 =∅ (K_3∪ K_4)∩ U=∅ε^|K_3∪ K_4|ψ^_>l(∪_i K_i)^2  ∑_0pt3K_3 ∩ K_4 =∅ (K_3∪ K_4)∩ U=∅  (ε^-1C^2a^2)^|K_3∪ K_4|
          = ∑_V⊂Ω_a \ U (2ε)^|V|ψ^_>l(U ∪ V)^2 ∑_V⊂Ω_a \ U
      (2ε^-1C^2a^2)^|V|
          = C_ε∑_V⊂Ω_a \ U (2ε)^|V|ψ^_>l(U ∪ V)^2
  
  where C_ε is a uniform bound on (1+ 2ε^-1C^2a^2)^|Ω_a| for  small. Substituting this bound together with (<ref>) into (<ref>) then gives
  
      [(Υ_^>l)^2] 
      ≤    ∑_U⊂Ω_a∑_0pt3K_1∩ K_2=K'_1∩ K'_2=∅ K_1∪ K_2=K_1'∪ K_2'=U C_ε (1+C ^7/4)^|K_1|+|K'_1| (C a)^|K_2|+|K'_2|∑_V⊂Ω_a \ U (2ε)^|V|ψ^_>l(U ∪ V)^2 
      ≤     C_ε∑_U∩ V=∅  (1+Ca)^|U| (2ε)^|V|ψ^_>l(U ∪ V)^2 
      
      =     C_ε∑_I⊂Ω_a  (1+η)^|I|ψ^_>l(I)^2,
  
  where η:=Ca+2ε can be made arbitrarily small by choosing ε small and let ↓ 0. This was shown to converge to 0 as → 0 then l→∞ in the proof of Theorem 3.14 in <cit.>.
  
  
  
  This follows from the moment bounds on tanh(ξ_x^+iϕ_x^) used in Lemma <ref> and a suitable representation of the kernel ψ^ on sets of size at most l. This is similar to <cit.>.
  We bound the difference of the real parts Υ_ϕ, ^≤ l-Ξ_ϕ, ^≤ l. The imaginary part follows similarly.
  
  Let _x:=ξ_x^/λ_x^, _x:=ϕ_x^/λ_x^, _x:=_x-[_x], _x:=_x-[_x]. Let ψ^_≤ l(K_1,K_2) be defined the same as ψ^_>l(K_1,K_2) in (<ref>), except that [_x] and [_x] are replaced by [_x] and [_x] respectively, and ψ_>l^(I) is replaced by ψ_≤ l^(I):=ψ^(I)1_|I|≤ l. By the same decomposition as in (<ref>), we have
  
      Ξ^≤ l_ϕ,  =  ∑_K_1, K_2 ⊂Ω_ (_·)^K_1 (_·)^K_2ψ^_≤ l(K_1,K_2).
  
  Therefore
  
      1/2[|Υ_ϕ, ^≤ l-Ξ_ϕ, ^≤ l|^2]
         ≤[|∑_K_1, K_2⊂Ω_((_·)^K_1(_·)^K_2-(_·)^K_1(_·)^K_2)ψ^_≤ l(K_1,K_2)|^2] 
            +[|∑_K_1, K_2⊂Ω_ (_·)^K_1(_·)^K_2(ψ^_≤ l(K_1,K_2)-ψ^_≤ l(K_1,K_2))|^2].
  
  The first term can be bounded in the same way as in the proof of Lemma <ref>. By Taylor expanding tanh(ξ^a_x+ i ϕ^a_x), we note that, uniformly in K_1, K_2⊂Ω_a with |K_1|+|K_2|≤ l, we have
  
      [((_·)^K_1(_·)^K_2-(_·)^K_1(_·)^K_2)^2] = o(1)[_·^2]^K_1[^2_·]^K_2     a↓ 0.
  
  It is then easily seen that the first term in (<ref>) tends to 0.
  
  The second term in (<ref>) can be bounded similarly, using that when we compare the definitions of ψ^_≤ l with that of ψ^_≤ l as in (<ref>), we note that uniformly in K_3, K_4⊂Ω_a with |K_3|+|K_4|≤ l, we have
  
      |[_·]^K_3[_·]^K_4- [_·]^K_3[_·]^K_4| = o(1)|[_·]^K_3[_·]^K_4|       a↓ 0,
  
  from which it follows that the second term of (<ref>) also tends to 0.
  
  
  
   Note that (W^ω,_φ)_φ∈ F, (Ξ^≤ l_ϕ,)_ϕ∈ G, (Ξ^≤ l_ϕ,)_ϕ∈ G are polynomial chaos expansions with respect to the random variables (ω^_x)_x∈Ω_, with kernels
  
      ψ^_φ(I)
          = φ(x)    if  I={x}, 
       0    otherwise,
      ψ^_ϕ,(I)
          = ∑_0pt3K_1, K_1 ⊂Ω_\ I K_1∩ K_2=∅, |K_2|∈ 2ψ_≤ l^(I∪ K_1∪ K_2) (h^_·/λ^_·)^K_1(ϕ^_·/λ^_·)^K_2(-1)^|K_2|/2, 
      ψ^_ϕ,(I)
          = ∑_0pt3K_1, K_1 ⊂Ω_\ I  K_1∩ K_2=∅, |K_2|+1∈ 2ψ_≤ l^(I∪ K_1∪ K_2) (h^_·/λ^_·)^K_1(ϕ^_·/λ^_·)^K_2(-1)^(|K_2|-1)/2,
  
  while (W^_φ)_φ∈ F, (Θ_ϕ,^≤ l)_ϕ∈ G, (Θ_ϕ,^≤ l)_ϕ∈ G are polynomial chaos expansions with respect to the Gaussian random variables (ϑ_x^)_a∈Ω_ with the same kernel. Lemma <ref> then follows from a  Lindeberg principle for polynomial chaos expansions. When only one polynomial chaos expansion is under consideration, such a Lindeberg principle was proved in <cit.>. We need here a multivariate version, which we formulate as follows.
  
  
  Let  be a finite index set, and let (ω_x)_x∈ and (ϑ_x)_x∈ be two different families of i.i.d. random variables with zero mean, unit variance, and finite third absolute moments. Let (ψ_i)_1≤ i≤ n be a family of kernels for polynomial chaos expansions of order at most l∈, and let Ψ(ω):=(Ψ_i(ω))_1≤ i≤ n and Ψ(ϑ):=(Ψ_i(ϑ))_1≤ i≤ n be the corresponding polynomial chaos expansions with respect to ω_· and ϑ_·, with kernel ψ_i. For all g: ^n → three times differentiable with sup_|α|≤ 3‖ D^α g‖ _∞<∞, there exists a constant C_g, l, n such that
  
      |[g(Ψ(ω))]-[g(Ψ(ϑ))]|
       ≤ C_g,l,n M^l ∑_i=1^n𝕍ar(Ψ_i) ( max_x∈ Inf_x[ψ_i])^1/2,
  
  where M=max{[|ω_x|^3], [|ϑ_x|^3]} and Inf_x[ψ_i] = ∑_I∋ xψ_i(I)^2.
  
  
  We defer the proof of Lemma <ref> and first apply it by verifying that for each of the kernels ψ^_* in (<ref>), we have that the variance stays bounded while
  
      lim_a↓ 0max_x∈Ω_a Inf_x(ψ^_*)=0.
  
  
  Note 𝕍 ar(W^ω, _φ)=∑_x∈Ω_^-2 (∫_S_(x)φ(x)  dx)^2→‖φ‖_2^2, while
  Inf_x[ψ^_φ]=a^2 φ(x)^2≤ a^2 |φ|_∞^2 which tends to 0 as a↓ 0. Therefore the conditions of Lemma <ref> hold for W^ω, _φ and its kernel ψ^_φ.
  
  Similarly, we have
  
      𝕍 ar(Ξ^≤ l_ϕ,) ≤[(Ξ^≤ l_ϕ,)^2]
      
      =    ∑_I (∑_0pt3K_1, K_1 ⊂Ω_\ I  K_1∩ K_2=∅, |K_2|∈ 2ψ_≤ l^(I∪ K_1∪ K_2) (h^_·/λ^_·)^K_1(ϕ^_·/λ^_·)^K_2(-1)^|K_2|/2)^2  
      ≤    ∑_I (∑_J⊂Ω_\ Iψ_≤ l^(I∪ J) (|h^_·| + |ϕ^_·|/λ^_·)^J)^2 
      ≤    ∑_I (∑_J⊂Ω_\ Iψ_≤ l^(I∪ J)^2  ) (∑_J⊂Ω_\ I(|h^_·| + |ϕ^_·|/λ^_·)^2J) 
      ≤     (1+Ca^2)^|Ω_|∑_U 2^|U|ψ^_≤ l(U)^2,
  
  where we used that by (<ref>) and (<ref>) and the assumption inf_x λ(x)>0, we can bound |h^_·| + |ϕ^_·|/λ^_· uniformly by Ca for some finite C. It follows from Lemma <ref> that  ^-kψ^(x_1, …, x_k) converges in L^2(Ω^k) for each k∈ and hence 𝕍 ar(Ξ^≤ l_ϕ,) is uniformly bounded as ↓ 0. Similarly,
  
      Inf_x(ψ^_ϕ,)     = ∑_I∋ x(∑_0pt3K_1, K_1 ⊂Ω_\ I K_1∩ K_2=∅, |K_2|∈ 2ψ_≤ l^(I∪ K_1∪ K_2) (h^_·/λ^_·)^K_1(ϕ^_·/λ^_·)^K_2(-1)^|K_2|/2)^2 
         ≤∑_U∋ x 2^|U|ψ^_≤ l(U)^2.
  
  Since for each k∈, ∑_|U|=kψ^(U)^2 converges to a finite integral as shown in <cit.>, it is not difficult to see that lim_↓ 0max_x ∑_U∋ xψ^_≤ l(U)^2=0. For more details, see <cit.>.
  
  In summary, the conditions of Lemma <ref> also hold for Ξ^≤ l_ϕ, and its kernel ψ^_ϕ,. The case
  of Ξ^≤ l_ϕ, is similar, which completes the proof of Proposition <ref>.
  
  
  
  Label the elements of  by 1, …, || according to any fixed order.  Write
  
      ζ^j:=(ζ_i^j)_1≤ i≤ || where ζ_i^j=ω_i    if  i≤ j, 
      ϑ_i     if  i>j,
  
  which replaces ω_· by ϑ_· one variable at a time. By a telescoping sum, we have
  
      [g(Ψ(ω))]-[g(Ψ(ϑ))] = ∑_j=1^||([g(Ψ(ζ^j))]-[g(Ψ(ζ^j-1))]).
  
  Writing g((y_i)_1≤ i≤ n) and Ψ_i((x_j)_1≤ j≤ ||), we note that g(Ψ(ζ^j)) and g(Ψ(ζ^j-1)) differ only in the variable x_j. Denote ϱ^ω, ϑ_j(x) = g(Ψ(ζ^j,x)) where ζ^j,x_j=x and ζ̅^j,x_i=ζ^j_i for i≠ j. By Taylor expansion,
  
      ϱ^ω, ϑ_j(x)= ϱ^ω, ϑ_j(0) + x (ϱ^ω, ϑ_j)'(0)+x^2/2(ϱ^ω, ϑ_j)”(0)+_j^ω, ϑ(x)
  
  where |_j^ω, ϑ(x)|≤ C|x|^3| (ϱ^ω, ϑ_j)”'| _∞. Note that (ϱ^ω, ϑ_j)'(0) and (ϱ^ω, ϑ_j)”(0) do not depend on ω_j or ϑ_j. Therefore using the fact that ω_j and ϑ_j have the same mean and variance, we obtain
  
      [g(Ψ(ζ^j))-g(Ψ(ζ^j-1))]    =[ϱ^ω, ϑ_j(ω_j) -ϱ^ω, ϑ_j(ϑ_j)]
      =[_j^ω, ϑ(ω_j)] - [_j^ω, ϑ(ϑ_j)].
  
  Substituting into (<ref>) then gives
  
      |[g(Ψ(ω^))]-[g(Ψ(ϑ^))] |
      ≤∑_j=1^||([|_j^ω, ϑ(ω_j)|] + [|_j^ω, ϑ(ϑ_j)|]).
  
  Let ∂_i g and ∂_j Ψ_i denote the derivative with respect to the i-th and j-th component of g and Ψ_i, respectively. Note that
  
      (ϱ^ω, ϑ_j)”'(x)     = ∑_i_1, i_2, i_3=1^n ∂^3_i_1, i_2, i_3 g(Ψ(ζ^j,x)) ∂_jΨ_i_1(ζ^j) ∂_jΨ_i_2(ζ^j) ∂_jΨ_i_2(ζ^j),
  
  where we used the fact that Ψ_i(ζ^j,x) is linear in x.  Bounding the derivatives of g by their sup-norm and applying the triangle inequality, we obtain
  
      |_j^ω, ϑ(ω_j)|
       ≤  C   |ω_j|^3| (ϱ_j^ω, ϑ)”'|_∞≤ C_g, n∑_i=1^n |ω_j ∂_j Ψ_i(ζ^j)|^3,
  
  Substituting this bound into (<ref>) then gives
  
      |[g(Ψ(ω^))]-[g(Ψ(ϑ^))] |
      ≤ C_g,n∑_i=1^n ∑_j=1^||[|ω_j ∂_j Ψ_i(ζ^j)|^3].
  
  Note that
  
      [|ω_j ∂_j Ψ_i(ζ^j)|^2] = ∑_I ∋ jψ_i(I)^2 =  Inf_j[ψ_i].
  
  The arguments to bound (<ref>) are exactly the same as in the proof of <cit.>, starting from (4.17) therein, by applying hypercontractivity to [|ω_j ∂_j Ψ_i(ζ^j)|^3] to get a bound of the form
  C_l  Inf_j[ψ_i]^3/2. The proof there assumed only finite second moments for ω and ϑ and hence required a truncation to apply hypercontractivity. Here we have assumed finite third absolute moments, which allows us to apply hypercontractivity directly. Comparing with the statement of <cit.> then gives the desired bound (<ref>).
  
  
  
  
  
  
  § TIGHTNESS
  
  
  In this section we prove that the laws of (W^ω,,Z^ω,_Ω;λ,h,μ^ω,_Ω;λ,h) are tight in ××_1(^α_loc(Ω)).
  Tightness of (W^ω,)_∈(0,1] is standard and can be checked using the tightness criterion of <cit.> similarly to below. Therefore we omit the proof.
  It was shown in Proposition <ref> that Z^ω,_Ω;λ,h converges in distribution as a↓ 0 and hence their laws are tight. 
  
  
  
  
  
  It remains to show that the family of laws of (μ^ω,_Ω;λ,h)_∈(0,1] is tight. As random probability measures on the complete separable metric space ^α_loc(Ω), where α<-1/8, tightness would follow if we show that
  
      ∀ ε>0   ∃  compact  K_ε⊂^α_loc(Ω)     sup_∈ (0,1][μ^ω,_Ω;λ,h(K_ε^c)]≤ε.
  
  This follows from the tightness criteria for random probability measures in <cit.>. Note that
  	
      [μ^ω,_Ω;λ,h(K_ε^c)] ≤( Z_Ω;λ,h^ω,≤η) +
      	[μ^ω,_Ω;λ,h(K_ε^c) _{Z_Ω;λ,h^ω,≥η}].
  
  Since Z_Ω;λ,h^ω, converges in distribution to ^W_Ω; λ,h, which is positive almost surely by Lemma <ref>, we can choose η>0 sufficiently small such that
  
      lim sup_↓ 0( Z_Ω;λ,h^ω,≤η) ≤ε.
  
  The tightness of (μ^ω,_Ω;λ,h)_∈(0,1] as _1(^α_loc(Ω))-valued random variables would then follow from the tightness of the average quenched laws ([μ^ω,_Ω;λ,h(·);Z_Ω;λ,h^ω,≥η]) for every η>0. To verify this, note that for any compact K⊂^α_loc(Ω),
  
      [μ^ω,_Ω;λ,h(K);Z_Ω;λ,h^ω,≥η]     =  [θ_/Z_Ω;λ,h^ω,_Ω^[ e^∑_x∈Ω_(λ_x^ω^_x+h_x^)σ_x_{Φ^_Ω∈ K }]_{Z_Ω;λ,h^ω,≥η}] 
         ≤1/η_Ω^[ θ_[e^∑_x∈Ω_λ_x^ω^_x σ_x] e^∑_x∈Ω_h_x^σ_x_{Φ^_Ω∈ K }] 
         ≤2/η_Ω^[e^∑_x∈Ω_h_x^σ_x_{Φ^_Ω∈ K }] 
         ≤2/η_Ω^[e^∑_x∈Ω_2h_x^σ_x]^1/2μ^_Ω(K)^1/2≤ C_ημ^_Ω(K)^1/2,
  
  where we used that θ_[e^∑_x∈Ω_λ_x^ω_x^σ_x]→ 1 uniformly in σ as → 0 by Lemma <ref>,  and
  
      _Ω^[e^∑_x∈Ω_2h_x^σ_x] ≤_Ω^[e^∑_x∈Ω_2|h_x^| σ_x] ≤_Ω^[e^2‖ h‖_∞^15/8∑_x∈Ω_σ_x]
  
  is bounded above as ↓ 0 by the exponential moment bound for the magnetisation random variable <cit.>. Since (μ^_Ω)_∈ (0,1] is a tight family of probability measures on ^α_loc(Ω) for any α<-1/8, as shown in <cit.>, it follows that ([μ^ω,_Ω;λ,h(·);Z_Ω;λ,h^ω,≥η]) is also tight for any η>0. This concludes the proof of tightness.
  
  
  
  
  
  § SINGULARITY
  
  
  
  
   §.§ Proof of Theorem <ref>
  
  In this section we prove Theorem <ref>, which shows that the disordered continuum limit μ_Ω;λ,h^W is almost surely singular with respect to the pure continuum limit μ_Ω, but its average with respect to W is absolutely continuous with respect to μ_Ω. We first prove the second statement which is straightforward and similar to <cit.>.
  
  
   Note that by (<ref>) and (<ref>) and computing characteristic functions, we find that
  
      [_Ω;λ,h^Wμ_Ω;λ,h^W(·)]=μ_Ω;0,h^W(·) = μ_Ω; h(·),
  
  which is the law of the continuum magnetisation field with external field h but without disorder. Since _Ω;λ,h^W>0 -a.s. by Lemma <ref>, for any measurable set A with μ_Ω;0,h^W(A)=0, we can find a set E_A with
  (W∈ E_A)=1 such that μ_Ω;λ,h^W(A)=0 for all W∈ E_A. Therefore μ_Ω;λ,h^W(A)=0 and hence μ_Ω;λ,h^W is absolutely continuous with respect to μ_Ω; h. By <cit.> we have that μ_Ω;h is absolutely continuous with respect to μ_Ω, which completes the proof of the absolute continuity.
  We remark that E_A depends on A, and there is no contradiction with the claim that almost surely,
  μ_Ω;λ,h^W is singular with respect to μ_Ω, because ∩ E_A will not be a measurable set with positive probability.
  
  
  
  Our proof is based on discrete approximations and techniques for bounding fractional moments that were first developed in the study of the disordered pinning model (see <cit.>). We will in fact show that the almost sure singularity of μ^W_Ω; λ, h with respect to μ_Ω is a local phenomenon, namely that if Ω is any open subset of Ω and χ_ΩΦ is the restriction of Φ∈^α_loc(Ω) to Ω, then almost surely, μ^W_Ω; λ, h∘χ_Ω^-1 is singular with respect to μ_Ω∘χ_Ω^-1.
  
  
  Let us assume for the moment that h≡0. To ease notation, we shall drop the subscripts Ω,λ,h when it is unambiguous. In particular, μ^W:=μ^W_Ω; λ, h and μ:=μ_Ω. Probability and expectation under μ and μ^W will be denoted by ,  and ^W, ^W respectively; probability and expectation under the law of white noise W will be denoted by  and  respectively.
  
  We first explain the proof strategy. Let Ω be any square sub-domain of Ω. Without loss of generality, we may assume Ω= (0,1)^2. To show that for  almost every W, μ^W∘χ_Ω^-1 is singular with respect to μ∘χ_Ω^-1, if suffices to pick a filtration (^Φ_N) on _loc^α, which will be generated by functions of the magnetisation field Φ restricted to Ω, and show that restricted to _N^Φ, the Radon-Nikodym derivative of μ^W with respect to μ,
  
      Q_N^W:=μ̣^W|__N^Φ/μ̣|__N^Φ,
  
  converges in -probability to 0 as N→∞. To prove this for -a.e. W, it suffices to show that
  
      lim_N→∞[ (Q_N^W)^1/2] = 0.
  
  We then try to approximate Q_N^W by its lattice analogue, Q_N^ω,. The difficulty lies in the fact that we only have the convergence of the law of the magnetisation fields, which does not imply the weak convergence of the Radon-Nikodym derivatives Q_N^ω, to Q_N^W. To overcome this difficulty and have a quantity more amenable to lattice approximations, we
  replace Q_N^W by
  
      _N:=[Q_N^W|_N^W],
  
  where ^W_N is a σ-algebra on _loc^α' generated by functions of the white noise W restricted to Ω. However, this is still not suitable for lattice approximation for reasons explained in Remark <ref>. We need to further approximate the random variables that generate ^W_N and ^Φ_N by random variables that take values in a discrete set 2^-m with m∈ chosen arbitrarily large.
  
  We now define the functions of W and Φ that will generate ^W_N and ^Φ_N, as well as their discrete approximations.
  Making the right choices for these functions is crucial for the fractional moment bounds to be carried out later.
  
  Let N=2^n. For i,j=1,...,N, denote B_i,j^N :=(i-1/N,i/N)×(j-1/N,j/N), which partitions Ω=(0,1)^2 into boxes of side length 1/N. Let Φ∈_loc^α be arbitrary, and let W be sampled from _loc^α' according to the probability measure . We then define
  
      Φ_λ, i,j^N := ⟨Φλ^2,_B_i,j^N⟩,     
      W_λ, i,j^N := ⟨ W,λ_B_i,j^N⟩, 
      _N^Φ :=σ({Φ_λ, i,j^N}_i,j=1^N),     _N^W :=σ({W_λ, i,j^N}_i,j=1^N),
  
  where ⟨Φλ^2,_B_i,j^N⟩ is well-defined and continuous in Φ∈^α_loc for α∈ (-1,0) by Lemma <ref> and the fact that Φ→λ^2 Φ is a continuous map from ^α_loc to itself because λ^2 ∈ C^1(Ω) (see e.g. <cit.>). This induces a σ-algebra _N^Φ on ^α_loc. As a stochastic integral, the mapping ⟨ W, λ_B_i,j^N⟩ is well-defined for -a.e. W∈^α', and hence it induces a σ-algebra _N^W on ^α'_loc if we complete the Borel σ-algebra on ^α' with sets of  measure 0. Let _N := _N^Φ×_N^W denote the product σ-algebra on _loc^α×_loc^α'. Note that if (W, Φ) are sampled according to ×, while given W, Φ^W is sampled according to ^W, then _N is simply the Radon-Nikodym derivative of the law of (W^N_λ, i,j, Φ^W,N_λ,i,j)_i,j=1^N with respect to the law of (W^N_λ, i,j, Φ^N_λ, i,j)_i,j=1^N.
  
  Next, we further approximate the random variables W^N_λ, i,j, Φ^N_λ, i,j, and Φ^W,N_λ, i,j by discrete valued functions
  as follows. For m∈, define
  
      Φ_λ, i,j^N, m := 2^-m⌊ 2^m Φ_λ, i,j^N⌋,      
      W_λ, i,j^N,m := 2^-m⌊ 2^m W_λ, i,j^N ⌋,  
      _N,m^Φ :=σ({Φ_λ, i,j^N,m}_i,j=1^N),     _N,m^W :=σ({W_λ, i,j^N,m}_i,j=1^N),
  
  and let _N,m := _N,m^Φ×_N,m^W. Note that (^Φ_N,m)_m∈ and (^W_N,m)_m∈ form filtrations that generate _N^Φ and _N^W respectively. Denote
  
      _N,m = [_N| _N,m]],
  
  which is just the Radon-Nikodym derivative of the law of (W^N,m_λ,i,j, Φ^W,N,m_λ, i,j)_i,j=1^N with respect to the law of (W^N,m_λ, i,j, Φ^N,m_λ, i,j)_i,j=1^N, where (W, Φ, Φ^W) are sampled according to ××^W. To prove that for  almost every W, μ^W is singular with respect to μ, it then suffices to show that
  
      lim_N→∞lim_m→∞[_N,m^1/2] = 0.
  
  
  We now define the lattice analogue of _N,m. We are free to choose the law of the disorder ω as long as Theorem <ref> can be applied. Therefore we let ω^_x:= ^-1∫_S_(x)W(ỵ) so that (ω^_x)_x∈Ω_ are standard Gaussian random variables defined from the white noise W.  Instead of working with the rescaled piecewise constant magnetisation field Φ^ defined in (<ref>), we need to work with
  
      W^ := ∑_x∈Ω_ω_x^δ_x         Φ^a := ∑_x∈Ω_^15/8σ_xδ_x,
  
  where the spin configuration σ is sampled according to ^a_Ω. If σ is sampled according to ^ω, _Ω; λ ,0, we will denote the resulting magnetisation field by Φ^ω, instead. We can apply the same operations as in (<ref>) and (<ref>) to obtain W^; N_λ, i,j, Φ^;N_λ, i,j, Φ^ω, ;N_λ, i,j, W_λ, i,j^a; N, m, Φ_λ, i,j^a; N, m and Φ_λ, i,j^ω, ; N, m.
  
  For N=2^n, m∈ and ∈(0,1], let _N^ and _N,m^ be the σ-fields generated by (W^; N_λ, i,j, Φ^;N_λ, i,j)_i,j=1^N and (W^; N, m_λ, i,j, Φ^; N, m_λ, i,j)_i,j=1^N, respectively then, define
  
      _N^:=[μ̣^ω,/μ̣^|_N^]     _N,m^:=[μ̣^ω,/μ̣^|_N,m^].
  
  Note that _N^ is the Radon-Nikodym derivative of the law of (W^; N_λ, i,j, Φ^ω, ; N_λ, i,j)_i,j=1^N with respect to the law of (W^; N_λ, i,j, Φ^; N_λ, i,j)_i,j=1^N, while _N,m^ is the Radon-Nikodym derivative of the law of (W^; N, m_λ, i,j, Φ^ω, ; N, m_λ, i,j)_i,j=1^N with respect to the law of (W^; N, m_λ, i,j, Φ^; N, m_λ, i,j)_i,j=1^N.
  
  
  To carry through the lattice approximation, we need the following convergence result.
  
  Denote the laws of (W^N,m_λ,i,j, Φ^N,m_λ, i,j)_i,j=1^N and (W^N,m_λ,i,j, Φ^W,N,m_λ, i,j)_i,j=1^N by  μ_N,m and ν_N,m respectively, where (W, Φ, Φ^W) is sampled from ××^W. Similarly, let μ^_N,m and ν^_N,m denote the law of (W^a; N,m_λ,i,j, Φ^; N,m_λ, i,j)_i,j=1^N and (W^a; N,m_λ,i,j, Φ^ω, ; N,m_λ, i,j)_i,j=1^N, where (W^, Φ^, Φ^ω, ) is sampled from ×^_Ω×^ω, _Ω; λ, 0. For any N and m, we have μ^_N,m⇒μ_N,m and ν^_N,m⇒ν_N,m as ↓ 0.
  
  Since μ_N,m, ν_N,m, μ_N,m^ and ν^_N,m are all supported on the discrete set (2^-m)^N^2, we have
  
      [_N,m^1/2]      = ∑_x∈ (2^-m)^N^2(ν_N,m(x)/μ_N,m(x))^1/2μ_N,m(x) 
         ≤lim inf_↓ 0∑_x∈ (2^-m)^N^2(ν^_N,m(x)/μ^_N,m(x))^1/2μ^_N,m(x) = lim inf_↓ 0[(_N,m^)^1/2].
  
  Therefore to prove (<ref>), it suffices to show that
   
  We have
  
      lim_N→∞lim_m→∞lim inf_↓ 0[(_N,m^)^1/2]=0.
  
  
  This would conclude the proof of the almost sure singularity of μ^W with respect to μ when h≡0. We will prove Lemmas <ref> and <ref> in the next two subsections.
  
  To extend the singularity result to general h∈ C^1(Ω), notice that the Radon-Nikodym derivative of the joint law of (W, Φ) under ^ω, _Ω; λ, h+Δ with respect to the joint law of (W, Φ) under ^ω, _Ω; λ, h is given by
  
      μ̣^ω,_Ω;λ,h+Δ/μ̣^ω,_Ω;λ,h(Φ)
        =  exp(⟨Φ, Δ⟩)Z^ω,_Ω;λ,h/Z^ω,_Ω;λ,h+Δ =: Y,
  
  which is uniformly integrable with respect to ^ω, _Ω; λ, h because for any C>0,
  
      ^ω, _Ω; λ, h[Y _Y>C]
          = ^ω, _Ω; λ, h+Δ[_Y>C] 
         ≤ C^-1^ω, _Ω; λ, h+Δ[e^⟨Φ, Δ⟩Z^ω,_Ω;λ,h/Z^ω,_Ω;λ,h+Δ] 
          = C^-1^_Ω[e^∑_x σ_x Δ^_x·e^∑_x σ_x (λ^_x ω^_x+ h^_x+Δ^_x)/Z^ω, _Ω;λ, h+Δ·Z^ω,_Ω;λ,h/Z^ω,_Ω;λ,h+Δ] 
          = C^-1[Z^ω,_Ω;λ,h+2Δ Z^ω,_Ω;λ,h/(Z^ω,_Ω;λ,h+Δ)^2] 
         ≤ C^-1[(Z^ω,_Ω;λ,h+2Δ)^3]^1/3[(Z^ω,_Ω;λ,h)^3]^1/3[(Z^ω,_Ω;λ, h+Δ)^-6]^1/3,
  
  where the moments are uniformly bounded by Lemma <ref>. Since h and Δ can be chosen arbitrarily, we can now apply Lemma <ref> to conclude that as ↓ 0, the limiting law of (W, Φ) under ^W_Ω; λ, h_1 is mutually absolutely continuous with respect to the law of (W, Φ) under ^W_Ω; λ, h_2. In particular, for -a.e. W, ^W_Ω; λ, h_1 is mutually absolutely continuous with respect to ^W_Ω; λ, h_2. Since ^W_Ω; λ, 0 is singular with respect to _Ω, so must be ^W_Ω; λ, h  for any h∈ C^1(Ω).
  
  
  
  
  
   §.§ Proof of Lemma <ref>
  
  
  
  By Theorem <ref>, (W^, Φ^, Φ^ω, ) converges in distribution to (W, Φ, Φ^W) as ↓ 0, where Φ^ and Φ^ω, are the rescaled piecewise constant magnetisation fields. Define (Φ^; N_λ, i,j, Φ^ω, ; N_λ, i,j)_i, j=1^N from (Φ^, Φ^ω, ) as in (<ref>) and (<ref>). By the continuous mapping theorem, (W^; N_λ, i,j, Φ^; N_λ, i,j, Φ^ω, ; N_λ, i,j)_i, j=1^N converges in distribution to (W^N_λ, i,j, Φ^N_λ, i,j, Φ^W,N_λ, i,j)_i,j=1^N.
  Since Φ^ and Φ^ω, are smeared out versions of Φ^ and Φ^ω, with the latter supported on Ω_, we have that
  
      |Φ^,N_λ,i,j-Φ^,N_λ,i,j|
      =|∑_x∈ B^N, _i,j^15/8σ_x(λ^2(x)-^-2∫_S_(x)λ^2(y)  dy)|
      
      
         ≤ 2^7/8 N^-2‖λ‖_∞‖λ'‖_∞
  
  converges to 0 deterministically. Therefore (W^; N_λ, i,j, Φ^; N_λ, i,j, Φ^ω, ; N_λ, i,j)_i, j=1^N also converges in distribution to (W^N_λ, i,j, Φ^N_λ, i,j, Φ^W,N_λ, i,j)_i,j=1^N.
  
  
  To show that the law of the discretised random variables also converge, it only remains to show that (W^N_λ, i,j, Φ^N_λ, i,j, Φ^W,N_λ, i,j)_i,j=1^N are continuous random variables. For Φ^N_λ, i,j, this follows similarly to <cit.>, which is based on bounds on its Fourier transform; for W^N_λ, i,j, this follows because it is Gaussian. We show in Lemma <ref> below that _N^, the Radon-Nikodym derivative of the law of (W^; N_λ, i,j, Φ^ω, ; N_λ, i,j) with respect to (W^; N_λ, i,j, Φ^; N_λ, i,j), is uniformly integrable as ↓ 0, and hence by Lemma <ref>, the law of the limit (W^N_λ, i,j, Φ^W, N_λ, i,j) is absolutely continuous with respect to the law of (W^N_λ, i,j, Φ^N_λ, i,j). Therefore (Φ^W,N_λ,i,j)_i,j=1^N are also continuous random variables, which concludes the proof.
  
  
  
  
  
  We now prove the uniform integrability of (_N^)_∈(0,1) defined in (<ref>).
  
  For each N∈, (_N^)_∈(0,1) is uniformly integrable with respect to .
  
  
  Abbreviate μ^ω, _Ω; λ, 0 and μ^ω, _Ω; 0, 0 by μ^ω, and μ^ respectively. We have that
  
      μ̣^ω,/μ̣^=(1+o(1))H^ω,/Z^ω,  where   H^ω,:=exp(∑_xσ_xω_x^λ_x^-1/2(λ_x^)^2)
  
  and 1+o(1)=θ_ e^1/2∑_x∈Ω_ (λ_x^)^2= e^-1/2^-1/4‖λ‖_L^2^2+1/2∑_x∈Ω_ (λ_x^)^2 is deterministic.
  
  Note that given ε>0, we can write
  
      _N^ = [H^ω,/Z^ω,_Z^ω,≤ε|_N^] + [H^ω,/Z^ω,_Z^ω,> ε|_N^],
  
  where the first term has expectation
  
      [[H^ω,/Z^ω,_Z^ω,≤ε|_N^]]
       = [H^ω,/Z^ω,_Z^ω,≤ε]
       = (1+o(1))(Z^ω,≤ε)
  
  since Z^ω,=(1+o(1))[H^ω,] by Lemma <ref>. This expectation can be made arbitrarily small uniformly in  by choosing ε>0 small, because Z^ω, converges in distribution to ^W_Ω; λ, h which is a.s. positive by Lemma <ref>. The second term in (<ref>) is bounded by
  ε^-1[H^ω,|_N^]. Combining these two observations, it follows that to prove uniform integrability of (_N^)_∈ (0,1), it suffices to prove that
  
      ([H^ω,|_N^])_∈(0,1)  .
  
  
  We will use the following lemma whose proof is standard and will be omitted.
  
  Suppose (X_k)_k=1^L, are independent, centred Gaussian random variables with variances (ς_k)_k=1^L. For M∈ fixed, (X_k)_k=1^L-1 conditioned on the event ∑_k=1^L X_k=M has a multivariate Gaussian distribution given by
  
      [X_j|∑_k=1^L X_k=M]=Mς_j/∑_k=1^Lς_k
  
  and
  
      [X_j X_l|∑_k=1^L X_k=M]=ς_j∑_k≠ jς_k/∑_k=1^Lς_k   if  j=l,
      ς_jς_l(M^2-∑_k=1^Lς_k)/(∑_k=1^Lς_k)^2   if  j≠ l.
  
  
  
  Write B_i,j^N, for the discretisation of the box B_i,j^N in ^2. Conditioning on the spin configuration σ and
  using independence of ω^_· in disjoint boxes, we have that [H^ω,|_N] equals
  
      [[exp(∑_x∈Ω_σ_xω_x^λ_x^-1/2(λ_x^)^2)|{a∑_x∈ B_i,j^N,λ(x)ω_x^=W_λ, i,j^,N}_i,j=1^N]|{Φ_λ, i,j^,N}_i,j=1^N] 
      
      =    [∏_i,j=1^N[exp(∑_x∈ B_i,j^N,σ_xω_x^λ_x^-1/2(λ_x^)^2)|  a∑_x∈ B_i,j^N,λ(x)ω_x^=W_λ, i,j^,N]|{Φ_λ, i,j^,N}_i,j=1^N].
  
  Let Λ_i,j^N:=∑_x∈ B_i,j^N,λ(x)^2 and
  
      _i,j^N(·):=(·|  a∑_x∈ B_i,j^N,λ(x)ω_x^=W_λ, i,j^,N).
  
  Fix any vertex z∈ B_i,j^N,. By Lemma <ref>, the laws of (λ(x)ω_x)_x∈ B_i,j^N,∖{z} with respect to _i,j^N are Gaussian with means and covariances given by
  
      _i,j^N[λ(x)ω_x^]     =λ^2(x)^-1W_λ, i,j^,N/Λ_i,j^N,
      Cov__i,j^N(λ(x)ω_x^,λ(y)ω_y^)     = λ(x)^2(Λ_i,j^N-λ(x)^2)/Λ_i,j^N   if  x=y, 
       -λ(x)^2λ(y)^2/Λ_i,j^N   if  x≠ y.
  
  Recall that
  
      Φ_λ,i,j^,N=⟨Φ^, λ^2_B_i,j^N⟩=∑_x∈ B_i,j^N^15/8λ^2(x)σ_x.
  
  Since under _i,j^N, ∑_x∈ B_i,j^N,σ_xω_x^λ_x^ (recall that λ^_x=^7/8λ(x)) is a Gaussian random variable, it has exponential moment
  
      _i,j^N[exp(∑_x∈ B_i,j^N,σ_xω_x^λ_x^)] 
      
      =    exp(∑_x∈ B_i,j^N,σ_x^7/8λ(x)^2^-1W_λ, i,j^,N/Λ_i,j^N+1/2∑_x∈ B_i,j^N,^7/4λ(x)^2
      -1/2∑_x,y∈ B_i,j^N,σ_xσ_yλ(x)^2λ(y)^2^7/4/Λ_i,j^N) 
      
      =      e^1/2∑_x∈ B_i,j^N,(λ_x^)^2exp(Φ_λ, i,j^,NW_λ, i,j^,N/^2Λ_i,j^N-1/2(Φ_λ, i,j^,N)^2/^2Λ_i,j^N).
  
  We note here that Λ_i,j^,N:=^2Λ_i,j^N converges to ‖λ‖_L^2(B_i,j^N)^2 as ↓ 0 and the exponential prefactor compensates with the tilting -(λ_x^)^2/2 appearing in the definition of H^ω,. In particular,
  
      [H^ω,|_N^]     = ∏_i,j=1^Nexp(Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N-1/2(Φ_λ, i,j^,N)^2/Λ_i,j^,N),
  
  where conditioned on σ, Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N is a centered Gaussian random variable with variance (Φ_λ, i,j^,N)^2/Λ_i,j^,N.
  
  For K>0 let
  
      _K:={[H^ω,|_N^]>K}  and  _K:= {∏_i,j=1^Nexp((Φ_λ, i,j^,N)^2/Λ_i,j^,N)>K^1/2}.
  
  We want to show that lim_K→∞sup_∈(0,1][[H^ω,|_N^]__K]=0. We have that
  
      [[H^ω,|_N^]__K] 
      ≤    [[H^ω,|_N^]__K] +[[H^ω,|_N^]__K__K^c].
  
  Using (<ref>) and that W_λ, i,j^,N are independent, centred Gaussians with variances Λ_i,j^,N, we can apply Markov's inequality to bound the final term in (<ref>) by
  
      1/K[∏_i,j=1^Nexp(2Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N-(Φ_λ, i,j^,N)^2/Λ_i,j^,N)__K^c] 
      
      =    1/K[∏_i,j=1^N[exp(2Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N)]exp(-(Φ_λ, i,j^,N)^2/Λ_i,j^,N)__K^c] 
      
      =    1/K[∏_i,j=1^Nexp((Φ_λ, i,j^,N)^2/Λ_i,j^,N)__K^c]
      ≤ K^-1/2
  
  uniformly in . Similarly, the first term in (<ref>) is equal to
  
      [__K∏_i,j=1^Nexp(Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N-1/2(Φ_λ, i,j^,N)^2/Λ_i,j^,N)] 
      
      =    [__K∏_i,j=1^N[exp(Φ_λ, i,j^,NW_λ, i,j^,N/Λ_i,j^,N)]exp(-1/2(Φ_λ, i,j^,N)^2/Λ_i,j^,N)]  
      
      =    (_K) ≤  2∑_i,j=1^N[(Φ_λ, i,j^,N)^2]/Λ_i,j^,Nlog(K)
  
  which converges to 0 uniformly in  as K→∞ since ⟨Φ^,λ^2_B^N_i,j⟩ has finite exponential moments uniformly in  as shown in <cit.>.
  
  
  
  
  
   §.§ Proof of Lemma <ref>
  
  Our proof of Lemma <ref> follows the fractional moment method first developed in the study of the disordered pinning model (see <cit.>).  It will become clear from the proof why we chose to work with W^,N_λ, i,j and Φ^,N_λ, i,j as defined in (<ref>).
  
  Recall that Φ^=∑_x∈Ω_^15/8σ_xδ_x and W^ = ∑_x∈Ω_ω_x^δ_x. We will choose a function f_N,m^=f_N,m^(W^,Φ^)≥ 0 that is
  measurable with respect to _N,m^, which can be interpreted as a tilting of the underlying measure. By the Cauchy-Schwarz inequality we have
  
      [(_N,m^)^1/2]
          =[(f_N,m^)^1/2(_N,m^)^1/2(f_N,m^)^-1/2] 
         ≤[f_N,m^_N, m^]^1/2[(f_N,m^)^-1]^1/2.
  
  Recalling the definition of _N,m^ and that f_N, m^ is _N, m^ measurable, we have that
  
  
  
  
  
  
  
      [f_N, m^_N, m^]
       = [f_N, m^[μ̣^ω,/μ̣^|^_N,m]]
       = [f_N, m^μ̣^ω,/μ̣^].
  
  Applying Fubini's theorem, it therefore suffices to show that
  
      lim_N→∞lim_m→∞lim_↓ 0[[f_N, m^μ̣^ω,/μ̣^]]^1/2[[(f_N, m^)^-1]]^1/2=0.
  
  To bound [f_N, m^μ̣^ω,/μ̣^], we will interpret μ̣^ω,/μ̣^ as a change of measure for ω^_· as follows. Write
  
      W^_λ:=∑_x∈Ω_λ(x)ω_x^δ_x
  
  and, with a slight abuse of notation, ⟨Φ^,W^_λ⟩ = ∑_x∈Ω_λ_x^σ_xω_x^= ∑_x∈Ω_^7/8λ(x)σ_xω_x^. Recall that θ_=e^-1/2^-1/4||λ||^2_L^2. By definition we have
  
      μ̣^ω,/μ̣^(Φ^)
       = θ_/Z^ω,exp(∑_x∈Ω_λ_x^σ_xω_x^)
       = θ_exp(⟨Φ^,W^_λ⟩)/Z^ω,.
  
  We now consider the change of measure  on the disorder defined by
  
      %̣ṣ/(ω^_·)=θ_exp(⟨Φ^,W^_λ⟩)/[θ_exp(⟨Φ^,W^_λ⟩)],
  
  where the denominator tends to 1 as a↓ 0. This change of measure shifts the disorder (ω^_x) into a field of independent variables with unit variance but spin-dependent centring as is stated in Lemma <ref>. We omit the proof which follows immediately by manipulating the Laplace transforms.
  
  Under  for σ fixed, the family {ω_x^}_x∈Ω_ are independent Gaussian random variables with means κ_x:=λ_x^σ_x and unit variance.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  To prove (<ref>), it suffices to find a suitable choice of f_N, m^ such that
  
      lim_N→∞lim_m→∞lim sup_↓ 0[[(f_N, m^)^-1]] <∞,
  
  and
  
      lim_N→∞lim_m→∞lim_↓ 0[[f_N, m^/Z^ω,]]=0.
  
  For this, we need to choose f_N, m^ to be typically close to 1 but small on the rare events that μ̣^ω,/μ̣^ is large.  We choose
  
      f_N, m^:=exp(-S_N^_{X_N,m^≥ M_N^})
  
  where M_N^, S_N^ are constant sequences to be chosen later, which do not depend on m and diverge as →0 then N→∞, and X_N,m^=X_N,m^(W^,Φ^) is an approximation of ⟨Φ^,W^_λ⟩ which is _N, m^ measurable. By doing this, we have the desired behaviour that f_N, m^ is typically close to 1 with respect to , but small when ⟨Φ^,W^_λ⟩ is large, which occurs with high probability with respect to .
  
  We wish to approximate the spin configuration σ by random variables measurable with respect to _N,m^. More precisely, we will replace σ_x for x∈ B_i,j^N, by the constant value
  
      ρ_i,j^N,:=1,    if ∑_x∈ B_i,j^N, (λ^_x)^2 σ_x  ≥0,
      
      -1,    if ∑_x∈ B_i,j^N, (λ^_x)^2 σ_x<0.
  
  Note that (ρ_i,j^N,)^2=1 which does not depend on Φ^. We then approximate ⟨Φ^,W^_λ⟩ by
  
      X_N, m^:= ^-1/8∑_i,j=1^Nρ_i,j^N, W^,N,m_λ, i,j
      = ^-1/8∑_i,j=1^Nρ_i,j^N, 2^-m⌊ 2^m ∑_x∈ B_i,j^N,λ_x^ω_x^⌋,
  
  where W^, N, m_λ, i, j is defined as in (<ref>). Note that
  
      lim_m→∞ X_N,m^  = X_N^:= ^-1/8∑_i,j=1^Nρ_i,j^N, W^,N_λ, i,j
      = ∑_i,j=1^Nρ_i,j^N,∑_x∈ B_i,j^N,λ^_x ω_x^, 
           |X_N,m^ - X_N^| ≤ 2^-mN^2 ^-1/8.
  
  
  
  We are now ready to prove (<ref>) for suitable choices of S_N^ and M_N^.
  
  Suppose that lim_N→∞lim_↓ 0S_N^=∞ and set M_N^:=(4S_N^∑_x∈Ω_(λ_x^)^2)^1/2. Then
  
      lim_N→∞lim_m→∞lim_↓ 0[(f_N, m^)^-1]<∞.
  
  
  
  By the definition of f_N, m^ and the bound in (<ref>), we have that
  
      [(f_N, m^)^-1]
         =1+(e^S_N^-1)(X_N,m^>M_N^)
         ≤ 1+(e^S_N^-1)(X_N^>M_N^ - 2^-mN^2 ^-1/8).
  
  From the definition of X^_N in (<ref>) and our choice of ω^_x, we note that X_N^ is a centred Gaussian random variable with variance
  
      ∑_i,j=1^N(ρ_i,j^N,)^2∑_x∈ B_i,j^N,(λ_x^)^2=∑_x∈Ω_(λ_x^)^2
      = ^-1/4 (1+o(1)) ‖λ‖_L^2(Ω)^2.
  
  It then follows by a Gaussian tail estimate that
  
      (X_N^>M_N^- 2^-mN^2 ^-1/8)
      ≤    exp(-(M_N^ - 2^-mN^2 ^-1/8)^2/2∑_x∈Ω_(λ_x^)^2)·(∑_x∈Ω_(λ_x^)^2)^1/2/(M_N^ - 2^-mN^2 ^-1/8) √(2π).
  
  In particular, by our choice of M_N^ we have the desired result.
  
  
  We are now ready to prove (<ref>) and thus conclude the proof of (<ref>).
  
  There exists S_N^ satisfying lim_N→∞lim_↓ 0S_N^=∞ such that
  
      lim_N→∞lim_m→∞lim_→ 0[[f_N, m^/Z^ω,]]=0
  
  where in the definition of f_N, m^ in (<ref>), M_N^:=(4S_N^∑_x∈Ω_(λ_x^)^2)^1/2.
  
  
  First note that for ε>0
  
      [f_N, m^/Z^ω,]
      ≤1/ε[f_N, m^_Z^ω,≥ε]
      + [_Z^ω,<ε/Z^ω,].
  
  By Fubini's theorem and the change of measure (<ref>), we have that
  
      [[_Z^ω,<ε/Z^ω,]]
       = (1+o(1))(Z^ω,<ε),
  
  which can be made arbitrarily small uniformly in >0 by choosing ε>0 sufficiently small since the weak limit of Z^ω,, ^W, does not have an atom at 0 by Lemma <ref>.
  
  We then have that
  
      [f_N, m^_Z^ω,≥ε]≤[f_N, m^]=1+(e^-S_N^-1)(X_N,m^>M_N^) ≤(X_N,m^≤ M_N^) + e^-S_N^.
  
  Since |X^_N,m-X^_N|≤ 2^-mN^2 ^-1/8 =:C_N,m^-1/8 by (<ref>), to conclude the proof, it suffices to show that
  
      lim_N→∞lim_m→∞lim_↓ 0[(X_N^≤ M_N^+C_N,m^-1/8)]=0.
  
  
  By Lemma <ref>, ω_x^ are independent Gaussian random variables with mean λ_x^σ_x and unit variance with respect to . It follows that X_N^ is a Gaussian random variable with mean and variance given by
  
      m_N, = ∑_i,j=1^Nρ_i,j^N,∑_x∈ B_i,j^N,(λ_x^)^2σ_x,     
      s_N,^2 = ∑_x∈Ω_(λ_x^)^2 = ^-1/4 (1+o(1)) ‖λ‖_L^2(Ω)^2
  
  respectively where we have used that (ρ^N, _i,j)^2=1.
  
  If M_N^ + C_N,m^-1/8 < m_N,, then by a Gaussian tail estimate,  we have that
  
      (X_N^≤ M_N^ +C_N,m^-1/8)≤exp(-1/2(m_N,-M_N^-C_N,m^-1/8/s_N,)^2)/√(2π)m_N,-M_N^-C_N,m^-1/8/s_N,.
  
  It therefore suffices to show that (m_N,-M_N^-C_N,m^-1/8)/s_N, tends to ∞ in -probability as ↓ 0, then m→∞ followed by N→∞.
  
  Recalling the definitions of M_N^ and s_N, and that C_N,m=2^-mN^2, we have that M_N^/s_N,=(4S_N^)^1/2 and lim_m→∞lim_↓ 0C_N,m^-1/8/s_N, =0.  Therefore, since we can choose S_N^ tending to ∞ as slowly as we like, it suffices to show that m_N,/s_N, tends to ∞, or equivalently,
  
      ^1/8 m_N,  = ^1/8∑_i,j=1^Nρ_i,j^N,∑_x∈ B_i,j^N,(λ_x^)^2σ_x= ∑_i,j=1^N|∑_x∈ B_i,j^N,^15/8λ(x)^2σ_x| →∞
  
  as ↓ 0,   m→∞,  N→∞
  where we have used the definition that ρ^N, _i,j is the sign of ∑_x∈ B_i,j^N,(λ_x^)^2σ_x.
  This divergence follows from Lemma <ref> below.
  
  
  
  In ^-distribution,
  
      lim_N→∞lim_→ 0∑_i,j=1^N|∑_x∈ B_i,j^N,^15/8λ(x)^2σ_x|= ∞.
  
  
  
  We first note that the sum we are interested in is equal to
  
      ∑_i,j=1^NN^-15/8|∑_x∈ B_i,j^N,( N)^15/8λ(x)^2σ_x|
      ∑_i,j=1^NN^-15/8|∑_x∈B_i,j^N,_N^15/8λ_N(x)^2σ_x|
  
  where we have rescaled space such that B_i,j^N, is the discretisation of the box [i-1,i]×[j-1,j] with mesh size _N= N, λ_N(x)=λ(x/N) and σ is an Ising configuration on ∪B_i,j^N,. Write
  
      Y_i,j^N,:=|∑_x∈B_i,j^N,_N^15/8λ_N(x)^2σ_x|
  
  then we will show that (Y_i,j^N,) stochastically dominates a family which obeys a law of large numbers.
  
  Write D_i,j^N,:=[i-2,i+1]×[j-2,j+1] for the box of side length 3 with B_i,j^N, at the centre. We can choose a set I_N⊂{(i,j):1≤ i,j≤ N} of size at least (N-2)^2/9 such that the boxes (D_i,j^N,)_(i,j)∈ I^N are contained within [0,N]^2 and do not intersect each other. This gives a collection of disjoint annuli A_i,j^N,:=D_i,j^N,∖B_i,j^N, with side length 3 and the centre box of side length 1 removed.
  
  We wish to show that, with high probability, a positive proportion of these annuli contain a + Ising loop. Let
  
      _i,j^N:={(i,j)∈ I_N: ∂B_i,j^N,-↮∂ D_i,j^N,}
  
  denote the event that B_i,j^N, is surrounded by a + loop in the annulus A_i,j^N,. A consequence of the RSW result <cit.> (see also <cit.>) is that, for >0 suitably small, there exists a constant p>0 independent of the boundary condition ξ on D_i,j^N, such that
  
      ^,ξ_D_i,j^N,(_i,j^N)>p.
  
  The number of disjoint annuli that contain such a loop stochastically dominates a binomial random variable with (N-2)^2/9 trials and success probability p. In particular, with high probability, there are at least cN^2 such loops for some c>0 and all N large. Write I_N⊂ I_N to be the collection with such a loop and γ_i,j the loop corresponding to (i,j) where we choose any such loop if there are more then one.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Conditioned on these loops, the random variables Y_i,j^N, for (i,j)∈I_N are independent and positive. It therefore suffices to show that ^(Y_i,j^N,>c_1|(i,j)∈I_N,γ_i,j)>c_2 for some c_1,c_2>0 which are independent of the pair (i,j) and the loop γ_i,j. By the Paley-Zygmund inequality
  
      ^(Y_i,j^N,>1/2^[Y_i,j^N,|(i,j)∈I_N,γ_i,j]  |  (i,j)∈I_N,γ_i,j)
      ≥    ^[Y_i,j^N,|(i,j)∈I_N,γ_i,j]^2/4^[(Y_i,j^N,)^2|(i,j)∈I_N,γ_i,j],
  
  therefore it suffices to show that independently of the pair (i,j) and the loop γ_i,j we have that [Y_i,j^N,|(i,j)∈I_N,γ_i,j] is bounded below and ^[(Y_i,j^N,)^2|(i,j)∈I_N,γ_i,j] is bounded above.
  
  For the lower bound, by the FKG inequality, we have that
  
      ^[Y_i,j^N,|(i,j)∈I_N,γ_i,j]
         ≥∑_x∈B_i,j^N,_N^15/8λ_N(x)^2^[σ_x|(i,j)∈I_N,γ_i,j]
         ≥inf_x∈Ωλ(x)^2∑_x∈B_i,j^N,^15/8_N^,+_D_i,j^N,[σ_x]
  
  which is bounded below by the convergence result <cit.>.
  
  For the upper bound we have that
  
      ^[(Y_i,j^N,)^2  |  (i,j)∈I_N,γ_i,j]
         =∑_x,y∈B_i,j^N,_N^15/4λ_N(x)^2λ_N(y)^2^[σ_xσ_y |  (i,j)∈I_N,γ_i,j] 
         ≤‖λ‖^4_∞∑_x,y∈B_i,j^N,_N^15/4^,+_B_i,j^N,[σ_xσ_y]
  
  which is bounded above uniformly in _N by <cit.> and <cit.>.
  
  
  
  
  
  
  
  
  § CONVERGENCE OF RADON-NIKODYM DERIVATIVES
   
  
  In this appendix, we prove an elementary measure theoretic result needed in the proof.
  
  
  Let (μ_n)_n∈ and (ν_n)_n∈ be two sequences of probability measures on a complete separable metric space B equipped with Borel σ-algebra . Suppose that μ_n⇒μ and ν_n⇒ν, and furthermore, ν_n is absolutely continuous with respect to μ_n with Radon-Nikodym derivative f_n(x) such that {f_n ∈ L^1(B, , μ_n)}_n∈ are uniformly integrable. Then ν is absolutely continuous with respect to μ.
  
   Let (X_n)_n∈ be random variables defined on a probability space (Γ, , ) such that X_n  has law μ_n. Denote Y_n=f_n(X_n). Then [Y_n]=1, and the sequence of random vectors (X_n, Y_n)_n∈ is tight and admits convergent subsequences. Let us restrict to such a subsequence and still denote it by (X_n, Y_n)_n∈ for notational simplicity. Furthermore, by Skorokhod's representation theorem, we can couple (X_n, Y_n)_n∈ and its limit (X, Y) with marginal laws μ and ν respectively, such that (X_n, Y_n)→ (X, Y) almost surely on the probability space (Γ, , ).
  
  For any bounded continuous function g on (B, ), we then have
  
      ∫ g(x) ν_n( dx) = ∫ g(x) f_n(x) μ_n( dx) = [g(X_n) Y_n].
  
  The left hand side converges to ∫ g(x) ν( dx), while the right hand side converges to [g(X) Y] by the uniform integrability assumption on (Y_n)_n∈. Since X  has law μ, it follows that ν is absolutely continuous with respect to μ with Radon-Nikodym derivative f(X)= [Y | X].
  
  
  
  In Lemma <ref>, we cannot deduce how singular ν is with respect to μ from an assumption such as sup_n ∫ f_n^α dμ_n≤ε for some α∈ (0,1) and ε small, because from the proof of Lemma <ref>, we obtain
  
      [Y^α] ≤lim inf_n→∞[Y_n^α] = ∫ f_n^α dμ_n ≤ε,
  
  but we cannot draw this conclusion if Y is replaced by f(X)=[Y|X], the Radon-Nikodym derivative of ν with respect to μ. Indeed, if μ is a delta measure, then so would be ν, and we would always have ∫ f^α dμ=1 regardless of the value of α and ε. However, if (μ_n)_n∈ and μ are atomic measures supported on the same discrete set containing no cluster points, then indeed,
  
      ∫ f^α dμ≤lim inf_n→∞∫ f_n^α dμ_n,     α∈ (0,1).
  
  Therefore, in order to prove the singularity in Theorem <ref>, it is not sufficient to show that
  
      lim_N→∞lim inf_↓ 0[(_N^)^1/2]=0      (_N^(<ref>)).
  
  Instead, we need to discretise the random variables as in (<ref>) and then prove their weak convergence in Lemma <ref>,
  so that (<ref>) can be applied.
  
  
  
  
  § POSITIVE MOMENTS OF THE PARTITION FUNCTIONS
   
  
  In this appendix, we give uniform bounds on positive moments of the partition function Z^ω, _Ω; λ, h as stated in Lemma <ref>. The proof is based on hyper-contractivity. Similar estimates have been obtained in <cit.> for the directed polymer model in dimension 2+1.
  
  We first formulate it explicitly as a lemma.
  
  Let (Z^ω,_Ω;λ,h)_∈ (0,1] be the RFIM partition function with + boundary condition as in Theorem <ref>. Then for any p>0,
  
      lim sup_↓ 0[(Z^ω,_Ω;λ,h)^p] <∞.
  
  
   We first recall from (<ref>) the following expansion, where ξ^_x:=λ_x^ω^_x+h_x^:
  
  Z_Ω;λ, h^ω,
      = θ_cosh(ξ^_·)^Ω_∑_I⊆Ω__Ω^[σ_·^I] tanh(ξ^_·)^I.
  
  By Lemma <ref>, the prefactor θ_cosh(ξ_·^)^Ω_ converges to 1 in probability and is uniformly bounded in L^p for any p>0. Therefore we will drop this prefactor from now on.
  
  By Taylor expansion and the assumption that ω has finite exponential moments, it is easily seen that
  
      μ^_x:=[tanhξ_x^]  = h_x^+O(^21/8),      (ϑ^_x)^2:= 𝕍 ar[tanhξ_x^]  = (λ_x^)^2+O(^7/2), 
      [(tanhξ_x^)^2k]  = (λ_x^)^2k+O(^7k/2)   ∀  k∈.
  
  Denoting η^_x:= (ξ^_x-μ^_x)/ϑ^_x, then we can write
  
      Ψ_(η^_·) := ∑_I⊆Ω__Ω^[σ_·^I]
      ∏_x∈ I (ϑ^_x η^_x + μ^_x) =: ∑_J⊂Ω_ψ_(J) ∏_x∈ Jη^_x,
  
  which is a polynomial chaos expansion with kernel ψ_ in the family of independent random variables (η^_x)_x∈Ω_, which has mean 0, variance 1, and uniformly bounded 2k-th moment for each k∈. Therefore we can apply hypercontractivity to bound its moments (see e.g. <cit.> for the exact form needed here, or the original reference <cit.> for a proof). More specifically,  for any p>2, we can find c_p∈ (1, ∞) such that
  
      [|Ψ(η^_·)|^p]^2/p≤∑_J⊂Ω_ c_p^2|J|ψ_a(J)^2
          = [ (∑_J⊂Ω_ψ_(J) ∏_x∈ J c_pη^_x)^2] 
          = [Ψ_(c_pη^_·)^2] = [Ψ_(c_pη^_·)]^2 + 𝕍 ar(Ψ_(c_pη^_·)).
  
  Note that
  
      [Ψ_(c_pη^_·)] = [Ψ_(η^_·)] = (1+o(1)) [Z_Ω;λ, h^ω,]
      = (1+o(1)) ^_Ω[e^∑_x σ_x h^_x] →_Ω[e^⟨Φ, h⟩]
  
  which is finite by <cit.> on the convergence of the exponential moment of the Ising magnetisation field. Therefore to bound
  [|Ψ(η^_·)|^p], it only remains to bound 𝕍 ar(Ψ_(c_pη^_·)).
  
  We can write
  
      Ψ_(c_pη^_·) = ∑_I⊆Ω__Ω^[σ_·^I] ∏_x∈ I (c_pϑ^_x η^_x + μ^_x) = ∑_I⊆Ω__Ω^[σ_·^I] ∏_x∈ I c_pϑ^_x (η^_x + μ̃^_x),
  
  where μ̃^_x := μ^_x/c_pϑ^_x = (+o()) h(x)/c_pλ(x). By <cit.>,
  in order to bound 𝕍 ar(Ψ_(c_pη^_·)), we can remove μ̃^_x from η^_x+μ̃^_x, and for any ε>0, bound
  
      𝕍 ar(Ψ_(c_pη^_·))    ≤ e^ε^-1∑_x∈Ω_ (μ̃^_x)^2∑_I ⊂Ω_ (1+ε)^|I|_Ω^[σ_·^I]^2 ∏_x∈ I (c_pϑ^_x)^2 
         ≤ e^ε^-1(c_p^-2+o(1)) ‖ h/λ‖_L^2(Ω)^2∑_I ⊂Ω_((1+ε)c_p^2|λ|_∞^2)^|I| f^2_Ω(I) ^2k
         ≤ C ∑_k=0^∞((1+ε)c_p^2|λ|_∞^2)^k‖ f_Ω‖_L^2(Ω^k)^2/k!
         ≤ C ∑_k=0^∞((1+ε)c_p^2|λ|_∞^2)^kC^k(k!)^1/4/k! <∞,
  
  where we have used the bounds (<ref>) and (<ref>). This shows that the right hand side of (<ref>) is uniformly bounded for  small, which concludes the proof of Lemma <ref>.
  
  
  
  
  § NEGATIVE MOMENTS OF THE PARTITION FUNCTIONS
   
  
  In this appendix, we give uniform bounds on negative moments of the partition function Z^ω, _Ω; λ, h as stated in Lemma <ref>, which implies that the continuum RFIM partition function ^W_Ω; λ, h has negative moments of all orders; in particular, it is almost surely strictly positive. The proof is based on a concentration of measure bound for convex and locally Lipschitz functions of i.i.d. random variables, which is applied to logZ^ω, _Ω; λ, h and then passed to the limit as ↓ 0. Similar estimates have been obtained for the directed polymer model <cit.> and the disordered pinning model <cit.>. We will follow closely <cit.> where such calculations were done for the directed polymer in dimension 2+1, using the inequality (<ref>) proved in <cit.>.
  
  
  Let (Z^ω,_Ω;λ,h)_∈ (0,1] be the RFIM partition function with + boundary condition as in Theorem <ref>. Assume that an i.i.d. sequence (ω_n)_n∈ with the same law as ω_x satisfies the following concentration of measure inequality:
  
      ∃ γ>1, C_1, C_2 ∈ (0,∞)  such that for all n∈ and g: ^n → convex and 1-Lipschitz, 
      ( | g(ω_1, …, ω_N) - M_g |
      	≥ t) ≤ C_1e^-C_2 t^γ ,    M_g g(ω_1, …, ω_N).
  
  Then there exists c=c(λ, h)∈ (0,∞) such that for all ∈ (0,1) sufficiently small, we have
  
      ∀  t ≥ 0:      (logZ^ω, _Ω; λ, h≤ -t) ≤ c   e^- t^γ / c.
  
  In particular, lim sup_a∈ (0,1][(Z^ω, _Ω; λ, h)^-p]<∞ for all p∈ (0,∞).
  
  
  We can improve (<ref>) to a uniform bound with respect to the boundary condition ξ∈{±1}^Ω_ if we had a uniform upper bound on [(Z^ω, , ξ_Ω;λ, h)^2], which unfortunately is missing because the spin correlation functions obtained in <cit.> is only for special boundary conditions and cannot be used to dominate spin correlation functions for general boundary conditions.
  
  
  Condition (<ref>) is satisfied if ω is either bounded, or Gaussian, or has a density of the form exp(-V(·) + U(·)) where V is uniformly strictly convex and U is bounded. See <cit.> for more details. In particular, by choosing (ω_x)_x∈^2 to be i.i.d. standard Gaussian and applying Theorem <ref> and Fatou's lemma, we obtain the following.
  
  
  Let ^W_Ω; λ, h be the continuum RFIM partition function with + boundary condition as in Theorem <ref>. Then [(^W_Ω; λ, h)^-p]<∞ for all p∈ (0,∞), and ^W_Ω; λ, h>0  -a.s.
  
  
  
  The first fact we need is that for (ω_n)_n∈ℕ that satisfy (<ref>), one can also obtain a lower deviation bound for f(ω_1, …, ω_n) where f is only convex and locally Lipschitz. More precisely, by <cit.>, for every n∈ and every convex differentiable function f^n→,
  
  the following bound holds for all b∈ and t,c∈ (0,∞),
  
      (f(ω)≤ b-t )   (f(ω)≥ b, |∇ f(ω) | ≤ c )
      	≤C_1exp( -(t/c)^γ/C_2 ),
  
  where ω = (ω_1, …, ω_n) and
  |∇ f(ω)|:=√(∑_i=1^n(∂_i f(ω))^2) is the norm of the gradient.
  We can then deduce (<ref>) by applying (<ref>) to f_(ω) = logZ^ω, _Ω; λ, h.
  
  It suffices to show that for some b to be chosen, there exist c=c(λ, h) ∈ (0,∞)
  and θ∈ (0,1) such that
  
      lim inf_a↓ 0( f_(ω) ≥ b  ,
      	|∇ f_(ω)|≤ c ) >0  .
  
  First note that for any c > 0, we have
  
      ( f_(ω)≥ b  ,|∇ f_(ω)|≤ c) = (f_(ω)≥ b) -
      	(f_(ω)≥ b  ,|∇ f_(ω)|> c).
  
  Note that by the definition of Z^ω, _Ω; λ, h in (<ref>), Lemma <ref>, and <cit.> on the convergence of the exponential moment of the Ising magnetisation field, we have
  
      lim_↓ 0 [Z^ω, _Ω; λ, h]=lim_↓ 0 (1+o(1))^_Ω[e^∑σ_x h^_x] = _Ω[e^⟨Φ, h⟩] =:C_1,h>0.
  
  On the other hand, we have lim_↓ 0 [ (Z^ω, _Ω; λ, h)^2 ] = [ (^W_Ω; λ, h)^2] =:C_2, λ h<∞ by <cit.>.
  
  Choosing b= log (C_1,h/2), we can bound the first probability in (<ref>) by Paley-Zygmund inequality:
  
      (f_(ω)≥ b) = ( Z^ω, _Ω; λ, h≥ C_1,h/2) =  	
      	( Z^ω, _Ω; λ, h≥1+o(1)2 [Z^ω, _Ω; λ, h] )
      	≥C_1,h^2/5C_2, λ, h>0.
  
  
  To bound the second term in (<ref>), we first compute that for each x∈Ω_,
  
      ∂ f_(ω)/∂ω_x = 1/Z^ω, _Ω; λ, h_Ω[ σ_x λ^_x e^∑_x σ_x(λ^_xω^_x+h^_x) ] =
      ^ω, _Ω;λ, h[λ^_x σ_x]
  
  
      |∇ f_(ω)|^2 = ∑_x∈Ω_(∂ f_/∂ω_x)^2 =
      ^ω, , ⊗ 2_Ω;λ, h [L_λ(σ, σ')],
  
  where L_λ(σ, σ'):= ∑_x∈Ω_ (λ^_x)^2 σ_x σ_x'≥ 0 for two spin configurations σ and σ' sampled independently according to _Ω^, which is the overlap between σ and σ' weighted by (λ^_·)^2.
  
  On the event that f_(ω)≥ b, that is Z^ω, _Ω; λ, h≥ C_1,h/2, we can bound
  
      |∇ f_(ω)|^2 ≤4θ_a^2/C_1,h^2 ^, ⊗2_Ω[L_λ(σ, σ') exp{∑_x(σ_x+σ_x')(λ^_x ω^_x + h^_x) }].
  
  Therefore we have
  
      (f_(ω)≥ b ,|∇ f_(ω)|> c)
      ≤    1/c^2  [ |∇ f_(ω)|^2  _{ f_(ω)≥ b }] 
      ≤    4 θ_^2/c^2 C_1,h^2[ ^, ⊗2_Ω[L_λ(σ, σ') exp{∑_x(σ_x+σ_x')(λ^_x ω^_x + h^_x) }] ] 
      
      =     4/c^2 C_1,h^2(θ_^2 e^∑_x (λ^_x)^2)
      ^, ⊗2_Ω[L_λ(σ, σ') e^L_λ(σ, σ') + ∑_x σ_x h^_x + ∑_x σ_x' h^_x ],
  
  where we note that θ_^2 e^∑_x (λ^_x)^2→ 1 as ↓ 0 by the same argument as in Lemma <ref>, while the expectation can be bounded by applying Cauchy-Schwarz and noting that
  
      ^, ⊗2_Ω[e^2∑_x σ_x h^_x + 2∑_x σ_x' h^_x ] = ^_Ω[e^2∑_x σ_x h^_x]^2 →_Ω[e^⟨Φ, 2h⟩]^2 <∞
  
  by <cit.>, and
  
      ^, ⊗2_Ω[L^2_λ(σ, σ') e^2L_λ(σ, σ')] ≤^, ⊗2_Ω[e^4L_λ(σ, σ')]     = e^-4∑_x (λ^_x)^2[_Ω^[e^∑_x 2λ^_x ω_x σ_x]^2] 
          = e^-4∑_x (λ^_x)^2 e^^-1/4‖ 2λ‖ _L^2^2[(Z^ω, _Ω; 2λ, 0)^2],
  
  where the product of the two exponentials converges to 1 by the same argument as in Lemma <ref>, and [(Z^ω, _Ω; 2λ, 0)^2]→ C_2, 2λ,0 <∞ by <cit.>.
  
  Choosing c large enough, we can make the right hand side of (<ref>) sufficiently small such that (<ref>) holds. The bound (<ref>) then follows from the concentration inequality (<ref>).
  
  
  
  
  
  § BESOV-HÖLDER DISTRIBUTIONS INTEGRATED OVER SUBDOMAINS
   
  
  In this appendix, we show that for any Φ∈^α_loc(Ω) with α∈ (-1,0), it is possible to define the integral of Φ over a subdomain B⊂Ω with a regular enough boundary, which can be applied in particular to the continuum RFIM magnetisation field. This is used in the proof of Theorem <ref> in Section <ref>.
  
  
  Let Ω be open, α∈ (-1,0), and B̅⊂Ω with ∂ B having upper box dimension d<2+α, which is defined by
  
      d=lim sup_ϵ↓ 0log N(ϵ)/log (1/ϵ)
  
  where N(ϵ) is the minimal number of boxes of side length ϵ needed to cover ∂ B. Then the map ⟨·,_B⟩:^α_loc(Ω)→ is well defined and continuous.
  
  
  We first consider the space ^α instead of ^α_loc(Ω). By definition, any f∈^α is the limit of a sequence (f_m)_m≥ 1⊂ C_c^∞ in ^α. We then define
  
      ⟨ f,_B⟩:=lim_m→∞⟨ f_m,_B⟩.
  
  We need to show that this limit exists and is unique. In particular,  if another sequence (f̃_m)_m≥ 1⊂ C_c^∞ converges to f in ^α, then ⟨ f_m,_B⟩-⟨f̃_m,_B⟩=⟨ f_m-f̃_m,_B⟩
  converges to 0 as m→∞. It then suffices to show that for any (f_m)_m≥ 1⊂ C_c^∞ converging to 0 in ^α, we have ⟨ f_m,_B⟩→ 0. The continuity of ⟨·, _B⟩ would also follow as a consequence.
  
  First choose d̅>d and n_0∈ such that for all n≥ n_0,
  
      log N(2^-n)/log 2^n≤d̅ <2+α.
  
  In other words, on the lattice Λ_n=^2/2^n, we need at most 2^nd̅+2 squares of side length 2^-n to cover ∂ B.
  Since f_m∈ L^2, by the multi-resolution analysis from (<ref>) and (<ref>), we have that
  
      ⟨ f_m,_B⟩   = ⟨_n_0f_m+∑_n=n_0^∞_nf_m,_B⟩
         = ∑_x∈Λ_n_0⟨ f_m,ϕ_n_0,x⟩_L^2⟨ϕ_n_0,x,_B⟩_L^2 + ∑_n=n_0^∞∑_x∈Λ_n, i=1,2,3⟨ f_m,ψ^(i)_n,x⟩_L^2⟨ψ^(i)_n,x,_B⟩_L^2.
  
  Since B is bounded and ϕ_n_0,x is supported on B(x,2^-n_0), there are only finitely many x∈Λ_n_0 such that ⟨ϕ_n_0,x,_B⟩_L^2≠ 0. Since for each x, ‖ f_m‖_^α→ 0 implies that ⟨ f_m,ϕ_n_0,x⟩_L^2→ 0, we have that the first term in (<ref>) converges to 0 as m→∞.
  
  Similarly, since each ψ^(i)_n,x is supported on B(x,2^-n) and ∫ψ^(i)_n,x(y) dy=0, only x∈Λ_n=^2/2^n within distance 2^-n from ∂ B can have ⟨ψ^(i)_n,x,_B⟩_L^2≠ 0. The number of such x is bounded by 2^nd̅+2, and for each such x, we can estimate more quantitatively
  
      |⟨ψ^(i)_n,x,_B⟩_L^2| = |∫_B 2^n ψ^(i)(2^n(y-x))  dy | ≤  2^-n∫_2^n(B-x) |ψ^(i)(z)|  dz
      ≤ C 2^-n
  
  uniformly in x and n. On the other hand, by the definition of ‖ f‖_^α in (<ref>),
  
      sup_x∈Λ_n|⟨ f_m,ψ^(i)_n,x⟩_L^2|
         = 2^-(1+α)nsup_x∈Λ_n|2^(2+α)n∫ f_m(y) ψ^(i)(2^n(y-x)) dy |
         ≤ 2^-(1+α)n‖ f_m‖_^α.
  
  Therefore, using the assumption that α>-1 and d̅<2+α, the second term in (<ref>) is bounded above by
  
      C∑_n=0^∞∑_i=1,2,3 2^n(d̅-1)· 2^-(1+α)n‖ f_m‖_^α≤ C' ‖ f_m‖_^α,
  
  which converges to 0 as m→∞. This concludes the proof that ⟨·,_B⟩:^α(Ω)→ is well defined and continuous. The proof for the case ^α_loc(Ω) is similar, where we replace f∈^α_loc(Ω) by f χ∈^α with χ∈ C_c^∞(Ω) and χ_B=χ.
  
  
  
  
  
  
  
  § ACKNOWLEDGEMENTS
  
  We wish to thank in particular F. Caravenna for fruitful discussions during various stages of this project, which helped us to overcome some of the technical difficulties. We would also like to thank C. Garban, Y. Le Jan, and N. Zygouras for helpful discussions, and an anonymous referee for suggesting a simplified proof of tightness. R. Sun and A. Bowditch are supported by NUS grants R-146-000-260-114 and R-146-000-300-114.
   
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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