Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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qðx  p

t Þ=

p

cosh½px  p

t þ 

½12

where p

, p

, , p, q are real constants.

Lumps

The KPI equation is

½q

þ 6qq

þ q

xxx

¼3q

½13

The 1-lump solution of this equation is give n by

qðx; y; tÞ¼2@

ln jLðx; y; tÞj

4



;

L ¼ x  2y þ 12

t þ a

 ¼ 

þ i

;

> 0

½14

where  and a are complex constants.

The focusing DSII equation is

þ q



 2q @

1



jqj

þ @

1

jqj





= 0 ½15

where z = x þ iy, and the operator @

1



is defined by

1





ðz;



zÞ¼

2i

f ð;



Þ

  z

d ^ d





The 1-lump solution of this equation is give n by

qðz;



z; tÞ¼

e

iðp



Þtþpz



jz þ  þ 2iptj

þjj

½16

where , , p are complex constants. A typical

1-lump solution is depicted in Figure 2.

Dromions

The DSI equation is

þ @



q þ qu ¼ 0

¼ 2 @

þ @



jqj

½17

The 1-dromion solution of this equation is given by

qðx; y; tÞ¼

e

X



e

Xþ



þ e

Y



þ e

Xþ



XY



þ 

X ¼ px þ ip

t; Y ¼ qy þ iq

jj

¼ 4p

ð  Þ

½18

where p, q are complex consta nts and , , ,  are

positive constants.

A Nonlinear Fourier Transform

The solution of the initial-value problem of an

integrable nonlinear evolution equation on the

infinite line is based on the spectral analysis of the

x-part of the Lax pair. Thus, for the KdV equation

one must analyze eqn [5a]. This equation is the

famous time-independent Schro¨ dinger equation. We

now give a physical interpretation of the relevant

spectral analysis. Let KdV describe the propagation

of a water wave and suppose that this wave is frozen

at a given instant of time. By bombarding this water

wave with quantum particles, one can reconstruct its

shape from knowledge of how these particles

scatter. In other words, the scattering data provide

an alternative description of the wave at fixed time.

abs u

abs u abs u

= –3

= 2.3

= 4.95

abs u

= –0.35

0.8

0.6

0.4

0.2

–20

0.8

0.6

0.4

0.2

–20

0.8

0.6

0.4

0.2

–20

0.8

0.6

0.4

0.2

–20

Figure 2 A typical 1-lump solution.

Integrable Systems and the Inverse Scattering Method 97

The mathematical expression of this description

takes the form of a linear integral equation found

by Faddeev (the so-called Gel’fand–Levitan–March-

enko equation) or equivalently the form of a 2  2

matrix RH problem uniquely specified by the

scattering data. This alternative description of the

shape of the wave will be useful if the evolution of

the scattering data is simple. This is indeed the case,

namely using eqn [5b], it can be shown that the

scattering data evolve linearly. Thus, this highly

nontrivial change of variables from the physical to

scattering space provides a linearization of the KdV

equation.

In what follows we will describe some of the

relevant mathematical formulas. We first

‘‘assume’’ that t here exists a real sol ution q(x, t)

of the initial-value problem which has sufficient

smoothness and which decays for all t as jxj!1.

We then discuss how this assumption can be

eliminated.

As it was mentioned earlier most of the analysis

of the inverse-scattering transform is carried out

on the x-part of the Lax pair, that is, on eqn [5a].

Hence, we first concentrate on eqn [5a] and for

convenience of notation we suppress the time

dependence.

The Direct Problem

As jxj!1, q ! 0, thus there exist solutions of eqn

[5a] which tend to exp[ ikx]asjxj!1. Let

(k, x)and

(k, x) denote solutions of eqn [5a]

with the following asymptotic property:

! e

ikx

;

! e

ikx

; as x !1; k 2 R ½19

Under the transformation k !k, eqn [5a] remains

invariant and the bounda ry condition for is mapped

to the boundary condition for

. Hence

ðk; xÞ¼ ðk; xÞ½20

We denote by (k, x) the solution of eqn [5a] which

tends to exp[ikx]asx !1,

 ! e

ikx

; as x !1; k 2 R ½21

It is more convenient to work with eigenfunctions

(i.e., solutions of [5a]) normalized to unity as x !1,

thus we introduce M(k, x)andN(k, x)asfollows:

M ¼ e

ikx

; N ¼ e

ikx

½22

The functions M and N can be expressed in terms of

q through the solution of linear Volterra integral

equations. Indeed, M satisfies

 2ikM

¼qM; k 2 R

M ! 1; x !1 ½23

The homogeneous version of [23] has solutions 1

and e

2ikx

. Thus,

M ¼ c

þ c

2ikx

þ M

½24

where c

, c

are constants and M

is given by

¼ u

ðxÞþu

ðxÞe

2ikx

½25

The functions u

, u

satisfy

þ e

2ikx

¼ 0; 2ike

2ikx

¼qM

Thus,

ðxÞ¼

2ik

1

dqðÞMðk;Þ;

ðxÞ¼

2ik

1

de

2ik

qðÞMðk;Þ

½26

Substituting [25] and [26] into [24] and using the

boundary condition [23], we find

Mðk; xÞ

¼ 1 þ

1

dð1 þ e

2ikðxÞ

ÞqðÞMðk;Þ½27

Similarly, one may establish that N sati sfied

Nðk; xÞ

¼ 1 þ

dð1 þ e

2ikðxÞ

ÞqðÞNðk;Þ½28

The kernel of eqn [27], as a function of k,is

bounded and analytic for Im k > 0. Thus, if q 2

, M(k, x) as a function of k is holomorphic for

Im k > 0. Similarly, N(k, x) as a function of k is

holomorphic for Im k > 0.

Thus, we have found particular solutions of eqn

[5a] which are holomorphic for Im k > 0. Further-

more, these solutions are simply related for k real.

Indeed, the linear independence of solutions of the

second-order ODE [5a] implies

ðk; xÞ¼aðkÞ

ðk; xÞþbðkÞ ðk; xÞ; k 2 R

Using [20] and replacing  and in terms of M and

N, we find

Mðk; xÞ

aðkÞ

¼ Nðk; xÞþðkÞe

2ikx

Nðk; xÞ

ðkÞ¼

bðkÞ

aðkÞ

; k 2 R ½29

98 Integrable Systems and the Inverse Scattering Method

The functions a(k) and b(k) are given by

aðkÞ¼1 

1

dqðÞMðk;Þ; k 2 R

bðkÞ¼

1

dqðÞMðk;Þe

2ik

; k 2 R

½30

Indeed as x !1, N ! 1, thus, eqn [29] implies

M ! aðkÞþbðkÞe

2ikx

as x !1 ½31

On the other hand, eqn [27] implies that

M ! 1 þ

1

dð1 þ e

2ikðxÞ

qðÞMðk;ÞÞ

x !1 ½32

Comparing eqns [31] and [32], we find eqns [30].

The expression for a(k) implies that this function

is also holomorphic for Im k > 0.

In summary, in the ‘‘direct pr oblem,’’ we have

found particular solutions of eqn [5a] which are

sectionally holomorphic :

Mðk; xÞ

Nðk; xÞ



and

Mðk; xÞ

Nðk; xÞ



are holomorphic for Im k > 0 and Im k < 0, respec-

tively. These solutions, which are characterized in

terms of q by eqns [27] and [28], are simply related

by eqn [29].

The Inverse Problem

Equation [28] expresses N in terms of q. Is it possible

to find an alternative expression for N in terms of

some appropriate ‘‘spectral data’’? The answer is

positive and is a direct consequence of the fact that

eqn [29] defines the ‘‘jump condition’’ of an RH

problem. Indeed, it can be shown that a(k) may have

simple zeros k

, ..., k

in the positive imaginary axis

of the k-complex plane. Hence, in general, M=a can

be expressed in the form

Mðk; xÞ

aðkÞ

¼Mðk; xÞþ

j¼1

ðxÞ

k  ip

; p

> 0

where M(k, x) as a function of k is holomorphic for

Im k > 0. It can also be shown that A

(x) = C

exp[2p

, x]N(k

, x). Hence eqn [29] becomes

Mðk; xÞNðk; xÞ

j¼1

2p

Nðip

; xÞ

k ip

þðkÞe

2ikx

Nðk; xÞ; k 2 R

Taking the () projection of this equation, and

using the fact that both M and N tend to 1 as k !1,

we find

Nðk; xÞ

2i

1

dlðlÞe

2ilx

Nðl; xÞ

l þ k þ i0

¼ 1 

j¼1

2p

k þ ip

Nðip

; xÞ½33

In summary, this equation expressed N(k, x)in

terms of the scattering data ((k), {C

, p

}

Since both eqns [28] and [33] are associated with

the same q, these equations can be used to obtain

the following expression for q:

q ¼2

2

1

dlðlÞe

2ilx

Nðl; xÞ

 i

j¼1

2p

Nðip

; xÞ

½34

Indeed, eqn [28] implies

lim

k!1

Nðk; xÞ¼1 

dqðÞ

Comparing this expression with the large-k behavi or

of eqn [33], we find [34].

Time Dependence of the Scattering Data

We now use eqn [5b] to compute the time

dependence of the scattering data by evaluating

eqn [5b] as x !1 we find  = 4ik

. Then,

evaluating it as x !1and using

  ae

ikx

þ be

ikx

; x !þ1

we find

¼ 0; b

¼ 8ik

Hence,

aðt; kÞ¼að0; kÞ;ðt; kÞ¼ð0; kÞe

8ik

½35

Thus,

ðtÞ¼p

ð0Þ; C

ðtÞ¼C

ð0Þe

½36

The above formal results motivate the follow-

ing definitions (for simplicity, we assume that a(k)

has no zeros). Given a decaying real function

Integrable Systems and the Inverse Scattering Method 99

(x), x 2 R, define M

(k, x) as the solution of the

linear Volterra integral equation

ðk;xÞ¼1 þ

1

dð1 þe

2ikðxÞ

qðÞM

ðk;ÞÞ

Im k  0

Given M

(k, x), define a

(k) and b

(k)by

ðk; xÞ!a

ðkÞþb

ðkÞe

2ikx

; x !1; k 2 R

Given a

and b

, define N(k, x, t) by the solution of

the linear integral equation

Nðk; x; tÞ

2

1

ðlÞ

8il

tþ2ilx

Nðl; x; tÞ

l þ k þ i0

¼ 1

A theorem of Gohberg and Krein implies that this

equation has a unique global solution. Given

, b

, N, define q(x, t)by

qðx; tÞ¼



1

ðkÞ

8ik

tþ2ikx

Nðk; x; tÞ

Then it can be shown that q(x, t) satisfies the KdV

equation and q(x,0)= q

(x).

A Unification

After the emergence of a method for solving the

initial-value problem for nonlinear integrable evolu-

tion equations in one and two space variables, the

most outstanding open problem in the analysis of

these equations became the solution of initial

boundary-value problems. A general approach for

solving such problems for evolution equations in one

space dim ension was provided by Foka s (1997).

This approach has already been used for the study of

nonlinear integrable evolution PDEs on the half-line

(Fokas 2002, 2005), on the interval, and in a time-

dependent domain. An important advantage of this

new method is that it yields the formulation of a

matrix RH problem (or a



@ problem in the case of a

convex time-dependent domain), which although has

more complicated jump matrices than the analogous

problem on the infinite line, it still has an explicit

exponential (x, t) dependence. This fact allows one to

describe effectively the asymptotic properties of the

solution, using the powerful Deift–Zhou method

(Deift and Zhou 1993). For example, the long-time

asymptotics of boundary-value problems on the half

line are discussed in Fokas and Its (1996).

It is remarkable that the above results have

motivated the discovery of a new method for solving

boundary-value problems, not only for linear evolu-

tion PDEs, but also for linear elliptic PDEs in two

dimensions. This includes the Laplace, the biharmonic

and the Helmholtz equations in a convex polygon

(Dassios and Fokas 2005). In a most recent develop-

ment, this method has also been applied to certain

classes of linear PDEs with variable coefficients. This

highly unexpected development unifies and extends

several classical branches of mathematics. In particu-

lar, it unifies the classical transform methods for

simple linear PDEs as well as the method of images,

the treatment of linear PDEs via certain ingenious

techniques such as the Wiener–Hopf technique, the

formulation of Ehrenpreis type integral representa-

tions, and the solution of integrable nonlinear PDEs

via the inverse-scattering transform. Furthermore, it

extends these results to arbitrary domains and to

certain classes of PDEs with variable coefficients.

Regarding linear equations we note the following:

Almost as soon as linear two-dimensional PDEs

made their appearance, d’Alembert and Euler discov-

ered a general approach for constructing large classes

of their solutions. This approach involved separating

variables and superimposing solutions of the resulting

ODEs. The method of separation of variables natu-

rally led to the solution of PDEs by a transform pair.

The prototypical such pair is the direct and the inverse

Fourier transforms; variations of this fundamental

transform include the Laplace, Mellin, sine, cosine

transforms, and their discrete analogs.

The proper transform for a given boundary-value

problem is specified by the PDE, by the domain, and

by the given boundary conditions. For some simple

boundary-value problems, there exists an algorithmic

procedure for deriving the associated transform. This

procedure involves constructing the Green’s function

of a single eigenvalue equation, and integrating this

Green’s function in the k-complex plane, where

k denotes the eigenvalue.

The transform method has been enormously

successful for solving a great variety of initial- and

boundary-value problems. However, for sufficiently

complicated problems the classical transform method

fails. For example, there does not exist a proper analog

of the sine transform for solving a third-order evolution

equation on the half-line. Similarly, there do not exist

proper transforms for solving boundary-value pro-

blems for elliptic equations even of second order and in

simple domains. The failure of the transform method

led to the development of several ingenious but

ad hoc techniques, which include: conformal mappings

for the Laplace and the biharmonic equations; the

Jones method and the formulation of the Wiener–Hop f

factorization problem; the use of some integral

representation, such as that of Sommerfeld; the

100 Integrable Systems and the Inverse Scattering Method

formulation of a difference equation, such as the

Malyuzhinet’s equation. The use of these techniques

has led to the solution of several classical problems in

acoustics, diffraction, electromagnetism, fluid

mechanics, etc. The Wiener–Hopf technique played a

central role in the solution of many of these problems.

A crucial role in the new method is played by the

global equation satisfied by the boundary values of q

and of its derivatives. For evolution equations and for

elliptic equations with simple boundary conditions, this

involves the solution of a system of algebraic equations,

while for elliptic equations with arbitrary boundary

conditions, it involves the solution of an RH problem.

For simple polygons, this RH problem is formulated on

the infinite line, thus it is equivalent to a Wiener–Hopf

problem. This explains the central role played by the

Wiener–Hopf technique in many earlier works.

For linear PDEs, the explicit x

, x

dependence of

q(x

, x

) is consistent with the Ehrenpreis formulation

of the solution. Thus, this method provides the

concrete implementation as well as the generalization

to concave domains of this fundamental principle. For

nonlinear equations, it provides the extension of the

Ehrenpreis principle to integrable nonlinear PDEs.

See also: Boundary value Problems for Integrable

Equations;



@-Approach to Integrable Systems; Integrable

Systems and Algebraic Geometry; Integrable Discrete

Systems; Integrable Systems and Discrete Geometry;

Integrable Systems in Random Matrix Theory; Integrable

Systems: Overview; Korteweg–de Vries Equation and

Other Modulation Equations; Partial Differential

Equations: Some Examples; Riemann–Hilbert Methods in

Integrable Systems; Sine-Gordon Equation; Toda

lattices; Twistor Theory: Some Applications [in Integrable

Systems, Complex Geometry and String Theory].

Further Reading

Ablowitz MJ and Segur H (1977) Exact linearization of a Painleve´

transcendent. Physical Review Letters 38: 1103–1106.

Ablowitz MJ, Yaakov DB, and Fokas AS (1983) On the inverse

scattering transform for the Kadomtsev–Petviashvili equation.

Studies in Applied Mathematics 69: 135–142.

Beals R and Coifman RR (1982) Scattering, transformations

spectrales, et equations d’evolution nonlineaire. I. In: Seminaire

Goulaouic–Meyer–Schwartz, Expose`21.E

cole Polytechnique,

Palaiseau.

Calogero F (1991) In: Zakharov VE (ed.) What Is Integrability?,

Springer.

Dassios G and Fokas AS (2005) The basic elliptic equations in an

equilateral triangle. Proceedings of the Royal Society of

London, Series A 461: 2721–2748.

Deift PA and Zhou X (1993) A steepest descent method for

oscillatory Riemann–Hilbert problems. Annals of Mathematics

137: 245–338.

Flaschka H and Newell AC (1980) Monodromy and spectrum-

preserving deformation I. Communications in Mathematical

Physics 76: 65–116.

Fokas AS (1997) A unified transform method for solving linear

and certain nonlinear PDE’s. Proceedings of the Royal Society

of London, Series A 453: 1411–1443.

Fokas AS (2002) Integrable nonlinear evolution equations on

the half-line. Communications in Mathematical Physics 230:

1–39.

Fokas AS (2005) A generalised Dirichlet to Neumann map for

certain nonlinear evolution PDEs. Communications on Pure

and Applied Mathematics 58: 639–670.

Fokas AS and Ablowitz MJ (1983a) On the initial value problem

of the second Painleve´ transcendent. Communications in

Mathematical Physics 91: 381–403.

Fokas AS and Ablowitz MJ (1983b) On the inverse scattering of

the time dependent Schro¨ dinger equation and the associated

KPI equation. Studies in Applied Mathematics 69: 211–228.

Fokas AS and Its AR (1996) The linearization of the initial-

boundary value problem of the nonlinear Schro¨ dinger equa-

tion. SIAM Journal on Mathematical Analysis 27: 738–764.

Fokas AS and Santini PM (1990) Dromions and a boundary value

problem for the Davey–Stewartson I equation. Physica D 44:

99–130.

Fokas AS and Zhou X (1992) On the solvability of Painleve´II

and IV. Communications in Mathematical Physics 144:

601–622.

Fokas AS, Its AR, Kapaev AA, and Novokshenov VY (2006)

Painleve´ Transcendents: The Riemann–Hilbert Approach.

Providence, RI: American Mathematical Society.

Gardner CS, Greene JM, Kruskal MD, and Miura RM (1967)

Method for solving the Korteweg–de Vries equation. Physical

Review Letters 19: 1095–1097.

Hietarinta J (2002) Scattering of solitons and dromions. In: Sabatier

P and Pike E (eds.) Scattering. San Diego: Academic Press.

Lax PD (1968) Integrals of nonlinear equations of evolution and

solitary waves. Communications on Pure and Applied Mathe-

matics 21: 467–490.

Zakharov V and Shabat A (1972) Exact theory of two-

dimensional self-focusing and one-dimensional self-modulation

of waves in nonlinear media. Soviet Physics – JEPT 34: 62–69.

Integrable Systems and the Inverse Scattering Method 101

Integrable Systems in Random Matrix Theory

C A Tracy, University of California at Davis, Davis,

CA, USA

H Widom, University of California at Santa Cruz,

Santa Cruz, CA, USA

ª 2006 Higher Education Press. Published by Elsevier Ltd.

An earlier version of this article was originally published as

‘‘Distribution functions for largest eigenvalues and their

applications’’. Proceedings of the International Congress of

Mathematics, Volume 1 (2002), pp. 587–596. Beijing, China:

Higher Education Press, with permission.

Random Matrix Models

A random matrix model is a probability space

(, P, F) where the sample space  is a set of

matrices. There are three classic finite N random

matrix models (see, e.g., Mehta (1991)):

1. Gaussian orthogonal ensemble ( = 1):

(a) =N  N real symmetric matrices;

(b) P= ‘‘unique’’ measure that is invariant under

orthogonal transformations and the matrix

elements are i.i.d. random vari ables; expli-

citly, the density is

exp trðA



dA ½1

where c

is a normalization constant and

dA =

i<j

, the product Lebesgue

measure on the independent matrix elements.

2. Gaussian unitary ensemble ( = 2):

(a) =N  N Hermitian matrices;

(b) P= ‘‘unique’’ measure that is invariant

under unitary transformations and the (inde-

pendent) real and imaginary matrix elements

are i.i.d. random variables; and

3. Gaussian symplectic ensemble ( = 4) (see Mehta

(1991) for a definition).

Generally speaking, the interest lies in the

N !1limit of these models. Here we concentrate

on one aspect of this limit. In all three models the

eigenvalues, which are random variables, are real

and with probability 1 they are distinct. If 

max

(A)

denotes the largest eigenvalue of the random

matrix A, then for each of the three Gaussian

ensembles we introduce the corresponding distri-

bution function

N;

ðtÞ :¼ P



ð

max

< tÞ;¼ 1; 2; 4

The basic limit laws (see Tracy and Widom

(1996) and references therein) state that



ðsÞ :¼ lim

N!1

N;

2

ﬃﬃﬃﬃﬃ

s

1=6



;¼ 1; 2; 4 ½2

exist and are given explicitly by

ðsÞ¼det I  K

Airy



¼ exp 

ðx  sÞq

ðxÞdx



where

Airy

AiðxÞAi

ðyÞAi

ðxÞAiðyÞ

x  y

acting on L

ðs; 1ÞðAiry kernel Þ

and q is the unique solution to the Painleve´II

equation

¼ sq þ 2q

satisfying the condition

qðsÞAiðsÞ as s !1

 in eqn [2] is the standard deviation of the

Gaussian distribution on the off-diagonal matrix

elements. For the normalization we have chosen

 = 1=

ﬃﬃﬃ

; however, for subseque nt comparisons, the

normalization  =

ﬃﬃﬃﬃﬃ

is perhaps more natural.

The orthogonal and symplectic distribution func-

tions are

ðsÞ¼exp 

qðxÞdx



ðF

ðsÞÞ

1=2

ðs=

ﬃﬃﬃ

Þ¼cosh

qðxÞdx



ðF

ðsÞÞ

1=2

Graphs of the densi ties dF



=ds are in the adjacent

figure and some statistics of F



can be found in

Figure 1.

The Airy kernel is an example of an integrable

integral operator and a general theory is developed in

Tracy and Widom (1994). A vertex operator approach

to these distributions (and many other closely related

distribution functions in random matrix theory) was

initiated by Adler, Shiota, and van Moerbeke (see the

review article var Moerbeke (2001) for further

developments of this latter approach).

Historically, the discovery of the connection

between Painleve´ functions (P

III

in this case) and

Toeplitz/Fredholm determinants appears in work

of Wu et al. (1976) on the spin–spin correlation

functions of the two-dimensional Ising model. Painleve´

functions first appear in random matrix theory in

102 Integrable Systems in Random Matrix Theory

Jimbo et al. (1980) where they prove that the Fredholm

determinant of the sine kernel is expressible in terms of

. Gaudin (using Mehta’s then newly invented

method of orthogonal polynomials (Porter 1965))

was the first to discover the connection between

random matrix theory and Fredholm determinants.

Universality Theorems

A natural question is to ask whether the above limit

laws depend upon the underlying Gaussian assump-

tion on the probability measure. To investigate this for

unitarily invariant measures ( = 2), one replaces in [1]

exp trðA



! exp trðVðAÞÞðÞ

Bleher and Its (1999) choose

VðAÞ¼gA

 A

; g > 0

and subsequently a large class of potentials V was

analyzed by Deift et al. (1999). These analyses

require proving new Plancherel–Rotach type formu-

las for nonclassical orthogon al polynomials. The

proofs use Rieman n–Hilbert methods. It was shown

that the generic behavior is GUE; hence, the limit

law for the largest eigenvalue is F

. However, by

finely tuning the potential new universality classes

will emerge at the edge of the spectrum . For  = 1, 4

a universality theorem was proved by Stojanovic

(2000) for the quartic potential.

In the case of noninvariant measures, Soshnikov

(1999) proved that for real symmetric Wigner matrices

(complex Hermitian Wigner matrices), the limiting

distribution of the largest eigenvalue is F

(respectively,

). (A symmetric Wigner matrix is a random matrix

whose entries on and above the main diagonal are

independent and identically distributed random vari-

ables with distribution function F. Soshnikov assumes

that F is even and all moments are finite.) The

significance of this result is that non-Gaussian Wigner

measures lie outside the ‘‘integrable class’’ (e.g., there

are no Fredholm determinant representations for the

distribution functions) yet the limit laws are the same as

in the integrable cases.

Appearance of F



in Limit Theorems

In this section we briefly survey the appearances of

the limit laws F



in widely differing areas.

Combinatorics

A major breakthrough occurred with the work of

Baik, Deift, and Johansson (see Baik et al. (2000) and

references therein) when they proved that the limiting

distribution of the length of the longest increasing

subsequence in a rando m permutation is F

. Precisely,

if ‘

() is the length of the longest increasing

subsequence in the permutation  2 S

, then

‘

 2

ﬃﬃﬃﬃﬃ

1=6

< s

! F

ðsÞ

as N !1. Here the probability measure on the

permutation group S

is the uniform measure.

Further discussion of this result can be found in

Johansson (2000b).

Baik and Rains (2001) showed by restricting the set

of permutations (and these restrictions have natural

symmetry interpretations) that F

and F

also appear.

Even the distributions F

and F

(Tracy and Widom

1999) arise. By the Robinson–Schensted–Knuth corre-

spondence, the Baik–Deift–Johansson result is equiva-

lent to the limiting distribution on the number of boxes

in the first row of random standard Young tableaux.

(The measure is the push-forward of the uniform

measure on S

.) These same authors conjectured that

the limiting distributions of the number of boxes in the

second, third, etc., rows were the same as the limiting

distributions of the next-largest, next-next-largest,

etc., eigenvalues in GUE. Since these eigenvalue

distributions were also found in Tracy and Widom

(1996), they were able to compare the then unpub-

lished numerical work of Odlyzko and Rains (2000)

with the predicted results of random matrix theory.

Subsequently, Baik et al. (2000) proved the conjecture

for the second row. The full conjecture was proved by

Okounkov (2000) using topological methods and by,

–4 –2 0 2

0.1

0.2

0.3

0.4

0.5

Probability densities

= 1

= 2

= 4

–1.20653

–1.77109

–2.30688

1.2680

0.9018

0.7195

0.293

0.224

0.166

0.165

0.093

0.050

Figure 1 The mean (



), standard deviation (



), skewness



), and kurtosis (K



)ofF



Integrable Systems in Random Matrix Theory 103

among others, Johansson (2001) using analytical

methods. For an interpretation of the Baik–Deift–

Johansson result in terms of the card game patience

sorting, see the very readable review paper by Aldous

and Diaconis (1999).

Growth Processes

Growth processes have an extensive history both in

the probability literature and the physics literature

(see, e.g., Meakin (1998) and references therein), but

it was only recently that Johansson (2002b) proved

that the fluctuations about the limiting shape in a

certain growth model (‘‘corner growth model’’) are

. Johansson further pointed out that certain

symmetry constraints (inspired from the Baik and

Rains (2001) work) lead to F

fluctuations (see

Growth Processes in Random Matrix Theory).

Subsequently, Baik and Rains (2000) and Gravner

et al. (2002) have shown the same distribution

functions appearing in closely related lattice growth

models. Pra¨hofer and Spohn (2000) reinterpreted the

work of Baik et al. in terms of the physicists’ poly-

nuclear growth (PNG) model thereby clarifying the role

of the symmetry parameter . For example,  = 2

describes growth from a single droplet, whereas  = 1

describes growth from a flat substrate. They also

related the distribution functions F



to fluctuations of

the height function in the KPZ equation (Kardar et al.

1986, Meakin 1998). (The connection with the KPZ

equation is heuristic.) Thus, one expects on physical

grounds that the fluctuations of any growth process

falling into the 1 þ 1KPZ universality class will be

described by the distribution functions F



or one of the

generalizations by Baik and Rains (2000).Sucha

physical conjecture can be tested experimentally. Ear-

lier Myllys et al. established experimentally that a slow,

flameless burning process in a random medium (paper!)

is in the 1 þ 1KPZ universality class. This sequence of

events is a rare instance in which new results in

mathematics inspire new experiments in physics.

In the context of the PNG model, Pra¨hofer and

Spohn have given a process interpretation, the Airy

process, of F

There is an extension of the growth model in

Gravner et al. (2002) to growth in a random

environment. In Gravner et al. (2002) the following

model of interface growth in two dimensions is

considered by introducing a height function on the

sites of a one-di mensional integer lattice with the

following update rule: the height above the site x

increases to the height above x  1, if the latter

height is larger; otherwise, the height above x

increases by 1 with probability p

. It is assumed

that the p

are chosen independently at random with

a common distribution function F, and that the initial

state is such that the origin is far above the other sites.

In the pure regime, Gravner–Tracy–Widom identify

an asymptotic shape and prove that the fluctuations

about that shape, normalized by the square root of

the time, are asymptotically normal. This contrasts

with the quenched version: conditioned on the

environment and normalized by the cube root of

time, the fluctuations almost surely approach the

distribution function F

. We mention that these same

authors find, under some conditions on F at the right

edge, a composite regime where now the interface

fluctuations are governed by the extremal statistics of

in the annealed case while the fluctuations are

asymptotically normal in the quenched case.

Random Tilings

The Aztec diamond of order n is a tiling by dominoes of

the lattice squares [m, m þ 1]  [‘, ‘ þ 1], m, n 2 Z,

that lie inside the region {(x, y):jxjþjyjn þ 1}. A

domino is a closed 1  2or2 1 rectangle in R

with

corners in Z

. A typical tiling is shown in Figure 2.One

observes that near the center the tiling appears random,

called the temperate zone, whereas near the edges the

tiling is frozen, called the polar zones. As n !1 the

boundary between the temperate zone and the polar

zones (appropriately scaled) converges to a circle

(‘ ‘arctic circle theorem’’). Johansson (2002a) proved

that the fluctuations about this limiting circle are F

Statistics

Johnstone (2001) considers the largest principal

component of the covariance matrix X

X where X

is an n  p data matrix all of whose entries are

independent standard Gaussian variables and proves

that for appropriate centering and scaling, the

limiting distribution equals F

in the limit n, p !1

with n=p ! 2 R

. Soshnikov has removed the

Gaussian assumption but requires that n  p =

O(p

1=3

). Thus, we can anticipate applications of

the distributions F



(and particularly F

) to the

statistical analysis of large data sets.

Figure 2 Random tilings.

104 Integrable Systems in Random Matrix Theory

Queuing Theory

Glynn and Whitt (1991) consider a series of n single-

server queues each with unlimited waiting space

with a first-in and first-out service. Service times are

i.i.d. with mean one and variance 

with distribu-

tion V. The quantity of interest is D(k, n), the

departure time of customer k (the last customer to

be served) from the last queue n. For a fixed number

of customers, k, they prove that

Dðk; nÞn



ﬃﬃﬃ

converges in distribution to a certain functional

of k-dimensional Brownian motion. They show that

is independent of the service time distribution V.

It was shown in, for example, Gravner et al. (2002)

that

is equal in distribution to the largest

eigenvalue of a k  k GUE random matrix. This

fascinating connection has been greatly clarified in

recent work of O’Connell and Yor (2002).

From Johansson (2002), it follows for V Poisson that

Dðbxnc; nÞc

1=3

< s



! F

ðsÞ

as n !1 for some explicitly known constants c

and c

(depending upon x).

Superconductors

Vavilov et al. (2001) have conjectured (based upon

certain physical assumptions supported by numer-

ical work) that the fluctuation of the excitation gap

in a metal grain or quantum dot induced by the

proximity to a superconductor is described by F

for

zero magnetic field and by F

for nonzero magnetic

field. They conclude their paper with the remark:

The universality of our prediction should offer ample

opportunities for experimental observation.

Acknowledgments

This work was supported by the National Science

Foundation through grants DMS-9802122 and

DMS-9732687.

See also: Determinantal Random Fields; Growth

Processes in Random Matrix Theory; Integrable Systems

and Algebraic Geometry; Integrable Systems and the

Inverse Scattering Method; Integrable Systems:

Overview; Quantum Calogero–Moser Systems; Random

Partitions; Random Matrix Theory in Physics; Symmetry

Classes in Random Matrix Theory; Toeplitz Determinants

and Statistical Mechanics.