Large deviations from the circular law

LARGE DEVIATIONS FROM THE CIRCULAR LAW

GERARD BEN AROUS AND OFER ZEITOUNI

Abstract. We prove a full large deviations principle, in the scale^N2, for the empirical measure of the eigenvalues of an ^{N N} (non self- adjoint) matrix composed of i.i.d. zero mean random variables with variance ^N;1. The (good) rate function which governs this rate func- tion possesses as unique minimizer the circular law, providing an alter- native proof of convergence to the latter. The techniques are related to recent work by Ben Arous and Guionnet, who treat the self-adjoint case. A crucial role is played by precise determinant computations due to Edelman and to Lehmann and Sommers.

1. Introduction

Let^XN =^fXijNgbe an^NN matrix whose entries are independent cen- tered normal random variables of variance ^N;1. We denote the (complex) eigenvalues of ^XN by ^Zii= 1^N, and form the empirical measure

^

N = 1

N N

X

i=1

Z

i :

The law of ^^N is denoted by^QN.

The celebrated circular law (c.f., e.g., Edelman A. (1997),Girko V.L.

(1984),Mehta M.L. (1991)) states that ^^N converges in distribution to the uniform law ^U on the disc ^D=^fZ : ^jZj1^g. Our goal in this paper is to study the corresponding large deviations. We follow a similar study which was carried out in Ben Arous G., Guionnet A. (1997) for the case of self- adjoint matrices. In that case, the large deviations uctuation have speed

N

;2 and a rate function related to Voiculescu's non commutative entropy.

More precisely, if ^YN =^fYijNg is an ^{N N} self-adjoint matrix with inde- pendent (for ^{j > i}) centered normal random variables of variance ^N;1=2 (variance ^N;1 if ⁱ=^j), and eigenvalues ^yi, and if ~^N = ^{N1 PNi=1y}i, then

~

N satises the large deviation principle with speed ^N;2 and rate function

I

R() = 12

Z

R x

2

(^dx)^;R()^; 3 8^;1

4 log 2 where

^R() =

Z

R Z

R

log^jx;yj(^dx)(^dy)^:

The work of the second author was partially supported by a US-Israel BSF grant, and by grant NCR 94-22513.

URL address of the journal: http://www.emath.fr/ps/

Received by the journal January 7, 1998. Revised June 2, 1998. Accepted for publica- tion September 28, 1998.

c

Societe de Mathematiques Appliquees et Industrielles. Typeset by L^ATEX.

Our main result is an analogous theorem in the non self-adjoint case. Let

M S

1(^C) denote the space of symmetric probability measures on^C (symmetric with respect to complex conjugation, that is (^A) =(^A) for ^A denoting the complex conjugate of ^A), equipped with the weak topology. We denote the Levy metric on^M1S(^C) by (), and recall that it is compatible with the weak topology. For^2M1S(^C), and Borel-measurable function^f :^{C2 !R}, dene

hfi:=

Z

C Z

C

f(^xy)(^dx)(^dy)^: Dene next

() =

Z

C Z

C

log^jx;yj(^dx)(^dy) :=^hlog^jx;yji and let

I() = 12

Z

C jxj

2

(^dx)^;()

;K

where

K := 12

Z

C jxj

2

U(^dx)^;(^U)

:

(Lemma 2.1 below shows that actually ^K = 3⁼8). The LDP for ^^N is characterized in the following theorem:

Theorem 1.1.

1. ^I is a good convex rate function on^M1S(^C).

2. ^I() is innite as soon as satises one of the following conditions:

a) ^Rjxj2(^dx) =¹

b) There exists an ^{A C} of positive -mass but null logarithmic capacity, i.e.

(^A) = exp

(

; inf

2M S

1

(C)supp^A

ZZ

log 1

jx;yj

(^dx)(^dy)

)

= 0 3. ^I() achieves its (unique) minimum value in ^M1S(^C) at ^U.

4. ^^N satises the large deviation principle (LDP) with speed ^N;2 and rate function^I, that is

; inf

2A o

I() liminf

N!1

1

N

2log^QN(^^{N 2A})

limsup

N!1

1

N

2log^QN(^^{N 2A})^;inf

2

A I()^: (Compare with Theorem 1.1 in Ben Arous G., Guionnet A. (1997)).

Technically, the main dierence between Theorem 1.1 and Ben Arous G., Guionnet A. (1997) is the lack of ordering in^C, which prevents us from using, in the proof of the lower bound, the approximation procedure presented in Ben Arous G., Guionnet A. (1997). Instead, we present an appropriate smoothing procedure (c.f., Lemma 2.2).

Our results extend naturally to allow for perturbations of the spectral measure ^^N, c.f. the remark at the end of Section 2. However, unlike in the self-adjoint or unitary cases, there does not appear to be a \natural" class of Gaussian ensembles tting these extensions. As in the self-adjoint case, the non-Gaussian situation remains largely open.

After this work was completed, we learnt of recent work of Hiai F., Petz D.

(1998), where similar questions for matrices with Gaussian complex entries are treated.

2. Proof of Theorem 1.1 We prove the theorem in a sequence of Lemmas.

Lemma 2.1. ^K = 3⁼8 and^I is a good convex rate function.

Proof. Note rst that, by a direct computation,

2(^U) =

Z

C Z

C

log^jx;yjdxdy=

Z

jxj>jy j

log^jx;yjdxdy

= 4

Z

1

0 r dr

Z

r

=0 d

Z

2

=0

dlog^{jr; ej j:} (2.1) Because log^jxjis harmonic, for ^<r,

Z

2

=0

dlog^{jr; ej j}= 2log^r hence, substituting back in (2.1),

2(^U) = 8²

Z

1

0

rlog^r

Z

r

0

d dr=^;4^2: (2.2) The conclusion ^K = 3⁼8 follows from (2.2) and the fact that^{RC jxj2U}(^dx) = 1⁼2.

The proof that the level sets of ^I() are compact follows the proof of 3) in property 2.1 of Ben Arous G., Guionnet A. (1997). Indeed, the closeness of the level sets and the boundedness below of^I() follow by truncating the integrands in the denition of () and using monotone convergence. The compactness of the level set then follows by noting that, for any^r>r0 with

r

0 large enough (such that ^;log() 2^r2;log(2^r2) for any 2^r2), and with ^Br denoting the centered disc of radius^r in the complex plane,

((^Bcr))²

= (^{jXj2rjYj2 r})

= (^jXj2+^jYj2;log(^jXj2+^jYj2)2^r2;log(2^r2))

2^r2;log(21 ^r2)

ZZ

(^jxj2+^jyj2;log(^jxj2+^jyj2))(^dx)(^dy)

2^r2;log(21 ^r2)

ZZ

(^jxj2+^jyj2;log(^jx;yj2)+log( ^jx;yj2

jxj

2+^jyj2)

(^dx)(^dy)

2^I() + 1

r 2

;log(2^r2)⁼2^:

This immediately implies the compactness of the level set, for

f:^I()^{Lg \ndr0ef}:(^Bnc)

s 2^L+ 1

n 2

;log(2ⁿ²)⁼2^g:

That ^I() is convex is proved exactly as the proof of 4) in property 2.1 of Ben Arous G., Guionnet A. (1997). Since we do not need this property,

we do not reproduce the proof. Thus one needs only show that^I 0. This follows from the lower bound below (Lemma 2.5), since ^K = 3⁼8 = ^K1 where ^K1 is dened in (2.7).

Lemma 2.2. Assume that ^{2 M1S}(^C) does not possess atoms, and that

I()^<1. Let ^"=^", where ^" is the standard centered Gaussian law on ^C2 of ^"I covariance ( denotes the convolution). Then^";!"!0 and

I(^")^;!

"!0

I()^: (2.3)

Proof. The rst assertion is obvious, for if (^{Z n"}) denote independent ran- dom variables distributed according to^", then^Z"=^Z+^n"is distributed according to^".

Turning to the proof of (2.3), note that by the lower semicontinuity of

I() and the convergence of the second moment of ^n", it suces to prove that

limsup

"!0

;(^")^;()^: (2.4) Toward this end, note that

;(^") =

Z

C Z

C

log 1

jx;yj

"(^dx)^"(^dy)

= ^Elog 1

jZ;Z

0+^n";n0"j

= ^Elog 1

jZ;Z 0

j

+^Elog 1

1 + ^n";nZ;Z00"

= ^;() +^Elog 1

1 + ^{Z;Zn"~ 0} (2.5) where (^{Z Z0}) are independent copies of^Z (^n"n0") are independent copies of

n

", and ~^n"=^p2^n". (Note that we have used here the fact that possesses no atoms). Thus, the lemma follows from (2.5) as soon as we show that

limsup

"!0 E

0

@

1

f1+

~ n"

Z;Z 0

1glog 1

1 + ^{Z;Zn"~ 0}

1

A

0^: (2.6)

Dene now, for ^a2C,

J

a=^E

0

@

1

f1+

~ n

"

a p

"

1glog 1

1 + ^a~n"p"

1

A

:

Fix ^>0 (eventually, we take ^!1). Then, for some universal constant

C

1 independent of^"a, and all large enough,

J

a = 12

Z

1

fj1+y =aj 1g e

;jy 2

j=2log 1

j1 +^y=ajdy

= ^a2 1 2

Z

1

fj1+z j 1g e

;a 2

jz 2

j=2log 1

j1 +^zjdz

1

f>jajg jaj

2 Z

jz +1j 1

log 1

jz+ 1^jdz +^{1f jajg jaj2}

Z

jz +1j 1

jz j jaj

;3=4

log 1

jz+ 1^jdz +^{1f jajg jaj2e;pjaj=4}

Z

jz +1j 1

jz j>jaj

;3=4

log 1

jz+ 1^jdz

C

1

1

f>jajg

2+^{1f jajg}

1

jaj

1=4 +^jaj2e;pjaj=4

where we have used that^jlog^j1+^zjj2^jzjfor^jzjsmall enough in bounding the second term. Hence,

limsup

"!0 E

0

@

1

f1+

~ n

"

Z;Z 0

1glog 1

1 +^{Z;Zn"~ 0}

1

A

= limsup

"!0 E

J

Z;Z 0

p

"

C

1limsup

"!0

2

(^{fZ Z0}: ^jZ;Z0j<p"g)+^;1=4+sup

>

2

e

; p

=4

= ^C1

;1=4+ sup

>

2

e

; p

=4

where we used in the last equality the fact that possesses no atoms and therefore (^{fZ Z0}: ^Z =^Z0g) = 0. Taking now ^{! 1} completes the proof of the lemma.

The key to the asymptotics of ^QN lies in an exact Jacobian computation.

For ^2M1S(^C), let ^k() =^Nfz : Im(^z) = 0^g. Clearly, 0^k(^^N)^N is an integer. Dene ^{`</sup>() = <sup>N;k ()2</sup> , then 0 <sup>`}(^^N) ^bN2c is again an integer.

For any sequence = ^{fjgkj=1 X} = ^{fXjg`j=1 Y</sup> = <sup>fYjg`j=1}, dene

XY(^dz) in ^M1S(^C) by

XY

= 1

N 0

@ k

X

j=1

j+^X`

j=1

X

j +iY

j+^Xj;iYj

1

A

:

Fix

C

N = 2^{;N(N+1)=4QN NN(N;1)=4}

i=1;^{;2i K1} := lim

N!1

1

N

2log^CN = 3⁼8 (2.7) where the equality ^K1 = 3⁼8 is obtained by directly evaluating the limit.

Lemma 2.3. For any measurable^{A M1S}(^C),

Q

N(^^{N 2A}) =

C

N b

N

2 c

X

`=0

(^pN)^N

Z

Z

k

Y

j=1 d

j

`

Y

j=1 dX

j

`

Y

j=1 dY

j 1

f XY

2A`(

XY

)=`g

2^2`exp

; N

2

2

Z

C jxj

2

XY(^dx)

`

Y

j=1

p

NY

jerfc(^Yjp2^N)^e2Yj2N

exp

N 2

2

ZZ

x6=y

x6=y

log^jx;yjXY(^dx)^XY(^dy)

!

:

Here,

erfc(^x) = 1^p

Z

1

x e

; 2

d :

Proof. see Edelman A. (1997) or Lehmann N, Sommers H.J. (1991).

We next turn to the proof of a preliminary upper bound. Dene ^f(^xy) =

jxj 2

+jy j 2

2

;log^jx;yj.

Lemma 2.4. Let^2M1S(^C) be given, and denote by^B() the (L evy) ball in ^M1S(^C) of radius centered at . Then,

lim

&0

limsup

N!1 N

;2log^QNB()^;

1

2 ^hfi;K1

:

Proof. The proof mimics the argument in Section 3.1 of Ben Arous G., Guionnet A. (1997). Note that

ZZ

orx=y^x=y ^^N(^dx)^^N(^dy) = ^N + 2^`(^^N)

N 2

2

N :

For ^{M 2 R+}, dene ^fM(^xy) = min(^Mf(^xy)). Using the bound

xerfc(^p2^x)^{e2x2 C1} for some universal constant ^C1, one concludes from Lemma 2.3 that

Q N

B()^CN(^pN)^NBNe2MN

b

N

2 c

X

`=0 Z

Z

k

Y

j=1 d

j

`

Y

j=1 dX

j

`

Y

j=1 dY

jexp

0

@

;

14

k

X

j=1

2

j

;

14

`

X

j=1

(^Xj2+^Yj2)

1

A

1

f XY

2B( )\`(

XY

)=`gexp

; N

2

2

1^;2

N

D

f

M

XY

XY

E

e 2MN

where log^BN =^O(^N). Hence, with log^BN =^o(^N2),

Q N

B()

C

N B

(1)

N N

2

X

`=0 Z

Z

k

Y

j=1 d

j

`

Y

j=1 dX

j

`

Y

j=1 dY

jexp

0

@

;

14

k

X

j=1

2

j

;

14

`

X

j=1

(^Xj2+^Yj2)

1

A

1

f XY

2B( )\`(

XY

)=`gexp

; N

2

2

1^; 2

N

inf

2B( ) hf

M i

implying that limsup

N!1

1

N

2 log^QNB() lim

N!1

1

N

2log^{CN ;}1

2 inf^{2B( )hfMi:} Since ^7!hfMi is continuous in the weak topology, we obtain that

lim

&0

limsup

N!1

1

N

2log^QNB()^K1;1

2 ^hfMi and the lemma follows by monotone convergence, taking ^{M !1}.

The complementary lower bound is given by:

Lemma 2.5. Let^2M1S(^C) be given, and denote by^B() the (L evy) ball of radius centered at . Then

lim

&0

liminf

N!1 N

;2log^QNB()^;

1

2 ^hfi;K1

:

Proof. By considering ^" =^" and using Lemma 2.2, it is clear that we may consider only ^2M1S which possesses no atoms and whose density ^g with respect to Lebesgue's measure is continuous and positive everywhere.

Let

X

; = max^nx: ^x1)(^;11)1^;8

o

X

+ = min^nx: (^;1x](^;11)1^;8

o

Y = min^ny: (^;11) 0^y]ⁱ1^;8

o

:

Let^D= ^X;X+] 0^Y],^DF = ^X;X+] ^;YY], and note that(^D)

1

2

;

2. Let ^C be such that, on ^D, the density ^g of with respect to Lebesgue's measure satises ^{C g} 1^=C.

For simplicity, we assume in the sequel that ^N is even, the case of ^N odd requiring only obvious modications. Fix ^{0 >} 0. Modifying slightly

X

; X

+ and ^Y if necessary, partition ^D into disjoint squares ^fB`gL`=1 of length ^pC=N0 and centers ^{Z`</sup>. Note that <sup>C2=N0</sup> (<sup>B`}) 1^=N0 and hence ^N0=2 ^{L N0=}2^C2. Fix ^{`</sup> = <sup>bN</sup>(<sup>B`})^c, then ^{dC2=0e `}

b1^=0c, and 1⁼2^;0;1^=NPL`=1` 1⁼2.

Next, for each ^{` 2 f</sup>1<sup>Lg</sup>, dene a sequence <sup>fZ`jgj=1`</sup> , <sup>Z`j 2 B`}, satisfying the following properties:

a) ^d(^Z`j@B`) ^14qNC0 b) ^jZ`j;Z`ij 4 ¹

p

NC i6=^j

(such a sequence always exists due to our construction of ^`).

Finally, for ^{k PL`=1`}, dene ^ik = max^{n`0</sup>: <sup>P``=10 ` <ko} and ~^Zk =

Z

i

k +1k ;

P

i

k

`=1

`

" Let ^K := ^fPL`=1` + 1^{:::N =}2^g, and let ^fZ~^{kgk 2K} be

K = ^{N2 ;PL`=1`}(+⁰)^N points equidistributed along the curve

D

B=

z2D

c : Im(^z)^> 1

2 ^d(^zD) = 1

: (2.8)

Let = ^{N1 PN =2k =1}(^Z~k+^Z~k), then () 2(⁰+). We remark that with this construction, for some ^C(⁰),

j^Z~^j;Z~^{ij C}(⁰)

N

i6=^j: (2.9)

Fix now ^{" >} 0 small enough. Returning to the proof of the lower bound, we have, by Lemma 2.3 (recall that ^N is assumed even, and reduce ⁰ if needed).

Q N

B()^CN

Z

Z

D

"

I N =2

Y

j=1 dx

j dy

jexp

0

@

;N N =2

X

j=1

(^x2j+^y2j)

1

A

N =2

Y

j=1 p

Ny

jerfc(^yjp2^N)^e;2y2jNexp

0

@ X

i6=j

log^{jzi;zjjjzi;zjj}

1

A

where ^DI"=ⁿ(^xjyj) : ^j(^xjyj)^;Z~^{jj "}

N

oand ^zj = (^xjyj).

Due to property a) and (2.8) we have that in ^DI", for^{"< 12} ,

p

Ny

jerfc(^yjp2^N)^{e;2yj2N C1}

N

for some universal constant ^C1. Furthermore, in ^DI"

N N =2

X

j=1 jz

j j

2

N + 1^p

N

N =2

X

j=1

j^Z~^jj2+ 1^:

Therefore, denoting by^BN a quantity (which may change from line to line) which satises ^N12 log ^{BN ;!}0, one nds

Q N

B()^CNBNexp(^;(^N+ 2))^{XN =2}

j=1 j^Z~^jj2

Z

Z

D

"

I

exp

0

@ X

i6=j

log^{jzi;zjjjzi;zjj}

1

A N =2

Y

j=1 e

;(x 2

j +y

2

j )=2

dx

j dy

j

: (2.10) Next, note that due to (2.9), property a) and the denition of ~^Zj, we have

jz

i

;z

j j

1^; 2^"

C(⁰)

j^Z~^i;Z~^{jj i6}=^j and

jz

i

;z

j j

1^; 2^"

C(⁰)

j^Z~^i;Z~j j

Therefore,

Q

N

B()^BNCNexp

0

@

;

N+ 1^p

N

N =2

X

j=1 j^Z~^jj2

1

A

exp

0

@ X

i6=j

log^jZ~^i;Z~^jjjZ~^i;Z~j j

1

A

exp

;N 2log

1^; 2^"

C(⁰)

ZZ

D

"

I N =2

Y

j=1 e

;(x 2

j +y

2

j )=2

dx

j dy

j

B (")

N C

Nexp

0

@

;

N + 1^p

N

N =2

X

j=1 j^Z~^jj2

1

A

exp

0

@ X

i6=j

log^jZ~i

;^Z~j jj^Z~i

;^Z~j j

1

A (2.11)

where

lim

"!0

lim

N!1

1

N

2log^BN(")= 0^: Next, for each ^`, let

G

"

` =

8

>

>

<

>

>

:

`

0: max

x2B

`

y 2B

` 0

jx;yj>(1 +^") min

~

Z

k 2B

`

~

Z

k 0

2B

` 0

j^Z~k

;^Z~k 0

j 9

>

>

=

>

>

Note that

jG

"

` j

2

"

2

:

Then, using the fact that for ^N large enough and for ^{x 2B`y 2B`0} with

` 0

2G

"

`, one has that log^jx;yjjx;yj0, 12

ZZ

D F

(^dx)(^dy)log^jx;yj=

ZZ

D

(^dx)(^dy)log^jx;yjjx;yj

= ^X

`

` 0

62G

"

` ZZ

x2B

`

y 2B

` 0

(^dx)(^dy)log^jx;yjjx;yj

+ ^X

`

` 0

2G

"

` ZZ

x2B

`

y 2B

` 0

(^dx)(^dy)log^jx;yjjx;yj

X

fij:

~

Z

i 2B

`

~

Z

j 62

` 0

2G

"

` B

` 0

g

(^{B`</sup>)(<sup>B`0})

`

` 0

log^jZ~^i;Z~^jjjZ~^i;Z~j j

+2log(1+)^: