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Introduction to the Random Matrix Theory:Gaussian Unitary Ensemble and Beyond

Yan V. Fyodorov

Department of Mathematical Sciences, Brunel University,Uxbridge, UB8 3PH, United Kingdom.

Abstract

These lectures provide an informal introduction into the notions and tools used to analyzestatistical properties of eigenvalues of large random Hermitian matrices. After developing thegeneral machinery of orthogonal polynomial method, we study in most detail Gaussian UnitaryEnsemble (GUE) as a paradigmatic example. In particular, we discuss Plancherel-Rotachasymptotics of Hermite polynomials in various regimes and employ it in spectral analysisof the GUE. In the last part of the course we discuss general relations between orthogonalpolynomials and characteristic polynomials of random matrices which is an active area ofcurrent research.

1 Preface

Gaussian Ensembles of random Hermitian or real symmetric matrices always played a prominentrole in the development and applications of Random Matrix Theory. Gaussian Ensembles areuniquely singled out by the fact that they belong both to the family of invariant ensembles, andto the family of ensembles with independent, identically distributed (i.i.d) entries. In general,mathematical methods used to treat those two families are very different.

In fact, all random matrix techniques and ideas can be most clearly and consistently introducedusing Gaussian case as a paradigmatic example. In the present set of lectures we mainly concentrateon consequences of the invariance of the corresponding probability density function, leaving asidemethods of exploiting statistical independence of matrix entries. Under these circumstances themethod of orthogonal polynomials is the most adequate one, and for the Gaussian case the relevantpolynomials are Hermite polynomials. Being mostly interested in the limit of large matrix sizeswe will spend a considerable amount of time investigating various asymptotic regimes of Hermitepolynomials, since the latter are main building blocks of various correlation functions of interest.In the last part of our lecture course we will discuss why statistics of characteristic polynomialsof random Hermitian matrices turns out to be interesting and informative to investigate, and willmake a contact with recent results in the domain.

The presentation is quite informal in the sense that I will not try to prove various statementsin full rigor or generality. I rather attempt outlining the main concepts, ideas and techniquespreferring a good illuminating example to a general proof. A much more rigorous and detailedexposition can be found in the cited literature. I will also frequently employ the symbol . In thepresent set of lectures it always means that the expression following contains a multiplicativeconstant which is of secondary importance for our goals and can be restored when necessary.

2 Introduction

In these lectures we use the symbol T to denote matrix or vector transposition and the asterisk

to denote Hermitian conjugation. In the present section the bar z denotes complex conjugation.

1

Let us start with a square complex matrix Z of dimensions N N , with complex entrieszij = xij + iyij , 1 i, j N . Every such matrix can be conveniently looked at as a point in a2N2-dimensional Euclidean space with real Cartesian coordinates xij , yij , and the length elementin this space is defined in a standard way as:

(ds)2 = Tr(dZdZ

)=ij

dzijdzij =ij

[(dx)2ij + (dy)

2ij

]. (1)

As is well-known (see e.g.[1]) any surface embedded in an Euclidean space inherits a naturalRiemannian metric from the underlying Euclidean structure. Namely, let the coordinates in andimensional Euclidean space be (x1, . . . , xn), and let a kdimensional surface embedded inthis space be parameterized in terms of coordinates (q1, . . . , qk), k n as xi = xi(q1, . . . , qk), i =1, . . . n. Then the Riemannian metric gml = glm on the surface is defined from the Euclidean lengthelement according to

(ds)2 =ni=1

(dxi)2 =

ni=1

(k

m=1

xiqm

dqm

)2=

km,l=1

gmndqmdql. (2)

Moreover, such a Riemannian metric induces the corresponding integration measure on the surface,with the volume element given by

d =|g|dq1 . . . dqk, g = det (gml)kl,m=1. (3)

For k = n these are just the familiar formulae for the lengths and volume associated withchange of coordinates in an Euclidean space. For example, for n = 2 we can pass from Cartesiancoordinates < x, y 0, 0 < 2pi by x = r cos , y = r sin , sothat dx = dr cos r sin d, dy = dr sin + r cos d, and the Riemannian metric is defined by(ds)2 = (dx)2 + (dy)2 = (dr)2 + r2(d)2. We find that g11 = 1, g12 = g21 = 0, g22 = r

2, and thevolume element of the integration measure in the new coordinates is d = rdrd; as it should be.As the simplest example of a surface with k < n = 2 embedded in such a two-dimensional spacewe consider a circle r = R = const. We immediately see that the length element (ds)2 restrictedto this surface is (ds)2 = R2(d)2, so that g11 = R

2, and the integration measure induced on thesurface is correspondingly d = Rd. The surface integration then gives the total volume ofthe embedded surface (i.e. circle length 2piR).

z

y

x

Figure 1: The spherical coordinates for a two dimensional sphere in the three-dimensional Euclideanspace.

2

Similarly, we can consider a two-dimensional (k = 2) sphere R2 = x2 + y2 + z2 embedded ina three-dimensional Euclidean space (n = 3) with coordinates x, y, z and length element (ds)2 =(dx)2 + (dy)2 + (dz)2. A natural parameterization of the points on the sphere is possible in termsof the spherical coordinates , (see Fig. 1)

x = R sin cos, y = R sin sin, z = R cos ; 0 pi, 0 < 2pi,which results in (ds)2 = R2(d)2 + R2 sin2 (d)2. Hence the matrix elements of the metric areg11 = R

2, g12 = g21 = 0, g22 = R2 sin2 , and the corresponding volume element on the sphere is

the familiar elementary area d = R2 sin dd.As a less trivial example to be used later on consider a 2dimensional manifold formed by 22

unitary matrices U embedded in the 8 dimensional Euclidean space of Gl(2;C) matrices. Everysuch matrix can be represented as the product of a matrix Uc from the coset space U(2)/U(1)U(1)parameterized by k = 2 real coordinates 0 < 2pi, 0 pi/2, and a diagonal unitary matrixUd, that is U = UdUc, where

Uc =

(cos sin ei

sin ei cos

), Ud =

(ei1 00 ei2

). (4)

Then the differential dU of the matrix U = UdUc has the following form:

dU =

( [d sin + i cos d1]ei1 ei(1+)[d cos + i(d1 + d) sin ]ei(+2)[d cos + i(d+ d2) sin ] [d sin + id2 cos ]ei2

), (5)

which yields the length element and the induced Riemannian metric:

(ds)2 = Tr(dUdU

)= 2(d)2 + (d1)

2 + (d2)2 + 2 sin2 (d)2 + 2 sin2 (d d1 + d d2). (6)

We see that the nonzero entries of the Riemannian metric tensor gmn in this case are g11 =2, g22 = g33 = 1, g44 = 2 sin

2 , g24 = g42 = g34 = g43 = sin2 , so that the determinant det [gmn] =

4 sin2 cos2 . Finally, the induced integration measure on the group U(2) is given by

d(U) = 2 sin cos d d d1 d2. (7)

It is immediately clear that the above expression is invariant, by construction, with respect tomultiplications U V U , for any fixed unitary matrix V from the same group. Therefore, Eq.(7)is just the Haar measure on the group.

We will make use of these ideas several times in our lectures. Let us now concentrate on theN2dimensional subspace of Hermitian matrices in the 2N2 dimensional space of all complexmatrices of a given size N . The Hermiticity condition H = H HT amounts to imposing thefollowing restrictions on the coordinates: xij = xji, yij = yji. Such a restriction from the spaceof general complex matrices results in the length and volume element on the subspace of Hermitianmatrices:

(ds)2 = Tr(dHdH

)=i

(dxii)2 + 2

i

where U U(N) is any given unitary N N matrix: U = U1. Therefore the correspondingintegration measure d(H) is also invariant with respect to all such rotations of the basis.

The above-given measure d(H) written in the coordinates xii, xi

The last factor dM(U) stands for the part of the measure depending only on the Uvariables. Amore detailed consideration shows that, in fact, dM(U) d(U), which means that it is given(up to a constant factor) by the invariant Haar measure on the unitary group U(N). This fact ishowever of secondary importance for the goals of the present lecture.

Having an integration measure at our disposal, we can introduce a probability density function(p.d.f.) P(H) on the space of Hermitian matrices, so that P(H)d(H) is the probability thata matrix H belongs to the volume element d(H). Then it seems natural to require for such a

probability to be invariant with respect to all the above automorphisms, i.e. P(H) = P(UHU

).

It is easy to understand that this postulate of invariance results in P being a function of N firsttraces TrHn, n = 1, . . . , N (the knowledge of firstN traces fixes the coefficients of the characteristicpolynomial of H uniquely, and hence the eigenvalues. Therefore traces of higher order can alwaysbe expressed in terms of the lower ones). Of particular interest is the case

P(H) = C expTrQ(H), Q(x) = a2jx2j + . . .+ a0, (15)where 2j N , the parameters a2l and C are real constants, and a2j > 0. Observe that if we take

Q(x) = ax2 + bx+ c, (16)

then eTr Q(H) takes the form of the product

ea[

ix2ii+2

i

where we kept only terms linear in . With the same precision we expand the factors in Eq.(18):

f1(x1) = f1(x1)

[1 + 2x12

1

f1

df1dx11

], f2(x

22) = f2(x22)

[1 2x12 1

f2

df2dx22

]

f(1)12 (x

12) = f

(1)12 (x12)

[1 + (x22 x11) 1

f(1)12

df(1)12

dx12

], f

(2)12 (y

21) = f

(2)12 (y12).

The requirements of statistical independence and invariance amount to the product of the left-handsides of the above expressions to be equal to the product of the right-hand sides, for any . Thisis possible only if:

2x12

[d ln f1dx11

d ln f2dx22

]+ (x22 x11)d ln f

(1)12

dx12= 0, (20)

which can be further rewritten as

1

(x22 x11)[d ln f1dx11

d ln f2dx22

]= const =

1

2x12

d ln f(1)12

dx12, (21)

where we used that the two sides in the equation above depend on essentially different sets ofvariables. Denoting const1 = 2a, we see immediately that

f(1)12 (x12) e2ax

212 ,

and further notice that

d ln f1dx11

+ 2ax11 = const2 =d ln f2dx22

+ 2ax22

by the same reasoning. Denoting const2 = b, we find:

f1(x11) eax211bx11 , f2(x22) eax

222bx22 , (22)

and thus we are able to reproduce the first two factors in Eq.(17). To reproduce the remaining

factors we consider the conjugation by the unitary matrix Ud =

(1 i 00 1 + i

), which cor-

responds to the choice = 0, 1 = 2 = = in Eq.(4), and again we keep only terms linearin the small parameter 1. Within such a precision the transformation leaves the diagonalentries x11, x22 unchanged, whereas the real and imaginary parts of the off-diagonal entries aretransformed as

x12 = x12 2y12, y12 = y12 + 2x12.In this case the invariance of the p.d.f. P(H) together with the statistical independence of theentries amount, after straightforward manipulations, to the condition

1

x12

d ln f(1)12

dx12=

1

y12

d ln f(2)12

dy12

which together with the previously found f(1)12 (x12) yields

f(2)12 (y12) e2ay

212 ,

6

completing the proof of Eq.(17).The Gaussian form of the probability density function, Eq.(17), can also be found as a result of

rather different lines of thought. For example, one may invoke an information theory approach a laShanon-Khinchin and define the amount of information I[P(H)] associated with any probabilitydensity function P(H) by

I[P(H)] = d(H)P(H) lnP(H) (23)

This is a natural extension of the corresponding definition I[p1, . . . , pm] = m

l=1 pm ln pm fordiscrete events 1, ...,m.

Now one can argue that in order to have matrices H as random as possible one has to findthe p.d.f. minimizing the information associated with it for a certain class of P(H) satisfyingsome conditions. The conditions usually have a form of constraints ensuring that the probabilitydensity function has desirable properties. Let us, for example, impose the only requirement thatthe ensemble average for the two lowest traces TrH,TrH2 must be equal to certain prescribed

values, say E[TrH

]= b and E

[TrH2

]= a > 0, where the E [. . .] stand for the expectation value

with respect to the p.d.f. P(H). Incorporating these constraints into the minimization procedurein a form of Lagrange multipliers 1, 2, we seek to minimize the functional

I[P(H)] = d(H)P(H)

{lnP(H) 1TrH 2TrH2

}. (24)

The variation of such a functional with respect to P(H) results in

I[P(H)] = d(H) P(H)

{1 + lnP(H) 1TrH 2TrH2

}= 0 (25)

possible only ifP(H) exp{1TrH + 2TrH2}

again giving the Gaussian form of the p.d.f. The values of the Lagrange multipliers are thenuniquely fixed by constants a, b, and the normalization condition on the probability density func-tion. For more detailed discussion, and for further reference see [2], p.68.

Finally, let us discuss yet another construction allowing one to arrive at the Gaussian Ensemblesexploiting the idea of Brownian motion. To start with, consider a system whose state at time tis described by one real variable x, evolving in time according to the simplest linear differentialequation ddtx = x describing a simple exponential relaxation x(t) = x0et towards the stableequilibrium x = 0. Suppose now that the system is subject to a random additive Gaussian whitenoise (t) function of intensity D 1, so that the corresponding equation acquires the form

d

dtx = x+ (t), E [(t1)(t1)] = D(t1 t2), (26)

1The following informal but instructive definition of the white noise process may be helpful for those not veryfamiliar with theory of stochastic processes. For any positive t > 0 and integer k 1 define the random function

k(t) =

2/pik

n=0an cosnt, where real coefficients an are all independent, Gaussian distributed with zero mean

E[an] = 0 and variances E[a20] = D/2 and E[a2n] = D for 1 n k. Then one can, in a certain sense, consider

white noise as the limit of k(t) for k . In particular, the Dirac (t t) is approximated by the limiting value

ofsin [(k+1/2)(tt)]

2pi sin (tt)/2

7

where E[. . .] stands for the expectation value with respect to the random noise. The main char-acteristic property of a Gaussian white noise process is the following identity:

E

[exp

{ ba

(t)v(t)dt

}]= exp

{D

2

ba

v2(t)dt

}(27)

valid for any (smooth enough) test function v(t). This is just a direct generalization of the standardGaussian integral identity:

dq2pia

e12a q

2+qb = eab2

2 . (28)

valid for Re a > 0, and any (also complex) parameter b.For any given realization of the Gaussian random process (t) the solution of the stochastic

differential equation Eq.(26) is obviously given by

x(t) = et[x0 +

t0

e()d

]. (29)

This is a random function, and our main goal is to find the probability density function P(t, x) forthe variable x(t) to take value x at any given moment in time t, if we know surely that x(0) = x0.This p.d.f. can be easily found from the characteristic function

F(t, q) = E[eiqx(t)

]= exp

{iqx0et Dq

2

4(1 e2t)

}(30)

obtained by using Eqs. (27) and (29). The p.d.f. is immediately recovered by employing theinverse Fourier transform:

P(t, x) =

dq

2pieiqxE

[eiqx(t)

]=

1piD(1 e2t) exp

{ (x x0e

t)2

D(1 e2t)

}. (31)

The formula Eq.(31) is called the Ornstein-Uhlenbeck (OU) probability density function, andthe function x(t) satisfying the equation Eq.(26) is known as the O-U process. In fact, such aprocess describes an interplay between the random kicks forcing the system to perform a kind ofBrownian motion and the relaxation towards x = 0. It is easy to see that when time grows the OUp.d.f. forgets about the initial state and tends to a stationary (i.e. time-independent) universalGaussian distribution:

P(t, x) = 1piD

exp

{x

2

D

}. (32)

Coming back to our main topic, let us consider N2 independent OU processes: N of themdenoted as

d

dtxi = xi + i(t), 1 i N (33)

and the rest N(N 1) given byd

dtxij = xij + (1)ij (t),

d

dtyij = yij + (2)ij (t), (34)

8

where the indices satisfy 1 i < j N . Stochastic processes (t) in the above equationsare taken to be all mutually independent Gaussian white noises characterized by the correlationfunctions:

E [i1(t1)i2(t2)] = 2Di1,i2(t1 t2), E[1ij (t1)

2kl (t2)

]= D1,2i,kj,l(t1 t2). (35)

As initial values xi(0), xij(0), yij(0) for each OU process we choose diagonal and off-diagonal entries

H(0)ii , i = 1, . . . , N and ReH

(0)i

Our main goal is to extract the information about these eigenvalue correlations in the limitof large size N . From this point of view it is pertinent to discuss a few quantitative measuresfrequently used to characterize correlations in sequences of real numbers.

3 Characterization of Spectral Sequences

Let < 1, 2, . . . , N

which can be interpreted asBB

R2(1, 2)d1d2 = expectation of the number of pairs of points in B (44)

where if, say, 1 and 2 are in B, then the pair {1, 2} and {2, 1} are both counted.To relate the two-point correlation function to the variance of NB we notice that in view of

Eq.(42) the one-point correlation function R1() coincides with the mean density N () of thepoints {i} around the point on the real axis. Similarly, write the mean square N2B

N2B =

B()B(

)N ()N () dd (45)

and notice that

N ()N () =ij

( i)( j) = ( )i

( i) +i6=j

( i)( j)

= ( )R1() +R2(, ). (46)In this way we arrive at the relation:

N2B = NB +

BB

R2(, ) dd. (47)

In fact, it is natural to introduce the so-called number variance statistics 2(B) = N2B [NB

]2describing the variance of the number of points of the sequence inside the set B. Obviously,

2(B) = NB +

BB

[R2(, )R1()R1()] dd NB BB

Y2(, ) dd (48)

where we introduced the so-called cluster function Y2(, ) = R1()R1()R2(, ) frequently

used in applications.Finally, in principle the knowledge of all npoint correlation functions provides one with the

possibility of calculating an important characteristic of the spectrum known as the hole probabil-ity A(L). This quantity is defined as the probability for a random matrix to have no eigenvaluesin the interval (L/2, L/2) 2. Define L() to be the characteristic function of this interval.Obviously,

A(L) =

. . .

P(1, . . . , N )

Nk=1

(1 L(k)) d1 . . . dN (49)

=

Nj=0

(1)j. . .

P(1, . . . , N )hj {L(1), . . . , L(N )} d1 . . . dN , (50)

where hj{x1, . . . , xN} is the j th symmetric function:

h0{x1, . . . , xN} = 1, h1{x1, . . . , xN} =Ni=1

xi ,

2Sometimes one uses instead the interval [L,L] to define A(L), see e.g. [3].

11

h2{x1, . . . , xN} =Ni

number of levels in the domain B and for its mean square:

NB = N

B

p() d, N2B = NB(N 1)/N +[NB

]2(58)

and for the hole probability

A(L) =

Nj=0

(1)jj!

N !

(N j)!

[ L/2L/2

p() d

]j=

[1

L/2L/2

p() d

]N. (59)

Finally, let us specify B to be the interval [L/2, L/2] around the origin, and being interestedmainly in large N 1 consider the length L comparable with the mean spacing between neigh-bouring points in the sequence {i} close to the origin, given by

[(0)

]1= 1/[Np(0)]. In

other words s = L/ = LNp(0) stays finite when N . On the other hand, for large enoughN the function p() can be considered practically constant through the interval of the lengthL = O(1/N), and therefore the mean number of points of the sequence {i} inside the interval[L/2, L/2] will be asymptotically given by N(s) N Lp(0) = s. Similarly, using Eq.(58) one caneasily calculate the number variance 2(s) = N2[L2 ,

L2 ][N[L2 ,

L2 ]

]2= (N1) L/2L/2 p() d s.

In the same approximation the hole probability, Eq.(59), tends asymptotically to A(s) es. Lateron we shall compare these results with the corresponding behaviour emerging from the randommatrix calculations.

4 The method of orthogonal polynomials

In the heart of the method developed mainly by Dyson, Mehta and Gaudin lies an integrating-outLemma [2]. In presenting this material I follow very closely [3], pp.103-105.

Let Jn = Jn(x) = (Jij)1i,jn be an n n matrix whose entries depend on a real vectorx = (x1, x2, . . . , xn) and have the form Jij = f(xi, xj), where f is some (in general, complex-valued) function satisfying for some measure d(x) the reproducing kernel property:

f(x, y)f(y, z) d(y) = f(x, z). (60)

Then det Jn(x) d(xn) = [q (n 1)]det Jn1 (61)

where q =f(x, x) d(x), and the matrix Jn1 = (Jij)1i,jn1 have the same functional

form as Jn with x replaced by (x1, x2, . . . , xn1).

Before giving the idea of the proof for an arbitrary n it is instructive to have a look on thesimplest case n = 2, when

J2 =

(f(x1, x1) f(x1, x2)f(x2, x1) f(x2, x2)

), hence det Jn = f(x1, x1)f(x2, x2) f(x1, x2)f(x2, x1).

Integrating the latter expression over x2, and using the reproducing kernel property, we imme-diately see that the result is indeed just (q 1)f(x, x) = (q 1) detJ1 in full agreement with thestatement of the Lemma.

13

For general n one should follow essentially the same strategy and expand the determinant as asum over n! permutations Pn() = (1, . . . n) of the index set 1, . . . , n as

detJn(x) d(xn) =Pn

(1)Pnf(x1, x1) . . . f(xn, xn) d(xn), (62)

where (1)Pn stands for the sign of permutations. Now, we classify the terms in the sum accordingto the actual value of the index n = k, k = 1, 2, . . . , n. Consider first the case n = n, wheneffectively only the last factor f(xn, xn) in the product is integrated yielding d upon the integration.Summing up over the remaining (n 1)! permutations Pn1() of the index set (1, 2, ..., n 1) wesee that:

Pn1

(1)Pn

f(x1, x1) . . . f(xn, xn) d(xn) = qPn1

(1)Pn1 f(x1, x1) . . . f(xn1, xn1),

which is evidently equal to q detJn1. Now consider (n 1)! terms with n = k < n, when wehave j = n for some j < n. For every such term we have by the reproducing property

f(x1, x1) . . . f(xj , xn) . . . f(xn, xk) d(xn) = f(x1, x1 ) . . . f(xj , xk) . . . f(xn1, xn1).

Therefore

detJn(x) d(xn) = q det Jn1 +

n1k=1

Pn:n=k)

(1)Pnf(x1, x1) . . . f(xj , xk) . . . f(xn1, xn1).

(63)It is evident that the structure and the number of terms is as required, and the remaining task isto show that the summation over all possible (n1)! permutation of the index set for fixed k yieldsalways detJn1, see [3]. Then the whole expression is indeed equal to [q (n 1)] detJn1 asrequired.

Our next step is to apply this Lemma to calculating the npoint correlation functions of theeigenvalues 1, . . . , n starting from the JPDF, Eq.(38).

For this we notice that

Ni

with any choice of the coefficients al, l = 0, . . . , k. Therefore:

Ni

for any indices i 1, j 1. Then we obviously haveKN (x, y)KN (y, z)dy =

N1j=0

N1k=0

e12 (Q(x)+Q(z))pij(x)pik(z)

pij(y)pik(y)e

Q(y)dy

=

N1j=0

e12 (Q(x)+Q(z))pij(x)pij(z) = KN(x, z) (73)

exactly as required by the reproducing property. Moreover, in this case obviously

qN =

KN (x, x)dx =

N1j=0

eQ(x)pij(x)pij(x) dx = N,

and therefore the relation (61) amounts todet (KN(xi, xj))1i,jN dxN = det (KN(xi, xj))1i,jN1 . (74)

Continuing this process one step further we see det (KN (xi, xj))1i, jN dxN1dxN =

det (KN (xi, xj))1i, jN1 dxN1 (75)

= [N (N 2)]det (KN(xi, xj))1i, jN2 (76)

and continuing by induction. . .

det (KN(xi, xj))1i, jN dxk+1 . . . dxN = (N k)! det (KN(xi, xj))1i, jk (77)

for k = 1, 2, . . ., and the result is N ! for k = 0. Remembering the expression of the JPDF, Eq.(68),in terms of the kernel KN(xi, xj) we see that, in fact, the theory developed provided simultane-ously the explicit formulae for all npoint correlation functions of the eigenvalues Rn(1, . . . , n),introduced by us earlier, Eq.(41):

Rn(1, . . . , n) = det (KN (i, j))1i,jn (78)

expressed, in view of the relations Eq.(71) effectively in terms of the orthogonal polynomials pik().In particular, remembering the relation between the mean eigenvalue density and the one-pointfunction derived by us earlier, we have:

N () = KN(, ) =

N1j=1

eQ()pij1()pij1(). (79)

The latter result allows to represent the connected (or cluster) part of the two-point correlationfunction introduced by us in Eq.(48) in the form:

Y2(1, 2) = N (1) N (2)R2(1, 2) = [KN (1, 2)]2 . (80)

16

Finally, combining the relation Eq.(55) between the hole probability A(L) and the n-pointcorrelation functions, and on the other hand the expression of the latter in terms of the kernelKN (,

), see Eq.(78), we arrive at

A(L) =

Nj=0

(1)jj!

L/2L/2

. . .

L/2L/2

det

KN(1, 1) . . . KN (1, j). . .. . .. . .

KN(j , 1) . . . KN (j , j)

d1 . . . dj . (81)

In fact, the last expression can be written in a very compact form by noticing that it is justa Fredholm determinant det (I KN ), where KN is a (finite rank) integral operator with thekernel KN(,

) =N1

i=0 i()i() acting on square-integrable functions on the interval

(L/2, L/2).

5 Properties of Hermite polynomials

5.1 Orthogonality, Recurrent Relations and Integral Representation

Consider the set of polynomials hk(x) defined as3

hk(x) = (1)keN x2

2dk

dxk

(eN

x2

2

)= Nkxk + , (82)

and consider, for k l

eNx2

2 hl(x)hk(x)dx = (1)k

dxhl(x)dk

dxk

(eN

x2

2

)(83)

= (1)k+1

dxhl(x)dk1

dxk1

(eN

x2

2

)= . . . = (1)2k

dx eNx2

2dk

dxkhl(x).

Obviously, for k > l we have dk

dxkhl(x) = 0, whereas for k = l we havedk

dxkhk(x) = k!Nk. In this

way we verified the orthogonality relations and the normalization conditions

eNx2

2 hl(x)hk(x)dx = kl (84)

for normalized polynomials

hk(x) =:1[

k!Nk

2piN

]1/2 hk(x). (85)3The standard reference to the Hermite polynomials uses the definition

Hk(x) = (1)kex

2 dk

dxk

(ex

2)= 2kxk + ,

Such a choice ensures Hk(x) to be orthogonal with respect to the weight ex2 . Our choice is motivated by random

matrix applications, and is related to the standard one as hk(x) = Hk

(N2x

).

17

In the theory of orthogonal polynomials an important role is played by recurrence relations:

hk+1(x) = (1)k+1eN x2

2dk

dxk

(ddxe

N x2

2

)= (1)k+2NeN x22 dkdxk

(xeN

x2

2

)(86)

= (1)k+2NeN x22[(

0k

)x d

k

dxk

(eN

x2

2

)+

(1k

)dk1

dxk1

(eN

x2

2

)]= N [xhk(x) k hk1(x)] ,

where we exploited the Leibniz formula for the kth derivative of a product. After normalizationwe therefore have [

k + 1

N

]1/2hk+1(x) = x hk(x)

[k

N

]1/2hk1(x). (87)

Let us multiply this relation with hk(y), and then replace x by y. In this way we arrive at tworelations: [

k + 1

N

]1/2hk+1(x)hk(y) = x hk(x)hk(y)

[k

N

]1/2hk1(x)hk(y), (88)

[k + 1

N

]1/2hk+1(y)hk(x) = y hk(x)hk(y)

[k

N

]1/2hk1(y)hk(x). (89)

The difference between the upper and the lower line can be written for any k = 1, 2, . . . as

(x y)hk(x)hk(y) = Ak+1 Ak , Ak =[k

N

]1/2{hk1(y)hk(x) hk1(x)hk(y)}.

Summing up these expressions over k:

(x y)n1k=1

hk(x)hk(y) = (A2 + . . .+An) (A1 + . . .+An1) = An A1

and remembering that A1 =

N2pi (x y) = (x y)h0(x)h0(y) we arrive at a very important

relation:n1k=0

hk(x)hk(y) =

n

N

hn1(y)hn(x) hn1(x)hn(y)x y , (90)

or, for the original (not-normalized) polynomials:

n1k=0

1

k!Nkhk(x)hk(y) =

1

(n 1)!Nnhn1(y)hn(x) hn1(x)hn(y)

x y , (91)

which are known as the Christoffel-Darboux formulae. Finally, taking the limit x y in the aboveexpression we see that

n1k=0

1

k!Nkh2k(x) =

1

(n 1)!Nn[hn1(x)h

n(x) hn1(x)hn(x)

]. (92)

Most of the properties and relations discussed above for Hermite polynomials have their ana-logues for general class of orthogonal polynomials. Now we are going to discuss another very useful

18

property which is however shared only by few families of classical orthogonal polynomials: Her-mite, Laguerre, Legendre, Gegenbauer and Jacoby. All these polynomials have one of few integralrepresentations which are frequently exploited when analyzing their properties. For the case ofHermite polynomials we can most easily arrive to the corresponding representation by using thefamiliar Gaussian integral identity, cf. Eq.(28):

eNx2

2 =

N

2pi

dq eN2 q

2+ixqN . (93)

Substituting such an identity to the original definition, Eq.(82), we immediately see that

hk(x) = (iN)kN

2pieN

x2

2

dq qk eN2 q

2+ixqN , (94)

which is the required integral representation, to be mainly used later on when addressing thelarge-N asymptotics of the Hermite polynomials. Meanwhile, let us note that differentiating theabove formula with respect to x one arrives at the useful relation ddxhk(x) = Nxhk(x)hk+1(x) =Nk hk1(x). This can be further used to simplify the formula Eq.(92) bringing it to the form

n2k=0

1

k!Nkh2k(x) =

1

(n 2)!Nn1[h2n(x) hn1(x)hn+1(x)

]. (95)

5.2 Saddle-point method and Plancherel-Rotach asymptotics of Her-

mite polynomials

In our definition, the Hermite polynomials hk(x) depend on two parameters: explicitly on theorder index k = 0, 1, . . . and implicitly on the parameter N due to the fact that the weight function

eNx2

2 contains this parameter. Invoking the random matrix background for the use of orthogonalpolynomials, we associate the parameter N with the size of the underlying random matrix. Fromthis point of view, the limit N 1 arises naturally as we are interested in investigating thespectral characteristics of large matrices. A more detailed consideration reveals that, from therandom matrix point of view, the most interesting task is to extract the asymptotic behaviour ofthe Hermite polynomials with index k large and comparable with N , i.e. k = N + n, where theparameter n is considered to be of the order of unity. Such behaviour is known as Plancherel-Rotachasymptotics.

To understand this fact it is enough to invoke the relation (79) expressing the mean eigenvaluedensity in terms of the set of orthogonal polynomials:

N () = KN (, ) = eN2

2N1j=0

h2j(), (96)

= eN2

2

N/2pi

(N 1)!NN[h2N () hN1()hN+1()

], (97)

where we used the expressions pertinent to the Gaussian weight: Q() N2 2 , pik() hk(), andfurther exploited the variant of the Christoffel-Darboux formula, Eq.(95). It is therefore evidentthat the limiting shape of the mean eigenvalue density for large random matrices taken fromthe Gaussian Unitary Ensemble is indeed controlled by the Plancherel-Rotach asymptotics of the

19

Hermite polynomials. In fact, similar considerations exploiting the original Christoffel-Darbouxformula, Eq.(90), show that our main object of interest -the kernel KN(,

) - can be expressedas

KN(, ) = e

N4 (

2+2) hN1()hN () hN1()hN () (98)

and therefore all the higher correlation functions are controlled by the Plancherel-Rotach asymp-totics as well.

For extracting the required asymptotics we are going to use the integral representation for theHermite polynomials. We start with rewriting the expression Eq.(94) as

hN+n(x) = (iN)N+nN

2pi

dq qN+n eN2 (qix)

2

(99)

= (iN)N+nN

2pi

[IN+n(x) + (1)N+nIN+n(x)

], (100)

where

IN+n(x) =

0

dq qn eNf(q), f(q) = ln q 12(q ix)2 . (101)

The latter form is suggestive of exploiting the so-called saddle-point method (also known as themethod of steepest descent or method of stationary phase) of asymptotic evaluation of integrals ofthe form

(z)eNF (z)dz, (102)

where the integration goes along a contour in the complex plane, F (z) is an analytic functionof z in some domain containing the contour of integration, and N is a large parameter. The mainidea of the method can be informally outlined as follows. Suppose that the contour is such that:(i) the value of ReF has its maximum at a point z0 , and decreases fast enough when we goalong away from z0, and (ii) the value of ImF stays constant along (to avoid fast oscillationsof the integrand). Then we can expect the main contribution for N 1 to come from a smallvicinity of z0 = x0 + iy0.

Z0

Figure 2: Schematic structure of a harmonic function in the vicinity of a stationary point z0.

Since the function ReF is a harmonic function of x = Rez, y = Imz, it can have only saddlepoints (see Fig. 2) found from the condition of stationarity F (z0) = 0. Let us suppose that thereexists only one such saddle point z = z0, close to which we can expand F (z) F (z0)+C(z z0)2,where C = 12F

(z0). Consider the level curves [ReF ](x, y) = [ReF ](x0, y0), which are known eitherto go to infinity, or end up at a boundary of the domain of analyticity. In the vicinity of the chosensaddle-point the equation for the level curves is Re[F (z) F (z0)] = 0, hence

Re[|C|ei(z z0)2] =[(x x0)2(y y0)2

]cos 2 (x x0)(y y0) sin () = 0,

20

which describes two orthogonal straight lines passing through the saddle-point

y = y0 + tan

(pi

4

2

)(x x0), y = y0 tan

(pi

4+

2

)(x x0)

partitioning the x, y plane into four sectors: two positive ones: ReF (z) > ReF (z0), and two

y

++

-

-

x

(/4/2)

-(/4/2)

z0

()/2

Figure 3: Partitioning of the xy plane in a vicinity of the stationary point z0 into four sectors bylevel curves (solid lines). Dashed line shows the bi-sector of the negative sectors: the directionof the steepest descent contour.

negative ones ReF (z) < ReF (z0), see Fig. 3. If the edge points of the integration contour (denoted z1 and z2) both belong to the same sector, and ReF (z1) 6= ReF (z2), one always candeform the contour in such a way that ReF (z) is monotonically increasing along the contour. Thenobviously the main contribution to the integral comes from the vicinity of the endpoint (of thelargest value of ReF (z)). Essentially the same situation happens when z1 belongs to a negative(positive) sector, and z2 is in a positive (resp., negative) sector. And only if the two endpointsbelong to two different negative sectors, we can deform the contour in such a way, that ReF (z) hasits maximum along the contour at z = z0, and decays away from this point. Moreover, it is easyto understand that the fastest decay away from z0 will occur along the bi-sector of the negativesectors, i.e. along the line y y0 = tan pi2 (x x0). Approximating the integration contour inthe vicinity of z0 as this bi-sector, i.e. by z = z0 + (x x0) e

ipi2

sin (/2) , we get the leading term of the

large-N asymptotics for the original integral by extending the limits of integration in the variablex = x x0 from to :

(z)eNF (z)dz (z0)eNF (z0) eipi2

sin (/2)

dxeN |C|x2

sin2 /2

= (z0)

2pi

N |F (z0)| exp {NF (z0) +i

2(pi Arg[F (z0)/2)])}. (103)

It is not difficult to make our informal consideration rigorous, and to calculate systematic correc-tions to the leading-order result, as well as to consider the case of several isolated saddle-points,the case of a saddle-point coinciding with an end of the contour, etc., see [7] for more detail.

After this long exposition of the method we proceed by applying it to our integral, Eq.(101).The saddle-point equation and its solution in that case amount to:

F (q) =1

q q + ix = 0, q = q = 1

2

(ix

4 x2

).

21

It is immediately clear that we have essentially three different cases: a) |x| < 2 (b) |x| > 2 and (c)|x| = 2.

1. |x| < 2. In this case we can introduce x = 2 cos, 0 < < pi, so that q = i cos sin, orq+ = e

i(pi/2), q = ei(+pi/2). It is easy to understand that we are interested only in q+

(see Fig.4) and to calculate that Ref(q+) =12 cos (2). On the other hand Ref(q)

when either q or q 0, so that both endpoints belong to negative sectors. Tounderstand whether they belong to the same or different sectors, we consider the values ofRef(q) = lnR 12 (R2 x2) along the real axis, q = R-real. As a function of the variable Rthis expression has its maximal value Ref(q = 1) = 12 + 2 cos2 at q = R = 1.

q- q+

Re q

Im q

s.d. contour

Figure 4: Structure of the saddle-points q and the relevant steepest descent contour for |x| < 2.

Noting that Ref(q = 1) Ref(q+) = cos2 > 0, we conclude that the point q = 1 belongsto a positive sector, and therefore the existence of this positive sector makes the endpointsq = 0 and q = belonging to two different negative sectors, as required by the saddle-pointmethod. Calculating

f (q+) = (1 +

1

q2+

)= 2i sinei

we see that |C| = sin, = + pi/2, and further

f (q+) =1

2cos (2) + i

[1

2sin (2) + pi/2

].

Now we have all the ingredients to enter in Eq.(103), and can find the leading order con-tribution to IN+n(x). Further using IN+n(x) = IN+n(x), valid for real x, we obtain therequired Plancherel-Rotach asymptotics of the Hermite polynomial:

hN+n(x) NN+n

2

sineN2 cos 2 cos

{(n+ 1/2) pi/4 +N

( 1

2sin 2

)}, (104)

where x = 2 cos, 0 < < pi, n N .Now we consider the opposite case:

2. |x| > 2. It is enough to consider explicitly the case x > 2 and parameterize x = 2 cosh, 0 <

One possible contour of the constant phase passing through both points is just the imaginaryaxis q = iy, where Imf(q) = pi/2 and Ref(q) = ln y + 12 (y x)2. Simple consideration givesthat y = e

corresponds to the maximum, and y+ = e to the minimum of Ref(q) along

such a contour. It is also clear that for q = iy+ the expression Ref(q) has a local maximumalong the path going through this point in the direction transverse to the imaginary axis.The topography of Ref(q) in the vicinity of the two saddle-points is sketched in Fig. 5

q-

q+

Re q

Im q

-1 1

Figure 5: The saddle-points q, the corresponding positive sectors (shaded), and the relevantsteepest descent contour (bold) for |x| > 2.

This discussion suggests a possibility to deform the path of integration to be a contourof constant phase Imf(q) consisting of two pieces - 1 = {q = iy, 0 y y+} and 2starting from q = iy+ perpendicular to the imaginary axis and then going towards q = .Correspondingly,

IN+n(x > 2) =

0

dq qn eNf(q) =

iy+0

dq qn eNf(q) +

2

dq qn eNf(q). (106)

The second integral is dominated by the vicinity of the saddle-point q = iy+, and its evalu-ation by the saddle-point technique gives:

2

dq qn eNf(q) 12

pie

N sinhin+Nen+N(+

12 e2),

where the factor 12 arises due to the saddle-point being simultaneously the end-point of thecontour. As to the first integral, it is dominated by the vicinity of iy, and can also beevaluated by the saddle-point method. However, it is easy to verify that when calculatinghN+n(x)

[IN+n(x) + (1)N+nIN+n(x)

]the corresponding contribution is cancelled out.

As a result, we recover the asymptotic behaviour of Hermite polynomials for x > 2 to begiven by:

hN+n(x = 2 cosh (2) > 2) =Nn+Ne

N2

sinhe(n+

12 )

N2 (sinh (2)2). (107)

Now we come to the only remaining possibility,

3. |x| = 2. It is again enough to consider only the case x = 2 explicitly. In fact, this is quite aspecial case, since for x 2 two saddle-points q degenerate into one: q+ q i. Undersuch exceptional circumstances the standard saddle-point method obviously fails. Indeed,

23

the method assumed that different saddle-points do not interfere, which means the distance|q+ q| =

|4 x2| is much larger than the typical widths W 1

N |f (q)|of the regions

around individual saddle-points which yield the main contribution to the integrand. Simplecalculation gives |f (q)| = |1 + q2 | =

|4 x2|, and the criterion of two separate saddle-

points amounts to |x 2| N2/3. We therefore see that in the vicinity of x = 2 such that|x 2| N2/3 additional care must be taken when extracting the leading order behaviourof the corresponding integral IN+n(x = 2) as N .To perform the corresponding calculation, we introduce a new scaling variable = N2/3(2x), and consider to be fixed and finite when N . We also envisage from the discussionabove that the main contribution to the integral comes from the domain around the saddle-point qsp = i of the widths |q i|

|2 x| N1/3. The integral we are interested in is

given by

JN () =

dq qN+n eN2 (qix)

2

(108)

= N1/3

dt

(i+

t

N1/3

)neN

[ln (i+ t

N1/3) 12

{i+ t

N1/3i(2

N2/3

)}2](109)

where we shifted the contour of integration from the real axis to the line q = i+ tN1/3

, 0 the mean densitydecays extremely fast, reflecting the typical absence of the eigenvalues beyond the spectral edge.

A very similar calculation shows that under the same conditions the kernel KN (, ) assumes

the form:

K(1, 2) =Ai(1)Ai

(2)Ai(2)Ai(1)1 2 (127)

known as the Airy kernel, see [8].

7 Orthogonal polynomials versus characteristic polynomials

Our efforts in studying Hermite polynomials in detail were amply rewarded by the provided pos-sibility to arrive at the bulk and edge scaling forms for the matrix kernel in the correspondinglarge-N limits. It is those forms which turn out to be universal, which means independent of the

29

particular detail of the random matrix probability distribution, provided size of the correspondingmatrices is large enough. This is why one can hope that the Dyson kernel would be relevant tomany applications, including properties of the Riemann -function. An important issue for manyyears was to prove the universality for unitary-invariant ensembles which was finally achieved, firstin [10].

In fact, quite a few basic properties of the Hermite polynomials are shared also by any other setof orthogonal polynomials pik(x). Among those worth of particular mentioning is the Christoffel-Darboux formula for the combination entering the two-point kernel Eq.(71), (cf. Eq.(91)):

n1k=0

pik(x)pik(y) = bnpin1(y)pin(x) pin1(x)pin(y)

x y , (128)

where bn are some constants. So the problem of the universality of the kernel (and hence, ofthe n-point correlation functions) amounts to finding the appropriate large-N scaling limit for theright-hand side of Eq.(128) (in the bulk of the spectrum, or close to the spectral edge).

The main dissatisfaction is that explicit formulas for orthogonal polynomials (most important,an integral representation similar to Eq.(94)) are not available for general weight functions dw() =eQ()d. For this reason we have to devise alternative tools of constructing the orthogonalpolynomials and extracting their asymptotics. Any detailed discussion of the relevant techniquegoes far beyond the modest goals of the present set of lectures. Nevertheless, some hints towardsthe essence of the powerful methods employed for that goal will be given after a digression.

Namely, I find it instructive to discuss first a question which seems to be quite unrelated,- thestatistical properties of the characteristic polynomials

ZN () = det(1N H

)=

Ni=1

( i) (129)

for any Hermitian matrix ensemble with invariant JPDF P(H) exp{NTrQ(H)}. Such objectsare very interesting on their own for many reasons. Moments of characteristic polynomials forvarious types of random matrices were much studied recently, in particular due to an attractivepossibility to use them, in a very natural way, for characterizing universal features of the Riemann-function along the critical line, see the pioneering paper[11] and the lectures by Jon Keating inthis volume. The same moments also have various interesting combinatorial interpretations, seee.g. [12, 13], and are important in applications to physics, as I will elucidate later on.

On the other hand, addressing those moments will allow us to arrive at the most natural wayof constructing polynomials orthogonal with respect to an arbitrary weight dw() = eQ()d. Tounderstand this, we start with considering the lowest moment, which is just the expectation valueof the characteristic polynomial:

E [ZN()] =

dw(1) . . .

dw(N )Ni

We first notice that

Ni

The integral in the right-hand side is obviously a polynomial of degree N in , which we denoteDN () and write in the final form as

DN() = det

dw()

dw() . . .

dw()N1 1

dw()

dw()2 . . .

dw()N

. . . .

. . . .

. . . . dw()

N1 dw()

N . . . dw()

2N2 N1 dw()

N dw()

N+1 . . . dw()

2N1 N

. (135)

The last form makes evident the following property. Multiply the right-hand side with dw()p

and integrate over . By linearity, the factor and the integration can be absorbed in the lastcolumn of the determinant. For p = 0, 1, . . . , N 1 this last column will be identical to one ofpreceding columns, making the whole determinant vanishing, so that

dw()pDN () = 0, p = 0, 1, . . . , N 1. (136)

Moreover, it is easy to satisfy oneself that the polynomial DN() can be written as DN () =

DN1N + . . ., where the leading coefficient DN1 = det

(

dw()i+j)Ni,j=0

is necessarily

positive: Dn1 > 0. The last fact immediately follows from the positivity of the quadratic form:

G(x1, . . . , xN ) =

dw()

(Ni=1

xii

)2=

Ni,j

xixj

dw()i+j

Finally, notice that

dw()D2N () =

dw()DN ()[DN1

N + lower powers]

(137)

= DN1

dw()DN ()N = DN1DN

where we first exploited Eq.(136) and at the last stage Eq.(135). Combining all these facts togetherwe thus proved that the polynomials piN () =

1DN1DN

D() form the orthogonal (and normal-

ized to unity) set with respect to the given measure dw(). Moreover, our discussion makes itimmediately clear that the expectation value of the characteristic polynomial ZN () for any givenrandom matrix ensemble is nothing else, but just the corresponding monic orthogonal polynomial:

E [ZN()] = pi(m)N (), (138)

whose leading coefficient is unity. Leaving aside the modern random matrix interpretation thecombination of the right hand sides of the formulas Eq.(138) and Eq.(130) goes back, according to[6], to Heine-Borel work of 1878, and as such is completely classical.

The random matrix interpretation is however quite instructive, since it suggests to consideralso higher moments of the characteristic polynomials, and even more general objects like thecorrelation functions

Ck(1, 2, . . . , k) = E [ZN (1)ZN (2) . . . ZN(k)] . (139)

32

Let us start with considering

C2(1, 2) =

dw(1) . . .

dw(N )Ni

This procedure can be very straightforwardly extended to higher order correlation functions[14,16], and higher order moments[15] of the characteristic polynomials. The general structure is alwaysthe same, and is given in the form of a determinant whose entries are orthogonal polynomials ofincreasing order.

One more observation deserving mentioning here is that the structure of the two-point corre-lation function of characteristic polynomials is identical to that of the Christoffel-Darboux, whichis the main building block of the kernel function, Eq.(71). Moreover, comparing the above for-mula (143) for the gaussian case with expressions (96,95), one notices a great degree in similaritybetween the structure of mean eigenvalue density and that for the second moment of the charac-teristic polynomial. All these similarities are not accidental, and there exists a general relationbetween the two types of quantities as I proceed to demonstrate on the simplest example. For thiswe recall that the mean eigenvalue density N () is just the one-point correlation function, seeEq.(42), and according to Eq.(39) and Eq.(38) can be written as

R1() = NP(, 2, . . . , N ) d2 . . . N

eQ()d2 . . . Ne

N

i=2Q(i)

Ni=2

( i)2

2i

(1N H)1, and statistics of its entries is of great interest. In particular, the familiar eigenvaluedensity () can be extracted from the trace of the resolvent as

() =1

pilim

Im0ImTr

1

1N H. (147)

It is easy to understand that one can get access to such an object, and more general correlationfunctions of the traces of the resolvent by using the identity:

Tr1

1N H=

ZN ()

ZN ()|= . (148)

We conclude that the products of ratios of characteristic polynomials can be used to extract themultipoint correlation function of spectral densities (see an example below). Moreover, distribu-tions of some other interesting quantities as, e.g. individual entries of the resolvent, or statisticsof eigenvalues as functions of some parameter can be characterized in terms of general correlationfunctions of ratios, see [21] for more details and examples. Thus, that type of the correlationfunction is even more informative than one containing products of only positive moments of thecharacteristic polynomials.

In fact, it turns out that there exists a general relation between the two types of the correlationfunctions, which is discussed in full generality in recent papers [17, 22, 23, 24]. Here we would liketo illustrate such a relation on the simplest example, Eq.(146). To this end let us use the followingidentity:

[ZN()]1

=1N

i=1 ( i)=

Nk=1

1

kNi6=k

1

i k (149)

and integrate the ratio of the two characteristic polynomials over the joint probability density of allthe eigenvalues. When performing integrations, each of N terms in the sum in Eq.(149) producesidentical contributions, so that we can take one term with k = 1 and multiply the result by N .Representing 2(1, . . . , N ) =

2i(1i)2

2i

The emerged functions fN () is a rather new feature in RandomMatrix Theory. It is instructiveto have a closer look at their properties for the simplest case of the Gaussian Ensemble, Q() =N2/2. It turns out that for such a case the functions fN () are, in fact, related to the so-called generalized Hermite functions HN which are second -non-polynomial- solutions of the samedifferential equation which is satisfied by Hermite polynomials themselves. The functions also havea convenient integral representations, which can be obtained in the most straightforward way bysubstituting the identity

1

0

dteitsgn[Im]()

into the definition (152), replacing the Hermite polynomial with its integral representation, Eq.(94),exchanging the order of integrations and performing the integral explicitly. Such a procedureresults in

fN+n() 0

dttN+neN(t2

2 itsgn[Im]).

Note, that this is precisely the integral (101) whose large-N asymptotics for real we studied inthe course of our saddle-point analysis. The results can be immediately extended to complex ,and in the bulk scaling limit we arrive to the following asymptotics of the correlation function(151) close to the origin

limN

KN (, ) = K [N(0)( )] , K(r) {eipir if Im() > 0eipir if Im() > 0

. (153)

In a similar, although more elaborate way one can calculate an arbitrary correlation functioncontaining ratios and products of characteristic polynomials [17, 23, 24]. The detailed analysis

shows that the kernel S(, ) = K(, )/( ) and its scaling form S(r) K(r)r play the roleof a building block for more general correlation functions involving ratios, in the same way as theDyson kernel (118) plays similar role for the n-point correlation functions of eigenvalue densities.This is a new type of kernel function with structure different from the standard random matrixkernel Eq.(69). The third type of such kernels - made from functions fN() alone - arises whenconsidering only negative moments of the characteristic polynomials.

To give an instructive example of the form emerging consider

KN (1, 2, 1, 2) = E[ZN (1)

ZN(1)

ZN (2)

ZN(2)

]=

(1 1)(1 2)(2 1)(2 2)(1 2)(1 2) (154)

det(S (1, 1) S (1, 2)S (2, 1) S (2, 2)

).

Assuming Im 1 > 0, Im 2 < 0, both infinitesimal, we find in the bulk scaling limit such thatboth N(0)1,2 = 1,2 and N(0)1,2 = 1,2 are finite the following expression (see e.g. [22], or[21])

limN

KN (1, 2, 1, 2) = K(1, 2, 1, 2) (155)

=eipi(12)

1 2

[eipi(12)

(1 1)(2 2)1 2 e

ipi(12)(1 2)(2 1)

1 2

]. (156)

This formula can be further utilized for many goals. For example, it is a useful exercise tounderstand how the scaling limit of the two-point cluster function (80) can be extracted from

36

such an expression (hint: the cluster function is related to the correlation function of eigenvaluedensities by Eq.(45); exploit the relations (147),(148)).

All these developments, - important and interesting on their own, indirectly prepared the groundfor discussing the mathematical framework for a proof of universality in the large-N limit. As wasalready mentioned, the main obstacle was the absence of any sensible integral representation forgeneral orthogonal polynomials and their Cauchy transforms. The method which circumvents thisobstacle in the most elegant fashion is based on the possibility to define both orthogonal polynomialsand their Cauchy transforms in a way proposed by Fokas, Its and Kitaev, see references in [3], aselements of a (matrix valued) solution of the following (Riemann-Hilbert) problem. The latter canbe introduced as follows. Let the contour be the real axis orientated from the left to the right.The upper half of the complex plane with respect to the contour will be called the positive oneand the lower half - the negative one. Fix an integer n 0 and the measure w(z) = eQ(z) anddefine the Riemann-Hilbert problem as that of finding a 2 2 matrix valued function Y = Y (n)(z)satisfying the following conditions:

Y (n)(z) analytic in C \

Y (n)+ (z) = Y (n) (z)(

1 w(z)0 1

), z

Y (n)(z) 7 (I +O(z1))( zn 00 zn

)as z 7

Here Y(n) (z) denotes the limit of Y

(n)(z) as z 7 z from the positive/negative side of thecomplex plane. It may be proved (see [3]) that the solution of such a problem is unique and isgiven by

Y (n)(z) =

(pin(z) fn(z)

n1pin1(z) n1fn1(z)

), Im z 6= 0 (157)

where the constants n are simply related to the normalization of the corresponding polynomials:n = 2pii[

dwpi2n]1.

On comparing formulae (151) and (157) we observe that the structure of the correlation functionKN (, ) is very intimately related to the above Riemann-Hilbert problem. In fact, for = = zthe matrices involved are identical (even the constant n1 in Eq.(151) emerges when we replace with exact equality sign). Actually, all three types of kernels can be expressed in terms of thesolution of the Riemann-Hilbert problem. The original works [3, 25] dealt only with the standardkernel built from polynomials alone. From that point of view the presence of Cauchy transformsin the Riemann-Hilbert problem might seem to be quite mysterious, and even superfluous. Now,after revealing the role and the meaning of more general kernels the picture can be consideredcomplete, and the presence of the Cauchy transforms has its logical justification.

The relation to the Riemann-Hilbert problem is the starting point for a very efficient methodof extracting the largeN asymptotics for essentially any potential function Q(x) entering theprobability distribution measure. The corresponding machinery is known as the variant of thesteepest descent/stationary phase method introduced for Riemann-Hilbert problems by Deift andZhou. It is discussed at length in the book by Deift[3] which can be recommended to the interestedreader for further details. In this way the universality was verified for all three types of kernelspertinent to the random matrix theory not only for bulk of the spectrum[22], but also for thespectral edges . In our considerations of the Gaussian Unitary Ensemble we already encounteredthe edge scaling regime where the spectral properties were parameterized by the Airy functions

37

Ai(x). Dealing with ratios of characteristic polynomials in such a regime requires second solutionof the Airy equation denoted by Bi(x), see [28].

We finish our exposition by claiming that there exist other interesting classes of matrix en-sembles which attracted a considerable attention recently, see the paper[26] for more detail on theclassification of random matrices by underlying symmetries. In the present framework we onlymention one of them - the so-called chiral GUE. The corresponding 2N 2N matrices are of theform Hch =

(0N J

J 0N

), where J of a general complex matrix. They were introduced to provide

a background for calculating the universal part of the microscopic level density for the EuclidianQCD Dirac operator, see [27] and references therein, and also have relevance for applications tocondensed matter physics. The eigenvalues of such matrices appear in pairs k , k = 1, ..., N .It is easy to understand that the origin = 0 plays a specific role in such matrices, and close tothis point eigenvalue correlations are rather different from those of the GUE, and described by theso-called Bessel kernels[29]. An alternative way of looking essentially at the same problem is toconsider the random matrices of Wishart type W = JJ , where the role of the special point isagain played by the origin (in such context the origin is frequently referred to as the hard spectraledge, since no eigenvalues are possible beyond that point. This should be contrasted with theAiry regime close to the semicircle edge, the latter being sometimes referred to as the soft edge ofthe spectrum. ). The corresponding problems for products and ratios of characteristic polynomialswere treated in full rigor by Riemann-Hilbert technique by Vanlessen[30], and in a less formal wayin [28].

7.1 Acknowledgement

My own understanding in the topics covered was shaped in the course of numerous discussionsand valuable contacts with many colleagues. I am particularly grateful to Gernot Akemann, JonKeating, Boris Khoruzhenko and Eugene Strahov for fruitful collaborations on various facets of theRandomMatrix theory which I enjoyed in recent years. I am indebted to Nina Snaith and FrancescoMezzadri for their invitation to give this lecture course, and for careful editing of the notes. Itallowed me to spend a few wonderful weeks in the most pleasant and stimulating atmosphere of theNewton Institute, Cambridge whose hospitality and financial support I acknowledge with thanks.Finally, it is my pleasure to thank Guler Ergun for her assistance with preparing this manuscriptfor publication.

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