Subsequential tightness of the maximum of two dimensional Ginzburg-Landau fields
Wei Wu Ofer Zeitouni





Subsequential tightness of the maximum of two dimensional
Ginzburg-Landau fields
     Wei Wu1, Ofer Zeitouni2
    February 26, 2018
================================================================================




We prove the subsequential tightness of centered maxima of two-dimensional
Ginzburg-Landau fields with bounded elliptic contrast.


[1]
Statistics department, University of Warwick, Coventry CV4 7AL, UK. E-mail:
W.Wu.9@warwick.ac.uk [2]
Department of Mathematics, Weizmann Institute of Science, Rehovot 7610001,
Israel, and Courant Institute, New York University, 251 Mercer St., New
York, New York 10012. E-mail: ofer.zeitouni@weizmann.ac.il. Supported in
part by the ERC Advanced grant LogCorrelatedFields.



§ INTRODUCTION


Let V∈ C^2( ℝ) satisfy 

    V( x) =V( -x) ,  
    0<c_-≤ V^''( x) ≤ c_+<∞,

where c_-,c_+ are positive constants. The ratio κ=c_+/c_-
is called the elliptic contrast of V. We assume (<ref>) and 
(<ref>) throughout this note without further mentioning it.

We treat V as a a nearest neighbor potential for a two dimensional
Ginzburg-Landau gradient field. Explicitly, let D_N:=[ -N,N]
^2∩ℤ^2 and let the boundary ∂ D_N consist of
the vertices in D_N that are connected to ℤ^2∖ D_N
by some edge. The Ginzburg-Landau field on D_N with zero boundary
condition is a random field denoted by ϕ ^D_N,0, whose distribution
is given by the Gibbs measure 

    dμ_N =Z_N^-1exp[ -∑_v∈ D_N∑_i=1^2V( ∇
    _iϕ( v) ) ] ∏_v∈ D_N\∂ D_Ndϕ( v) ∏_v∈∂ D_Nδ
    _0( ϕ( v) ) ,

where ∇ _iϕ( v) =ϕ( v+e_i) -ϕ( v), e_1=(1,0) and e_2=(0,1), and Z_N is the
normalizing constant ensuring that μ_N is a probability measure, i.e. μ_N(ℝ^|D_N|)=1. We denote expectation with respect to μ_N
by 𝔼_N, or simply by 𝔼 when no confusion can occur.

Ginzburg-Landau fields with convex potential, which are natural
generalizations of the standard lattice Gaussian free field corresponding to
quadratic V (DGFF), have been extensively studied since the seminal works 
<cit.>. Of particular relevance to this paper is Miller's coupling,
described in Section <ref> below, which shows that certain
multi-scale decompositions that hold for the Gaussian case continue to hold,
approximately, for the Ginzburg-Landau model.

In this paper, we study the maximum of Ginzburg-Landau fields. Given U⊂ D_N, let 

    M_U:=max_x∈ Uϕ ^D_N,0( x) ,

and set M_N=M_D_N. For the Gaussian case, we write M_N^G for M_N. Much is known about M_N^G, following a long succession of
papers starting with <cit.>. In particular, see <cit.> and <cit.>, M_N^G-m_N^G converges in distribution to a randomly shifted
Gumbel, with m_N^G=c_1log N-c_2loglog N and explicit constants 
c_1,c_2.

Much less is known concerning the extrema in the Ginzburg-Landau setup, even
though linear statistics of such fields converge to their Gaussian
counterparts <cit.>. A first step toward the study of the maximum was
undertaken in <cit.>, where the following law of large numbers is proved: 



    M_D_N/log N→ 2√(g) in L^2,  for
    some g=g( c_+,c_-) .


In this note we prove that the fluctuations of M_D_N around its mean
are tight, at least along some (deterministic) subsequence.


There is a deterministic sequence { n_k} with n_k→_k→∞∞ such that the sequence of random variables { M_D_n_k-𝔼M_D_n_k} is tight.


As will be clear from the proof, the sequence { n_k} can
be chosen with density arbitrarily close to 1. Theorem <ref> is the
counterpart of an analogous result for the Gaussian case proved in <cit.>, building on a technique introduced by Dekking and Host <cit.>.
The Dekking-Host technique is also instrumental in the proof of Theorem <ref>. However, due to the fact that the Ginzburg-Landau field does not
possess good decoupling properties near the boundary, significant changes
need to be made. Additional crucial ingredients in the proof are Miller's
coupling and a decomposition in differences of harmonic functions introduced
in <cit.>.



§ PRELIMINARIES




 §.§ The Brascamp-Lieb inequality


One can bound the variances and exponential moments with respect to the
Ginzburg-Landau measure by those with respect to the Gaussian measure, using
the following Brascamp-Lieb inequality. Let ϕ be sampled from the
Gibbs measure (<ref>). Given η∈ℝ^D_N, set 

    ⟨ϕ ,η⟩ :=∑_v∈ D_Nϕ _vη( v) .



Assume that V∈ C^2( ℝ) satisfies inf_x∈ℝV^''( x) ≥ c_->0. Let 𝔼_ and _ denote the expectation and
variance with respect to the  measure (that is, (<ref>)
with V( x) =x^2/2). Then for any η∈ℝ^D_N, 

    ⟨ϕ ,η⟩     ≤    c_-^-1_⟨ϕ ,η⟩ ,
    
    𝔼[ exp( ⟨ϕ ,η⟩ -𝔼
    ⟨ϕ ,η⟩) ]        ≤   exp( 1/2c_-^-1_⟨ϕ ,η⟩) .





 §.§ Approximate harmonic coupling


By their definition, the Ginzburg-Landau measures satisfy the domain Markov
property: conditioned on the values on the boundary of a domain, the field
inside the domain is again a gradient field with boundary condition given by
the conditioned values. For the discrete GFF, there is in addition a nice
orthogonal decomposition. More precisely, the conditioned field inside the
domain is the discrete harmonic extension of the boundary value to the whole
domain plus an independent copy of a zero boundary
discrete GFF.

While this exact decomposition does not carry over to general
Ginzburg-Landau measures, the next result due to Jason Miller, see <cit.>,
provides an approximate version.


 Let D⊂ℤ^2 be a simply connected domain
of diameter R, and denote D^r={ x∈ D:(x,∂
D)>r}. Let Λ be such that f:∂ D→
ℝ satisfies max_x∈∂ D| f( x) |≤Λ|log R| ^Λ. Let ϕ be
sampled from the Ginzburg-Landau measure (<ref>) on D with zero
boundary condition, and ϕ ^f be sampled from Ginzburg-Landau measure
on D with boundary condition f. Then there exist constants c,γ
,δ ^'∈( 0,1), that only depend on V, so that
if r>cR^γ then the following holds. There exists a coupling ( ϕ ,ϕ ^f), such that if ϕ̂:D^r→ℝ is discrete harmonic with ϕ̂|_∂ D^r=ϕ
^f-ϕ |_∂ D^r, then 

    ℙ( ϕ ^f=ϕ +ϕ̂ in D^r) ≥
    1-c( Λ) R^-δ ^'.



Here and in the sequel of the paper, for a set A⊂ℤ^2 and a
point x∈ℤ^2, we use (x,A) to denote the
(lattice) distance from x to A.



 §.§ Pointwise tail bound


We also recall the pointwise tail bound for the Ginzburg-Landau field (<ref>), proved in <cit.>.


 Let g be the constant as in (<ref>). For all u>0 large enough and all v∈ D_N we have


    ℙ( ϕ _v≥ u) ≤exp( -u^2/
    2glog( v,∂ D_N) +o( u) ) .



This allows us to conclude that the maximum of ϕ ^D_N,0 does not
occur within a thin layer near the boundary.


 Given δ <1, there exists δ ^'>0 such that

    ℙ( M_A_N,N^δ>( 2√(g)-δ ^') log N) ≤ N^δ -1/2,

where

    A_N,N^δ:={ v∈ D_N:( x,∂ D_N)
    <N^δ} .




Let Δ =( x,∂ D_N). For δ ^'
small enough, applying Theorem <ref> with u=( 2√(g)
-δ ^') log N yields

    P( ϕ _v≥( 2√(g)-δ ^') log
    N)    ≤   exp( -2( log N) ^2/logΔ+
    2δ ^'/√(g)( log N) ^2/logΔ+o( log N) ) 
       ≤   N^-2+2δ ^'/√(g)+o( 1) ,  for
    all v∈ A_N,N^δ.

Therefore a union bound yields 

    P( M_A_N,N^δ≥( 2√(g)-δ ^')
    log N) ≤ N^δ -1+2δ ^'/√(g)+o( 1)
    .

It suffices to take δ ^' such that 2δ ^'/√(g
)<1-δ/2.




§ THE RECURSION AND PROOF OF THEOREM <REF>


We prove Theorem <ref> by establishing a recursion for some random
variable M_Y_N, where Y_N⊂ D_N is a specific subset
defined below. Denote by T_N=[-N,N]×{N}⊂ D_N the top
boundary of D_N. For fixed ε >0, define

    Y_N={ v∈ D_N:( v,∂ D_N) ≥ε
    N}∪{ v∈ D_N:( v,∂ D_N) =
    (v,T_N) }.

For δ∈( 0,1), we also define Y_N,δ⊂
Y_N as 

    Y_N,δ={ v∈ Y_N:( v,T_N) >N^1-δ} ,

see Figure <ref>. 




For the constant g=g( c_+,c_-) in (<ref>),
we have 

    M_Y_N,δ/log N→ 2√(g) in L^2.




Let D_N^ε:={ v∈ D_N:(v,∂
D_N) ≥ε N}. Since 

    M_D_N^ε/log N≤M_Y_N,δ/log N
    ≤M_D_N/log N,

the claim (<ref>) follows from <cit.>, since the upper control on 
M_D_N/log N follows from (<ref>) while the lower control on M_D_N^ε/log N follows from the display below (5.19) in 
<cit.>.


We now switch to dyadic scales. For n∈ℕ, set N=2^n and m_n:=M_Y_2^n,δ. We set up a recursion for m_n. Clearly, 

    𝔼m_n+2 =𝔼M_Y_4N,δ≥𝔼max{max_v∈ Y_N,δ^( 1) ϕ _v^D_4N,0,max_v∈
    Y_N,δ^( 2) ϕ _v^D_4N,0} ,

where Y_N,δ^( i) are the translations of Y_N,δ, defined by Y_N,δ^( 1) =Y_N,δ+( -1.1N,3N), Y_N,δ^( 2) =Y_N,δ+( 1.1N,3N), see Figure <ref> 


The next two lemmas will allow us to control the difference between ϕ^D_4N,0 and ϕ ^D_N,0 (and as a consequence, between m_n+2 and m_n).


There exist δ ^',1>δ >γ >0, such that the
following statement holds. Set D_N^( 1) =D_N+(
-1.1N,3N), D_N^( 2) =D_N+( 1.1N,3N).
Let D_N^γ ,( i) :={ v∈ D_N^( i) :
(v,∂ D_N^( i) ) ≥ N^γ}. Then there exists a coupling ℙ of ( ϕ ^D_4N,0,ϕ ^D_N^( 1) ,0,ϕ
^D_N^( 2) ,0) and an event 𝒢 with ℙ( 𝒢^c) ≤ N^-δ ^', such
that with h_v^( i) being harmonic functions in D_N^( i) with boundary conditions ϕ ^D_4N,0-ϕ
^D_N^( i) ,0, on the event 𝒢, we have

    ϕ _v^D_4N,0=ϕ _v^D_N^( i) ,0+h_v^(
    i) , for all v∈ Y_N,γ^( i) , for 
    i=1,2.

Moreover, there is a constant C_0=C_0( δ), such
that, for any 1>δ >γ, 

    max_ i=1,2  
     v∈ Y_N,δ^( i) ( h_v^( i) ) ≤ C_0( δ) .




 With notation as in Lemma <ref>, there exists a constant C_1<∞, such that

    𝔼min_imin_v∈ Y_N,δ^( i) h_v^(
    i) =-𝔼max_imax_v∈ Y_N,δ^( i)
    h_v^( i) ≥ -C_1.



The proof of Lemmas <ref> and <ref> are postponed to Section <ref>. In the rest of this section, we bring the proof of Theorem <ref>.


Denote by m_n^∗ an independent copy of m_n. We combine Lemmas 
<ref> and <ref> to conclude

    𝔼m_n+2   ≥   𝔼[ 1_𝒢max_imax_v∈
    Y_N,δ^( i) ( ϕ _v^D_N^( i)
    ,0+h_v^( i) ) ] 
       ≥   𝔼max{ m_n,m_n^∗} +2𝔼
    min_imin_v∈ Y_N,δ^( i) h_v^( i)
    -2𝔼[ 1_𝒢^cm_n] .

We apply (<ref>) to conclude that 

    𝔼[ 1_𝒢^cm_n] ≤ℙ( 
    𝒢^c) ^1/2𝔼[ m_n^2] ^1/2≤ C
    log N/N^δ ^'/2

Thus for all large n, we can apply Lemma <ref> to get 

    𝔼m_n+2≥𝔼max{ m_n,m_n^∗}
    -3C_1.

Using max{ a,b} =1/2( a+b+|
a-b|) and Jensen's inequality, we obtain

    𝔼m_n+2-𝔼m_n≥1/2𝔼|
    m_n-m_n^∗| -3C_1≥1/2𝔼|
    m_n-𝔼m_n^∗| -3C_1.

We need the following lemma.


There exists a sequence { n_k} and a
constant K<∞ such that

    𝔼m_n_k+2≤𝔼m_n_k+K.




Let I_n,K={ j∈{n,n+2,… ,2n}:𝔼m_j+2≥
𝔼m_j+K}. Then Lemma <ref> implies that 𝔼
m_n/n→ 2√(g), while (<ref>) gives 𝔼
m_n+2≥𝔼m_n-3C_1. Therefore, for any fixed η >0 and
all large n, 

    K|I_n,K|-3C_1(n/2-|I_n,K|)≤ 2√(g)n(1+η )≤ 4√(g
    )n,

giving that for all large n, 

    |I_n,K|≤ n4√(g)+3/2C_1/K+3C_1.

Choosing K>16√(g)+3C_1 gives that for all n large, |I_n,K|≤
n/4. 



It follows that for all large n, there exists n^'∈[ n,2n
], such that

    𝔼m_n^'+2≤𝔼m_n^'+K,

This completes the proof of Lemma <ref>.


We continue with the proof of Theorem <ref>. Using the subsequence { n_k} from Lemma <ref>, we have from (<ref>)
that 

    𝔼| m_n_k-𝔼m_n_k^∗|≤
    2K+6C_1,

which implies, using Jensen's inequality, that { m_n_k-𝔼
m_n_k^∗} is tight. This implies that the the sequence of
random variables

    M̅_D_N_k^δ:=max{ϕ _v^D_N_k,0:v∈
    D_N_k,(v,∂ D_N_k) ≥ N_k^1-δ}

is tight around its mean because M̅_D_N_k^δ is the
maximum of 4 rotated copies of m_n_k.

Finally, combining (<ref>) and Lemma <ref> we obtain

    ℙ( M_D_N_k>M̅_D_N_k^δ) ≤
    2^n_k(δ -1)/2,

and 

    𝔼M_D_N_k-𝔼M̅_D_N_k^δ   ≤   
    𝔼M_D_N_k1_{ M_D_N_k>M̅_D_N_k^δ}
       ≤   ℙ( M_D_N_k>M̅_D_N_k^δ)
    ^1/2( 𝔼M_D_N_k^2) ^1/2
       ≤   2^n_kδ -1/4O( log N_k) → 0.

We conclude that the sequence { M_D_N_k-𝔼
M_D_N_k} is tight.




§ PROOF OF LEMMA <REF> AND <REF>





The existence of the harmonic decomposition is implied by the Markov
property and Theorem <ref> (with δ ^',γ taken
as the constants in Theorem <ref>). It thus suffices to obtain an
upper bound for Var( h_v^( i) ). Write h_v^( i) =ĥ_v^( i) -h̃
_v^( i), where ĥ_v^( i) is the
harmonic function in D_N^γ ,( i) with boundary value ϕ ^D_4N,0, and h̃_v^( i) is the harmonic
function in D_N^γ ,( i) with boundary value ϕ
^D_N^( i) ,0. Without loss of generality we set i=1.
Applying the Brascamp-Lieb inequality (<ref>) we get

    ( h_v^( 1) ) ≤ c_-^-1_( h_v^( 1) ) .

The orthogonal decomposition for GFF implies

    _( ĥ_v^( 1) )    =   
    _( 𝔼_[ ϕ
    _v^D_4N,0|ℱ_∂ D_N^γ ,( 1) 
    ] ) 
       =   _[ ϕ _v^D_4N,0] -_[ ϕ _v^D_N^γ ,( 1) ,0
    ]

and

    _( h̃_v^( 1) ) =
    _[ ϕ _v^D_N^( 1) ,0
    ] -_[ ϕ _v^D_N^γ ,(
    1) ,0] .

We now estimate the last two expressions for different regions of v∈
Y_N,δ^( 1). First of all, it suffices to control h_v^( 1) for v∈∂ Y_N,δ^( 1). Let 

    Q    :   ={ v∈∂ Y_N,δ^( 1) :
    (v,∂ D_N) =(v,T) } , 
    
    R    :   ={ v∈∂ Y_N,δ^( 1) :
    (v,∂ D_N)  =ε N} .

We first show that

    max_v∈ R_( ĥ_v^( 1)
    )    ≤   C( ε) ,  
    max_v∈ Q∪ R_( h̃_v^(
    1) )    ≤   C_0N^γ -δ.

Indeed, standard asymptotics for the lattice Green's function (following
e.g. from <cit.>) give, for some constant g_0, 

    _[ ϕ _v^D_4N,0]
    -_[ ϕ _v^D_N^γ ,(
    1) ,0] 
       =   g_0( log(v,∂ D_4N) -log
    (v,∂ D_N^γ ,( 1) ) ) +o_N( 1)
    
       ≤   g_0log4N/ε N-N^γ+o_N( 1)
    ≤ C(ε ),

and similarly,

    _[ ϕ _v^D_N^(
    1) ,0] -_[ ϕ
    _v^D_N^γ ,( 1) ,0] 
       =   g_0( log(v,∂ D_N^^( 1) )
    -log(v,∂ D_N^γ ,( 1) ) )
    +O( N^-1) 
       ≤   g_0logN^δ/N^δ-N^γ+O(
    N^-1) ≤ C_0N^γ -δ.


To conclude the proof, we also claim for δ∈( γ ,1)

    max_v∈ Q_( ĥ_v^( 1)
    ) ≤ CN^γ -δ.

Indeed, denote by T_γ the top boundary of D_N^γ, we
apply asymptotics for lattice Green's function to obtain

    _[ ϕ _v^D_4N,0]
    -_[ ϕ _v^D_N^γ,0] 
          =g_0( log(v,∂ D_4N) -log
    (v,∂ D_N^γ) ) +O( N^-1) 
          =g_0( log(v,T) -log( v,T_γ)
    ) +O( N^-1) .

Since 

    log(v,T) /(v,T_γ) ≤logN^δ/N^δ-N^γ≤ CN^γ -δ,

we obtain (<ref>).



Recall that h_v^(i)=ĥ_v^(i)-h̃_v^(i). We will prove
that there exist C_0<∞ and α >0, such that for all C_1>C_0, 

    ℙ( max_v∈ Qĥ_v^( 1) >C_1)
       ≤   e^-α C_1,  
    ℙ( max_v∈ Rĥ_v^( 1) >C_1)
       ≤   e^-α C_1,  
    ℙ( min_v∈ Q∪ Rh̃_v^( 1)
    <-C_1)    ≤   e^-α C_1.

Indeed, (<ref>) follows from (<ref>) and the exponential
Brascamp-Lieb inequality (<ref>):

    ℙ( max_v∈ Qĥ_v^( 1) >C_1)
       ≤   | Q|max_v∈ Qℙ( ĥ
    _v^( 1) >C_1) 
       ≤   C_3Nexp( -C_1^2/C_2_( ĥ_v^( 1) ) ) 
       ≤   C_3Nexp( -C_1^2/C_2N^δ -γ) ,

where C_2,C_3 are some fixed constants. The same argument using (<ref>) gives (<ref>).

We now prove (<ref>) using chaining. Omitting the superscripts (
1) in ĥ^( 1) and D_N^γ ,(
1), we claim that there exists K<∞, such that for u,v∈ R
, 

    _[ ĥ_u-ĥ_v] ≤ K
    | u-v|/ε N.

Applying the orthogonal decomposition of the DGFF we obtain

    ϕ _u^D_4N,0-ϕ _v^D_4N,0=ϕ _u^D_N^γ,0-ϕ
    _v^D_N^γ,0+ĥ_u-ĥ_v,

and therefore, by the independence of ϕ _u^D_N^γ,0-ϕ
_v^D_N^γ,0 and ĥ_u-ĥ_v under the DGFF
measure, 

    _[ ĥ_u-ĥ_v] =_[ ϕ _u^D_4N,0-ϕ _v^D_4N,0] - 
    _[ ϕ _u^D_N^γ,0-ϕ
    _v^D_N^γ,0] .

We now apply the representation of the lattice Green's function, 


see, e.g., <cit.>,

    G^D_N( u,v) =∑_y∈∂ D_NH_∂ D_N( u,y) a( y-v)
    -a( u-v) ,

where H_∂ D_N( u,·) is the harmonic measure of 
D_N seen at u and a is the potential kernel on ℤ^2
which satisfies the asymptotics

    a( x) =2/πlog| x| +D_0+O(
    | x| ^-2) ,

where D_0 is an explicit constant (see e.g. <cit.> for a
slightly weaker result which nevertheless is sufficient for our needs).
Substituting into (<ref>), we see that 

    _[ ϕ _u^D_4N,0-ϕ
    _v^D_4N,0] - _[ ϕ
    _u^D_N^γ,0-ϕ _v^D_N^γ,0] 
       =   G^D_4N( u,u) +G^D_4N( v,v)
    -2G^D_4N( u,v) 
          -( G^D_N^γ( u,u) +G^D_N^γ(
    v,v) -2G^D_N^γ( u,v) ) 
       =   ∑_z∈∂ D_4NH_∂ D_4N( u,z) a(
    u-z) +∑_z∈∂ D_4NH_∂ D_4N( v,z)
    a( v-z) 
                 -2∑_z∈∂ D_4NH_∂ D_4N(
    u,z) a( v-z) 
          -∑_z∈∂ D_N^γH_∂ D_N^γ(
    u,z) a( u-z) -∑_z∈∂ D_N^γH_∂ D_N^γ( v,z) a( v-z) 
                 +2∑_z∈∂ D_N^γH_∂
    D_N^γ( u,z) a( v-z) 
       :   =A_D_4N-A_D_N^γ

We now apply the Harnack inequality, see <cit.>,

    | H_∂ D_4N( u,z) -H_∂ D_4N(
    v,z) |≤| u-v|/4N

to obtain

    A_D_4N   =   ∑_z∈∂ D_4NH_∂ D_4N(
    u,z) ( a( u-z) -a( v-z) ) 
          +∑_z∈∂ D_4N( H_∂ D_4N( v,z)
    -H_∂ D_4N( u,z) ) a( v-z) 
       ≤   | u-v|/4N∑_z∈∂
    D_4NH_∂ D_4N( u,z) 
          +∑_z∈∂ D_4N( H_∂ D_4N( v,z)
    -H_∂ D_4N( u,z) ) ( a( v-z) -
    2/πlog N-D_0) 
       ≤   K| u-v|/N,   for some K<∞.

The same argument gives | A_D_N^γ|≤ K
| u-v|/ε N, thus (<ref>) is
proved.

Now fix a large k_0. For k≥ k_0 let P_k be subsets of R
that plays the role of dyadic approximations: P_k contains O(
2^k) vertices that are equally spaced and the graph distance
between adjacent points is ε N2^-k. For v∈ R, denote by P_k( v) the k^th dyadic approximation of v, namely the
vertex in P_k that is closest to v. Then for v∈ R,

    ĥ_v=P_k_0( v) +∑_k≥ k_0ĥ
    _P_k+1( v) -ĥ_P_k( v) .

We now apply the exponential Brascamp-Lieb inequality (<ref>
), (<ref>), and a union bound to obtain 

    ℙ( max_v∈ R[ ĥ_P_k+1(
    v) -ĥ_P_k( v) ] >√(K( 3/2
    ) ^-k)) 
          ≤ C_32^kexp( -K( 3/2) ^-kC_4/
    2· 2^-k) 
          ≤ C_32^kexp( -C_4( 4/3) ^kK/2
    ) ,

for some constant C_4. Since both √(K( 3/2) ^-k)
and the tail probability are summable in k, we conclude that (<ref>)
holds.




Acknowledgment O.Z. thanks Jason Miller for suggesting,
years ago, that the coupling in <cit.> could be useful in carrying out the
Dekking-Host argument for the Ginzburg-Landau model. W.W. thanks the
Weizmann Institute for its hospitality. This work was started while both
authors were at the Courant Institute, NYU.

alpha


