Moderate deviations for the spectral measure of certain random matrices

2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved 10.1016/S0246-0203(03)00024-4/FLA

MODERATE DEVIATIONS FOR THE SPECTRAL MEASURE OF CERTAIN RANDOM MATRICES

A. DEMBO^a,1, A. GUIONNET^b,∗,2, O. ZEITOUNI^c,3

aDepartments of Mathematics and of Statistics, Stanford University, Stanford, CA 94305, USA bUMPA, École normale superieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, France cDepartments of Electrical Engineering and of Mathematics, Technion, Haifa 32000, Israel

Received 14 February 2002, accepted 25 February 2003

ABSTRACT. – We derive a moderate deviations principle for matrices of the form XN= DN +WN where WN are Wigner matrices and DN is a sequence of deterministic matrices whose spectral measures converge in a strong sense to a limit µD. Our techniques are based on a dynamical approach introduced by Cabanal-Duvillard and Guionnet.

2003 Éditions scientifiques et médicales Elsevier SAS MSC: 60F99; 15A52

Keywords: Limit theorems; Random matrices

RÉSUMÉ. – Nous démontrons un principe de déviations modérées pour la mesure spectrale de matrices de la formeXN=DN+WN oùWN sont des matrices de Wigner etDN une suite de matrices déterministes dont la mesure spectrale converge fortement vers une loi limiteµD. Nous utilisons pour cela des techniques basées sur l’approche dynamique introduite par Cabanal- Duvillard et Guionnet.

2003 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

LetMNdenote the set ofN×NHermitian matrices, and letWN∈MNbe a Gaussian Wigner matrix, that is, a symmetric or Hermitian matrix with real (respectively, complex) i.i.d. Gaussian entries of covarianceN⁻¹above the diagonal. We consider

XN=DN+WN

∗Corresponding author.

E-mail address: [email protected] (A. Guionnet).

1Research partially supported by NSF grant #DMS-0072331.

2Research partially supported by NSF grant #DMS-9631278.

3Research partially supported by a grant from the Israel Science Foundation, administered by the Israel Academy of Sciences.

for a diagonal matrix DN with diagonal elements d_i^N and spectral measure µˆ^N_DN :=

N^−1N_i=1δ_d^N

i converging towards a compactly supported probability measure µD, in such a way that

˜

c(ε):=max

i=1,2max

N sup

z∈C\R,|(z)|ε

NtrN(z−DN)⁻ⁱ−

(z−x)⁻ⁱdµD(x)<∞ ∀ε >0 (A) (where trNdenotes the trace of the matrix, normalized by its sizeN).

Denote byµˆ^N_X

Nthe spectral measure ofXN. Recall (see, e.g., [19]) thatµˆ^N_X

N converges weakly to the compactly supported probability measure µ^∗₁ where µ^∗_t =µD σt, σt

denoting the Wigner semi-circular distribution σt(dx)=(2π t)⁻¹√

4t−x²dx and denoting free convolution of measures.

Large deviations (in the scaleN²) and CLT’s forµˆ^N_X

N are obtained in [5,6,9,12], by a dynamical approach based on the observation thatWN can be constructed as a Hermitian or symmetric Brownian motion at time one. These large deviations are essential tools in the study of so-called “matrix models” in physics, see [10]. It is our goal in this work to extend this analysis to study moderate deviations ofµˆ^N_XN. Note that since exponentially good approximations at the scaleN²are no longer such for the moderate deviation scales considered here, this study is far from being a straight forward extension of the previous analysis mentioned above. Our work can be considered as a non-commutative partial analogue of the moderate deviations principle for the empirical measure of i.i.d. random variables, see [21].

In order to state our results, we first introduce some notations. LetStieljes(C)be the complex vector space generated by the Stieljes functions{f (x)=(z−x)⁻¹, z∈C\R}

(withx∈R), and denote byStieljes(R)⊂Cb^∞(R)the subset of real valued functions in Stieljes(C). Note that(z−x)⁻¹∈Stieljes(R)forz∈C\R.

To any f =f1∈ Stieljes(C), we associate the function s → fs that solves the differential equation

∂sfs(x)= − ∂xfs(x)−∂xfs(y)

x−y dµ^∗_s(y), f1(x)=f (x). (1.1) In Section 4 below we show that (1.1) has a unique solution wheneverf (x)=c(z−x)⁻¹, z∈C\R(given by (4.1) and (4.2)). By the linearity of (1.1), the same applies for any f ∈Stieljes(C).

For anyf, g∈Stieljes(R)and the corresponding solutionsfs,gs of (1.1), define Vt(f, g)=

t 0

(∂xfs)(x)(∂xgs)(x) dµ^∗_s(x) ds. (1.2)

In what follows, we letStieljes(R):= {f: f ∈Stieljes(R)}and L^p_c(R)denote the subset ofL^p(R)consisting of functions of compact support. Let

FN(x):=µ^∗₁(−∞, x]− ˆµ^N_XN(−∞, x],

which is inL^p_c(R)for any p1 wheneverµD (and henceµ^∗₁) is of compact support.

Integration by parts shows that for allf ∈C¹b(R), XN(f ):=

f (x) dµˆ^N_XN(x)−

f (x) dµ^∗₁(x)=

f(x)FN(x) dx.

With some abuse of notation we let g, G denote the value of a linear functional G atg which is in a vector subspace of Cb(R) (to be understood from the context of the statement), using alsog, G = g(x)G(x) dxin the caseG∈L¹(R).

Our main result now reads

THEOREM 1.1. – For any aN →0 such that N aN → ∞, the sequence of random variables {a⁻_N¹FN}N in L¹_c(R) equipped with the Stieljes(R)-topology and the cor- responding cylinder σ-field, satisfies the Large Deviation Principle (LDP) with speed (N aN)⁻²and good rate functionI (·)defined by

I (F ):=β

2 sup

h∈Stieljes(R)

h, F −1

2V1(h, h)

(1.3) withβ=1 (resp.β=2) in the symmetric (resp. Hermitian) case.

(We refer to [8] for standard terminology concerning the LDP. Because the rate function is the same for a range of speeds, we refer to the LDP in Theorem 1.1 as a moderate deviation principle (MDP).)

Note that the topology for the MDP in Theorem 1.1 is weaker than theCb(R)-topology of convergence in law. Some strengthening of the former topology can be achieved by considering theN-dependent centering

FN(x):=Eµˆ^N_XN(−∞, x]− ˆµ^N_XN(−∞, x].

Specifically, withC_b(R):= {f: f ∈C¹_b(R)}denoting the space of bounded continuous functions which possess a bounded primitive, we have:

THEOREM 1.2. – For any aN →0 such that N aN → ∞, the sequence of random variables{a⁻_N¹FN}NinL¹_c(R)equipped with theCb(R)-topology and the corresponding cylinder σ-field, satisfies the LDP with speed (N aN)⁻² and good rate function I (·) of (1.3).

As the rate functionI (·)is not particularly transparent to work with, we provide next some useful information about it. First, it follows from the CLT of [9, Section 6] that for anyh∈Stieljes(R),

V1(h, h)= lim

N→∞N²EXN(h)²= lim

N→∞VarN XN(h). (1.4) Our proof of Theorems 1.1 and 1.2 provides also that for any aN → 0 such that N aN→ ∞,

V1(h, h)=2 lim

N→∞(N aN)⁻²logEe^N2aNXN(h)

=2 lim

N→∞(N aN)⁻²logEe^N2aNXN(h). (1.5)

Recall that µ^∗₁ has a density p1 (see [2, Corollary 2]), and for F = f p1 with f ∈L²(µ^∗₁)set

J (F ):=β 4

f²(x) dµ^∗₁(x)+β 4

f (x)−f (y) x−y

2

dµ^∗₁(x) dµ^∗₁(y) (1.6) (which is well defined, though possibly infinite), settingJ (F )= ∞ for allF ∈L¹_c(R) not of the above form. LetL^1,_c^∗(R)denote the subset ofL¹_c(R)consisting of functions whose support is contained in the support ofµ^∗₁. In Section 6 we prove that

LEMMA 1.3. – The functionI (·)is finite only for linear functionals onStieljes(R) that are of the formh, F =F (x)h(x) dx for someF ∈L^1,_c^∗(R), in which case

β 4

F (x)dx 2

β

4

F (x)²

p1(x) dxI (F ). (1.7) Further,I (F )J (F )forF =f p1withf ∈Cb¹(R), and more generally, for all

F ∈P:=f p1:f ∈L²(µ^∗₁), ∃Q^δpolynomials such thatQ^δ−→^L

δ→0f, lim inf

δ→0 JQ^δp1

J (f p1).

Here, Q^δ−→^L

δ→0f iff g(x)Q^δ(x)p1(x) dx converges towards g(x)f (x)p1(x) dx for any bounded continuous functiongonR.

In the special caseµD=0, one can make the rate function more explicit. Indeed, in this caseµ^∗_t =σt andp1(y)=(2π )⁻¹4−y², and one obtains

LEMMA 1.4. – SupposeµD=0. Then, for everyh∈Stieljes(R), V1(h, h)= − 1

2π² 2

−2

2

−2

h(x)h(y) y−x

4−y²

√4−x²dy dx. (1.8)

Moreover, assumeF =f p1∈L¹_c(R)for somef ∈Cb³(R). Then

I (F )=J (F )= −β 2

2

−2

2

−2

F(x)F(y)log|x−y|dx dy. (1.9) The expression in the right-hand side of (1.9) resembles Voiculescu’s non-commutative entropy ofDN+WNtaken at the measureF(x) dx.

The structure of the article is as follows. In Section 2, we introduce the (matrix val- ued) Brownian motion WN(t) and recall the elements of stochastic calculus we need.

Section 3 is devoted to the proof of a CLT type approximation for the empirical Stieljes transform M_t^N(z) of XN(t)=DN +WN(t). Section 4 controls the influence of other centerings on the convergence properties ofM_t^N(z). Our moderate deviations results are proved in Section 5. We present first in Theorem 5.1 a finite-dimensional moderate de-

viations principle (forXN(f⁽ⁱ⁾), i=1, . . . , d) which among other things implies (1.5), and then the projective limits argument leading to Theorems 1.1 and 1.2, deferring the proofs of Lemma 1.3 (via free probability theory) and Lemma 1.4 to Section 6.

2. Itô’s calculus

Let HN(·) (respectively, SN(·)) be a N × N Hermitian (respectively, symmet- ric) Brownian motion constructed via independent real valued Brownian motions (βi,j,β˜k,l)¹₁^k<l_ij^N_N by

H_N^k,l=























√1

2N(βk,l+iβ˜k,l) if k < l,

√1

2N(βl,k−iβ˜l,k) ifk > l,

√1

Nβl,l ifk=l and

S_N^k,l=

√1+δk=l

√N βk∧l,k∨l,

respectively. Take WN(t)=HN(t) in the Hermitian case and WN(t) =SN(t) in the symmetric case. Then,WN(1)is a complex (respectively, real), Wigner matrix. Letµˆ^N_t denote the spectral measure ofXN(t)=DN+WN(t) (note thatµˆ^N₁ = ˆµ^N_X

N), thenµˆ^N_t can be studied by use of Itô’s calculus as we now explain.

It was proved in [3,6] thatµˆ^N_. satisfies an Itô’s formula (in the special case where ˆ

µ^N_D =δ0, assumption which is in fact clearly irrelevant). Then, if we denote, for any f, g∈Cb^2,1(R× [0,1]), anyst∈ [0,1], and anyν_·∈C([0,1],P(R)),

S^s,t(ν, f )=

f (x, t) dνt(x)−

f (x, s) dνs(x)− t s

∂uf (x, u) dνu(x) du

−1 2 t

s

∂xf (x, u)−∂xf (y, u)

x−y dνu(x) dνu(y) du, (2.1) and

f, g^νs,t= t

s

∂xf (x, u)∂xg(x, u) dνu(x) du, (2.2) we have

THEOREM 2.1 [6]. – In the Hermitian case, for anyN∈N, anyf ∈Cb^2,1(R× [0,1]) and any s ∈ [0,1),(S^s,t(µˆ^N, f ), st1)is a bounded continuous martingale with quadratic variation

S^s,·µˆ^N, f_t= 1

N²f, f^µs,t^ˆN. (2.3)

In the symmetric case, for any N ∈N, any f ∈Cb^2,1(R× [0,1])and anys ∈ [0,1), (S^s,t(µˆ^N, f ):=S^s,t(µˆ^N, f )−(2N )⁻¹_s^tf(x) dµˆ^N_u(x) du, s t1) is a bounded continuous martingale with quadratic variation

S^s,·µˆ^N, f_t= 2

N²f, f^µs,t^ˆN.

Note that it is not hard to see (see [6] or [11]) that the law of µˆ^N_· is tight in both Hermitian and symmetric settings. The limit points are characterized by

S^s,t(µ^∗_·, f )=0 (2.4)

for all functionsf ∈Cb^2,1(R×[0,1]). It can be shown (see [6, Corollary 1.4] or [12]) that such an equation has a unique solutionµ^∗_·, given by the free convolutionµ^∗_t =µDσt.

In the sequel, we shall be interested in specific test functions of the Stieljes type:

f (x, t)= ct

zt −x (2.5)

with a complex-valued differentiable function z_·:[0,1] →C\R with non-vanishing imaginary part and a complex-valued differentiable function ct:[0,1] →C. Observe that in this case, for anyν∈C([0,1],P(R)),

1 2

∂xf (x, u)−∂xf (y, u)

x−y dνu(x) dνu(y)

=cu

1

zu−x dνu(x)

1

(zu−x)²dνu(x). (2.6) 3. Central limit approximation

Following Israelsson (see [13, Proposition 1]), we prove the following central limit type approximation. Throughout, we set β=1 in the symmetric case andβ=2 in the Hermitian case.

LEMMA 3.1. – Consider Hermitian or symmetric matrices such that (A) holds. Then, for anyη >0, there exists a finite constantC(η)such that for allN andz∈C\R, z= a+ib, |b|η,

jmax=1,2 sup

τ∈[0,1]EtrN

z−XN(τ )^−j−

(z−x)^−jdµ^∗_τ(x)

21/2

C(η)

N . (3.1)

Proof. – Israelsson [13] considers only the symmetric case with Ornstein–Uhlenbeck entries. Hence, for completeness, we next adapt his approach to the context of the lemma.

The main idea, used also by [5] and [9] is to choose (c_·, z_·) in such a way that the finite variation term in Theorem 2.1 is negligible. Whereas Cabanal-Duvillard [5] and Guionnet [9] choose a non-random (c_·, z_·) that is independent of N and then control

the remainder term of finite variation, we follow Israelsson [13] in choosing (c_·, z_·) randomly and depending onN in such a way that this term completely vanishes.

We first consider the Hermitian case andj=1 in (3.1). Forz∈C\Rdenote M_t^N(z)=trN

zI−XN(t)⁻¹, Mt(z)=

(z−x)⁻¹dµ^∗_t(x).

Applying Theorem 2.1 tof (x, t)of (2.5), and using (2.6) it is easy to check that for any continuously differentiable functions(c_·, z_·)such thatzt stays uniformly away from the real axis,

ctM_t^N(zt)=c0M₀^N(z0)+ t 0

(∂scs)M_s^N(zs)+cs(∂szs)∂zM_s^N(zs)

−csM_s^N(zs)∂zM_s^N(zs)ds+m^N_t (z_·, c_·)

with the bounded, complex valued, martingalem^N(z_·, c_·)having the quadratic variations m^N(z_·, c_·)_t= 1

N² t 0

cu

(zu−x)² 2

dµˆ^N_u(x) du, m^N(z_·, c_·)_t= 1

N² t 0

cu

(zu−x)² 2

dµˆ^N_u(x) du. (3.2) Since|z−x||(z)|forx∈R, it follows thatM_t^N(·)andMt(·)are uniformly Lipschitz continuous onC\R× [−|b|,|b|]. Specifically, there

M_t^N(z)−M_t^N(˜z)∨Mt(z)−Mt(˜z)|b|⁻²|z− ˜z|. (3.3) Fixing τ ∈ [0,1], following Israelsson [13], we choose (c_·, z_·)=(c^N_· , z^N_· ) to be the solution of

∂tz^N_t =1 2

M_t^Nz^N_t +Mt

z^N_t , z^N_τ =z, (3.4)

∂tc_t^N=1 2

∂zM_t^Nz_t^N+∂zMt

z_t^Nc^N_t , c^N_τ =1, (3.5) whose existence and uniqueness we next prove. Indeed, the sign of(M_t^N(ξ )+Mt(ξ )) is opposite to that of (ξ ). Thus, taking u^(l)t , l =0,1, . . ., such that u^(l)_τ =z for all l, u⁽⁰⁾t =zfort∈ [0, τ]and

∂tu^(l_t⁺¹⁾=1 2

M_t^Nu^(l)_t +Mt

u^(l)_t ,

it is easy to see by induction that|(u^(l)t )||b|for allt∈ [0, τ]andl0. The uniform Lipschitz property of M_t^N(·)+Mt(·) implies by Gronwall’s lemma that the sequence u^(l) converges uniformly on [0, τ] to the unique solution of (3.4). The existence of a

unique solution of (3.5) is then clear. Note thatt→ |(z^N_t )|is monotone non-increasing on[0, τ]. Moreover, by (2.4), it is easy to check that fort∈ [0, τ],

c^N_t Mt

z_t^N=c^N₀M0

z^N₀+ t

0

−c^N_s Ms

z_s^N∂zMs

z^N_s +∂sc^N_s Ms

z_s^N +c^N_s ∂sz^N_s ∂zMs

z^N_s ds

=c^N₀M0

z^N₀+ t

0

c^N_s 2

Ms

z^N_s ∂zM^Nz_s^N+M_s^Nz^N_s ∂zMz^N_s ds, (3.6)

where (3.6) is a consequence of (3.4) and (3.5). Similarly, we find that c^N_t M_t^Nz^N_t =c₀^NM₀^Nz^N₀+

t 0

c^N_s 2

M_s^Nz^N_s ∂zMz^N_s +Ms

z^N_s ∂zM^Nz^N_s ds

+m^N_t z^N_· , c^N_· . (3.7)

Subtracting (3.6) from (3.7) we find that c^N_t M_t^Nz^N_t −Mt

z_t^N=c₀^NM₀^Nz^N₀−M0

z^N₀+m^N_t z_·^N, c^N_· . (3.8) Let b_s^N = (z^N_s ), noting that |b_t^N| |b| for t ∈ [0, τ]. Let v_s^N = |c_s^N|² and a_s^N = (∂zM_s^N(z^N_s )+∂zMs(z^N_s )), noting that

a_s^N∂zM_s^Nz^N_s +∂zMs

z^N_s 2

|b_s^N|² 2

|b|², (3.9)

whereas∂tv_t^N=a_t^Nv_t^N,v_τ^N=1, by (3.5). Sincev^N_t =exp(−_t^τa_s^Nds), it follows that sup

t∈[0,τ]

c^N_t exp|b|⁻². (3.10)

Thus, by (3.2), for anyt∈ [0, τ],

m^Nz_·^N, c^N_{· t}+m^Nz^N_· , c_·^N_t

= 1 N²

t 0

c^N_s (z_s^N−x)²

²dµˆ^N_s (x) ds e^2|b|−2

N²|b|⁴ (3.11) implying that

Em^N_t z^N_· , c_·^N2=Em^Nz^N_· , c_·^N_t+Em^Nz^N_· , c_·^N_t e^2|b|−2 N²|b|⁴. We have by (3.10) and assumption (A) that

c₀^NM₀^Nz^N₀−M0

z^N₀=c^N₀trN

z^N₀ −DN

₋1

− z^N₀ −x⁻¹dµD(x)

e^|b|−2c(˜ |b|)

N .

Withc^N_τ =1, we thus see that (3.8) yields the bound EM_τ^Nz^N_τ−Mτ

z^N_τ ^21/2e^|b|−2c(˜ |b|)

N + e^|b|−2 N|b|².

Since z∈C\R,N and τ ∈ [0,1] are arbitrary, this completes the proof of (3.1) when j =1, in the Hermitian case. Still in the Hermitian case, let us now prove it forj =2 and, without loss of generality, forz∈C+\R. First, observe that

trN

z−XN(τ )⁻²+N M_τ^N z+ 1 N

−M_τ^N(z) 1

N|b|³, (3.12)

(z−x)⁻²dµ^∗_τ(x)+N Mτ z+ 1 N

−Mτ(z) 1

N|b|³. (3.13) Therefore, it is enough to bound

η^N_τ :=N M_τ^N z+ 1 N

−M_τ^N(z)

−N Mτ z+ 1 N

−Mτ(z)

.

We proceed as above by considering a martingale representation of η^N_τ , given, if (z^N_t (z), c^N_t (z)) are the functions constructed in (3.4) and (3.5) with terminal data (z^N_τ(z), c^N_τ (z))=(z,1), by

η^N_t =N c^N_t z+ 1

N M_t^N z_t^N z+ 1 N

−Mt z^N_t z+ 1 N

−N c^N_t (z)M_t^Nz^N_t (z)−Mt

z^N_t (z).

By (3.8),

η^N_t =η₀^N+N m^N_t z^N_· z+ 1 N

, c^N_· z+ 1 N

−m^N_t z^N_· (z), c^N_· (z):=η^N₀ + m^N_t (z).

Note that since (z^N_t (z))(z)=b >0 for z=z orz=z+N⁻¹, (3.3) and (3.4) imply that

∂t z^N_t z+ 1 N

−z^N_t (z)|b|⁻²z_t^N z+ 1 N

−z_t^N(z), whereas (3.5), (3.9) and (3.10) imply that

∂t c^N_t z+ 1 N

−c^N_t (z) e^|b|−2|b|⁻³z^N_t z+ 1

N

−z^N_t (z)+ |b|⁻²c_t^N z+ 1 N

−c_t^N(z). Therefore, Gronwall’s lemma gives fortτ

z_t^N z+ 1 N

−z_t^N(z) 1

Ne^{(τ−t )/|b|2}, c_t^N z+ 1

N

−c_t^N(z) 1

N|b|³e^{(2(τ−t )+1)/|b|2}. (3.14) Using the above control, we can boundη₀^N. To this end, let

η₀^N=N c^N₀ z+ 1 N

M₀^N z^N₀ z+ 1 N

−M₀^Nz^N₀(z)− M0 z^N₀ z+ 1 N

−M0

z₀^N(z)+N c^N₀ z+ 1 N

−c^N₀(z)M₀^Nz^N₀(z)−M0

z^N₀(z)

=I+II.

The first term can be decomposed as follows I= −N c^N₀ z+ 1

N z₀^N z+ 1 N

−z₀^N(z)

×

trN

z^N₀(z)−DN

₋2

− z₀^N(z)−x⁻²dµD(x)

+N c₀^N z+ 1

N z^N₀ z+ 1 N

−z^N₀(z) 2

× z^N₀ z+ 1 N

−x ₋1

z₀^N(z)−x⁻²dµˆ^N₀ −µD

(x)

=I1+I2. By (3.10), (3.14) and assumption (A) we find that

|I1|Ne^|b|−2 1

Ne^τ/|b|2c(|b|)˜ N

and

|I2|Ne^|b|−2e^2τ/|b|2 N²

1

|b|³ so that we have found a finite constantC1(b)such that

|I|C1(b)

N . (3.15)

Moreover, by (3.14) and assumption (A),

|II|N 1

N|b|³e^(2τ+1)/|b|2c(˜ |b|) N

resulting, with (3.15), with the existence of a finite constantC2(b)such that η^N₀C2(b)

N . (3.16)

Moreoverm^N_· (z)is a martingale whose martingale bracket can be computed. Following (3.11), we find

m^N(z)_τ+m^N(z)_τ

= 1 N²

τ 0

N c^N_s (z+N⁻¹)

(z^N_s (z+N⁻¹)−x)² − N c_s^N(z) (z^N_s (z)−x)²

²dµˆ^N_s (x) ds

1

N² τ 0

(N|c^N_s (z+N⁻¹)−c^N_s (z)|)²

|b^N_s (z+N⁻¹)|⁴

ds

+ 1 N²

τ 0

|c^N_s (z)|²|N (z^N_s (z+N⁻¹)−z^N_s (z))|² (|b^N_s (z)| ∧ |b_s^N(z+N⁻¹)|)⁶

ds

C3(b)

N² , (3.17)

where C3(b) is a finite constant derived from (3.10) and (3.14). From (3.12), (3.13), (3.16) and (3.17), we conclude that

EtrN

z−XN(τ )⁻²− (z−x)⁻²dµ^∗(x)

21/2

2

N|b|³+C2(b)

N +C3(b)^1/2

N =C4(b) N

finishing the proof of the lemma in the Hermitian case. When we consider the symmetric case (studied already by [13]), an extra term of the form (2N )⁻¹₀^τc_s^N∂_z²M_s^N(z^N_s ) ds appears in (3.8). This term is in turn bounded byCN⁻¹log(1+_|¹_b|)(see [13, p. 9] for details), completing the proof of the lemma. ✷

4. A martingale representation forM^N(z)

In Section 3, we used the martingale representation (3.8) ofM_·^N(z_·^N)to estimate its rate of convergence asN → ∞. Here, we shall follow more closely [9] and [5] to get a similar representation but for deterministic functions (ct, zt), independent of N, in order to study the moderate deviations of the sequence{M^N(z)−M(z)}N. To this end, let(c_·, z_·)be the solution of

∂tzt=Mt(zt), z1=z=a+ib, (4.1)

∂tct=∂zMt(zt)ct, c₁=c. (4.2) By the same arguments as above, we see that |(zt)|is non-increasing on [0,1], with existence and uniqueness of (ct, zt)as a result. Further, in analogy with (3.8) one finds that

cM₁^N(z1)−M1(z1)=c0

M₀^N(z0)−M0(z0)+r₁^N(z_·, c_·)+m^N₁(z_·, c_·) (4.3) withm^N₁(z_·, c_·)the martingale of (3.2) and