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

Подождите немного. Документ загружается.

projection of the wave functions onto a certain state

in Hilbert space. If the latter are represented by a

fixed matrix A or a fixed vector a, respectively, one

can calculate the RMT prediction for the probability

densities of the matrix elements u

or the

widths a

from the probability density of the

eigenvectors.

Scattering Systems

It is important that RMT can be used as a powerful

tool in scattering theory, because the major part of

the experimental information about quantum sys-

tems comes from scattering experiments. Consider

an example from compound nucleus scattering. In

an accelerator, a proton is shot on a nucleus, with

which it forms a compou nd nucleus. This then

decays by emitting a neutron. More generally, the

ingoing channel  (the proton in our example)

connects to the interaction region (the nucleus),

which also connects to an outgoing channel  (the

neutron). There are  channels with channel wave

functions which are labeled  = 1, ...,  . The

interaction region is described by an N  N

Hamiltonian matrix H whose eigenvalues x

are

bound-state energies labeled n = 1, ..., N. The

dimension N is a cutoff which has to be taken to

infinity at the end of a calculation. The   

scattering matrix S contains the information about

how the ingoing channels are transformed into the

outgoing channels. The scattering matrix S is

unitary. Under certain and often justified assump-

tions, a scattering matrix element can be cast into

the form



¼



 i2W



1



½16

The couplings W

n

between the bound states n and

the channels  are collected in the N   matrix W,



is its th column. The propagator G

1

is the

inverse of

G ¼z1

 H þ i

 open



½17

Here, z is the scattering energy and the summati on

is only ov er chan nels which are open, that is,

accessible. Formula [16] has a clear intuitive inter-

pretation. The scattering region is entered through

channel , the bound states of H become resonances

in the scattering process according to eqn [17], the

interaction region is left through channel . This

formulation applies in many areas of physics. All

observables such as transmission coefficients, cross

sections, and others can be calculated from the

scattering matrix S.

We have not made any statistical assumptions yet.

Often, one can understand generic features of a

scattering system by assum ing that the Hamiltonian

H is a random matrix, taken from one of the three

classical ensembles. This is one RMT approach used

in scattering theory.

Another RMT approach is based on the scattering

matrix itself, S is modeled by a    unitary

random matrix. Taking into account additional

symmetries, one arrives at the three circular ensem-

bles, circular orthogonal (COE) , unitary (CUE) and

symplectic (CSE ). They correspon d to the three

classical Gaussian ensembles and are also labeled

with the Dyson index  = 1, 2, 4. The eigenphases of

the random scat tering matrix correspond to the

eigenvalues of the random Hamiltonian matrix. The

unfolded correlation funct ions of the circular

ensembles are identical to those of the Gaussian

ensembles.

Supersymmetry

Apart from the symmetries, random matrices con-

tain nothing but random numbers. Thus, a certain

type of redundancy is present in RMT. Remarkably,

this redundancy can be removed, without losing any

piece of information by using supersymmetry, that

is, by a reformulation of the random matrix model

involving commuting an d anticommuting variables.

For the sake of simplicity, we sketch the main ideas

for the GUE, but they apply to the GOE and the

GSE accordingly.

One defines the k-level correlat ion functions by

using the resolvent of the Schro¨ dinger equation,

ð2Þ

ðx

; ...; x



ð2Þ

ðHÞ

p¼1



 H

d½H½18

The energies carry an imaginary increment x



= x



i" and the limit " ! 0 has to be taken at the end of

the calculation. The k-level correlation functions

(2)

, ..., x

) as defined in eqn [8] can always be

obtained from the functions [18] by constructing a

linear combination of the

(2)

, ..., x

)inwhich

the signs of the imaginary increments are chosen

such that only the imaginary parts of the traces

contribute. Some trivial -distributions have to be

removed. The k-level correlation functions [18]

can be written as the k-fold derivative

ð2Þ

ðx

; ...; x

ð2Þ

p¼1

ð2Þ

ðx þ JÞ



J¼0

½19

342 Random Matrix Theory in Physics

of the generating function

ð2Þ

ðx þ J Þ

ð2Þ

ðHÞ

p¼1

det x



þ J

 H



det x



 J

 H



d½H½20

which depends on the energies and k new source

variables J

, p = 1, ..., k, ordered in 2k  2k diag-

onal matrices

x ¼ diagðx

; x

; ...; x

; x

J ¼ diagðþJ

; J

; ...; þJ

; J

½21

We notice the normalization Z

(2)

(x) = 1atJ = 0. The

generating function [20] is an integral over an

ordinary N  N matrix H. It can be exactly rewritten

as an integral over a 2k  2k supermatrix  contain-

ing commuting and anticommuting variable s,

ð2Þ

ðx þ JÞ

ð2Þ

ðÞsdet

N

ðx



þ J  Þd½½22

The integrals over the commuting variable s are of

the ordinary Riemann–Stiltjes type, while those over

the anticommuting variables are Berezin integrals.

The Gaussian probability density [5] is mapped onto

its counterpart in superspace

ð2Þ

ðÞ¼c

ð2Þ

exp 

str 



½23

where c

(2)

is a normalization constant. The supertrace

str and the superdeterminant sdet generalize the

corresponding invariants for ordinary matrices. The

total number of integrations in eqn [22] is drastically

reduced as compared to eqn [20]. Importantly, it is

independent of the level number N which now only

appears as the negative power of the superdeterminant

in eqn [22], that is, as an explicit parameter. This most

convenient feature makes it possible to take the limit of

infinitely many levels by means of a saddle point

approximation to the generating function.

Loosely speaking, the supersymmetric formulation

can be viewed as an irreducible representation of RMT

which yields a clearer insight into the mathematical

structures. The same is true for applications in

scattering theory and in models for crossover transi-

tions to be discussed below. This explains why super-

symmetry is so often used in RMT calculations.

It should be emphasized that the roˆ le of super-

symmetry in RMT is quite different from the one in

high-energy physics, where the commuting and

anticommuting variables represent physical parti-

cles, bosons and fermions, respectively. This is not

so in the RMT context. The commuting and

anticommuting variables have no direct phy sics

interpretation; they appear simply as helpful math-

ematical devices to cast the RMT model into an

often much more convenient form.

Crossover Transitions

TheRMTmodelsdiscusseduptonowdescribe

four extreme situations, the absence of correla-

tions in the Poisson case and the presence of

correlations as in the three fully rotational

invariant models GOE, GUE, and GSE. A real

physics system , however, is often between t hese

extreme situations. The corresponding RMT mod-

els can vary considerably, depending on the

specific situation. Nevertheless, those models in

which the random matrices for two extreme

situations are simply added with some weight are

useful in so many applications that they acquired a

rather generic standing. One writes

HðÞ¼H

ð0Þ

þ H

ð Þ

½24

where H

(0)

is a random matrix drawn from an

ensemble with a completely arbitrary probability

density P

(0)

). The case of a fixed matrix is

included, because one may choose a product of

-distributions for the probability density. The

matrix H

()

is random and drawn from the classical

Gaussian ensembles with probability density

()

) for  = 1, 2, 4. One requires that the

group diagonalizing H

(0)

is a subgroup of the one

diagonalizing H

()

. The model [24] describes a

crossover transition. The weight  is referred to as

transition parameter. It is useful to choo se the

spectral support of H

(0)

and H

()

equal. One can

then view  as the root-mean-square matrix element

of H

()

.At = 0, one has the arbitrary ensemble.

The Gaussian ensembles are formally recovered in

the limi t  !1, to be taken in a proper way such

that the energies remain finite.

We are always interested in the unfolded correla-

tion functions. T hus,  hastobemeasuredinunits

of the mean level spacing D such that  = =D is

the physically relevant transition parameter. It

means that, depending on the numerical value of

D,evenasmalleffectontheoriginalenergyscale

can have sizeable i mpact on the spectral statistics.

This is referred to as statistical enhancement.The

nearest-neighbor spacing distribution is already

very close to p

()

(s) f or the Gaussian ensembles if

 is larger than 0.5 or so. In the l ong-range

observables such as the level number variance



(L), the deviation from the Gaussian ensemble

statistics becomes visible at interval lengths L

comparable to .

Random Matrix Theory in Physics 343

Crossover transitions can be interpreted as diffu-

sion processes. With the fictitious time t = 

=2, the

probability density P

(x, t) of the eigenvalues x of

the total Hamilton matrix H = H(t) = H() satisfies

the diffusion equation



ðx; tÞ¼



ðx; tÞ½25

where the probability density for the arbitrary

ensemble is the initial condition P

(x,0)= P

(0)

(x).

The Laplacian



n¼1

n<m



 x





½26

lives in the curved space of the eigenvalues x.This

diffusion process is Dyson’s Brownian motion in

slightly simplified form. It has a rather general meaning

for harmonic analysis on symmetric spaces, connecting

to the spherical functions of Gelfand and Harish-

Chandra, Itzykson–Zuber integrals, and to Calogero–

Sutherland models of interacting particles. All this

generalizes to superspace. In the supersymmetric

version of Dyson’s Brownian motion the generating

function of the correlation functions is propagated,



ðs; tÞ¼



ðs; tÞ½27

where the initial condition Z

(s,0)= Z

(0)

(s) is the

generating function of the correlation functions for

the arbitrary ensemble. Here, s denotes the eigenva-

lues of some supermatrices, not to be confused with

the spacing between adjacent levels. Since the

Laplacian 

lives in this curved eigenvalue space,

this diffusion process establishes an intimate con-

nection to harmonic analysis on superspaces. Advan-

tageously, the diffusion [27] is the same on the

original and on the unfolded energy scales.

Fields of Application

Many-Body Systems

Numerous studies apply RMT to nuclear physics

which is also the field of its origin. If the total

number of nucleons, that is, protons and neutrons, is

not too small, nuclei show single-particle and

collective motion. Roughly speaking, the former is

decoherent out-of-phase motion of the nucleons

confined in the nucleus, while the latter is coherent

in-phase motion of all nucleons or of large groups of

them such that any additional individual motion of

the nucleons becomes largely irrelevant. It has been

shown empirically that the single-particle excitations

lead to GOE statistics, while collective excitations

produce different statistics, often of the Poisson type.

Mixed statistics as described by crossover transitions

are then of particular interest to investigate the

character of excitations. For example, one applies

the model [24] with H

(0)

drawn from a Poisson

ensemble and H

()

from a GOE. Another application

of crossover transitions is breaking of time-reversal

invariance in nuclei. Here, H

(0)

is from a GOE and

()

from a GUE. Indeed, a fit of spectral data to this

model yields an upper bound for the time-reversal

invariance violating root-mean-square matrix element

in nuclei. Yet another application is breaking of

symmetries such as parity or isospin. In the case of

two quantum numbers, positive and negative parity,

say, one chooses H

(0)

= diag(H

(þ)

, H

()

) block-

diagonal with H

(þ)

and H

()

drawn from two

uncorrelated GOE and H

()

from a third uncorre-

lated GOE which breaks the block structure. Again,

root-mean-square matrix elements for symmetry

breaking have been derived from the data.

Nuclear excitation spectra are extracted from

scattering experiments. An analysis as described

above is only possible if the resonances are isolated.

Often, this is not the case and the resonance widths

are comparable to or even much larger than the mean

level spacing, making it impossible to obtain the

excitation energies directly from the cross sections.

One then analyzes the latter and their fluctuations as

measured and applies the concepts sketched above

for scattering systems. This approach has also been

successful for crossover transitions.

Due to the complexity of the nuclear many-body

problem, one has to use effective or phenomenological

interactions when calculating spectra. Hence, one often

studies whether the statistical features found in the

experimental data are also present in the calculated

spectra which result from the various models for nuclei.

Other many-body systems, such as complex atoms

and molecules, have also been studied with RMT

concepts, but the main focus has always been on nuclei.

Quantum Chaos

Originally, RMT was intended for modeling systems

with many degrees of freedom such as nuclei. Surpris-

ingly, RMT proved useful for systems with few degrees

of freedom as well. Most of these studies aim at

establishing a link between RMT and classical chaos.

Consider as an example the classical motion of a point-

like particle in a rectangle billiard. Ideal reflection at the

boundaries and absence of friction are assumed,

implying that the particle is reflected infinitely many

times. A second billiard is built by taking a rectangle

and replacing one corner with a quarter circle as shown

in Figure 5. The motion of the particle in this Sinai

344 Random Matrix Theory in Physics

billiard is very different from the one in the rectangle.

The quarter circle acts like a convex mirror which

spreads out the rays of light upon reflection. This effect

accumulates, because the vast majority of the possible

trajectories hit the quarter circle infinitely many times

under different angles. This makes the motion in the

Sinai billiard classically chaotic, while the one in the

rectangle is classically regular. The rectangle is separ-

able and integrable, while this feature is destroyed in the

Sinai billiard. One now quantizes these billiard systems,

calculates the spectra, and analyzes their statistics. Up

to certain scales, the rectangle (for irrational squared

ratio of the side lengths) shows Poisson behavior, the

Sinai billiard yields GOE statistics.

A wealth of such empirical studies led to the Bohigas–

Giannoni–Schmit conjecture. We state it here not in its

original, but in a frequently used form: spectra of

systems whose classical analogues are fully chaotic

show correlation properties as modeled by the Gaussian

ensembles. The Berry–Tabor conjecture is complemen-

tary: spectra of systems whose classical analogs are fully

regular show correlation properties which are often

those of the Poisson type. As far as concrete physics

applications are concerned, these conjectures are well-

posed. From a strict mathematical viewpoint, they have

to be supplemented with certain conditions to exclude

exceptions such as Artin’s billiard. Due to the defnition

of this system on the hyperbolic plane, its quantum

version shows Poisson-like statistics, although the

classical dynamics is chaotic. Up to now, no general

and mathematically rigorous proofs could be given.

However, semiclassical reasoning involving periodic

orbit theory and, in particular, the Gutzwiller trace

formula, yields at least a heuristic understanding.

Quantum chaos has been studied in numerous

systems. An especially prominent example is the

Hydrogen atom put in a strong magnetic field,

which breaks the integrability and drives the

correlations towards the GOE limit.

Disordered and Mesoscopic Systems

An electron moving in a probe, a piece of wire, say, is

scattered many times at impurities in the material.

This renders the motion diffusive. In a statistical

model, one writes the Hamilton operator as a sum of

the kinetic part, that is, the Laplacian, and a white-

noise disorder potential V(r)withsecondmoment

hVðrÞVðr

Þi¼c



ðdÞ

ðr  r

Þ½28

Here, r is the position vector in d dimensions. The

constant c

determines the mean free time between

two scattering processes in relation to the density of

states. It is assumed that phase coherence is present

such that quantum effects are still significant. This

defines the mesoscopic regime. The average over the

disorder potential can be done with supersymmetry.

In fact, this is the context in which supersymmetric

techniques in statistical physics were developed,

before they were applied to RMT models. In the

case of weak disorder, the resulting field theory in

superspace for two-level correlations acquires the

form

dðQÞf ðQÞexp SðQÞðÞ ½29

where f (Q) projects out the observable under

consideration and where S(Q) is the effective

Lagrangian

SðQÞ¼

str DðrQðrÞÞ

þ i2rMQðrÞ



r ½30

This is the supersymmetric nonlinear  model.Itis

used to study level correlations, but also to obtain

information about the conductance and conduc-

tance fluctuations when the probe is coupled to

external leads. The supermatrix field Q(r) is the

remainder of the disorder average, its matrix

dimension is four or eight, depending on the

symmetry class. This field is a Goldstone mode. It

does not directly represent a particle as often the

case in high-energy physics. The matrix Q(r) lives

in a coset space of certain supergroups. A tensor M

appears in the calculation, and r is the energy

difference on the unfolded scale, not to be confused

with the position vector r.

The first term in the effective Lagrangian invol-

ving a gradient squared is the kinetic term, it stems

from the Laplacian in the Hamiltonian. The con-

stant D is the classical diffusion constant for the

motion of the electron through the probe. The

second term is the ergodic term. In the limit of

zero dimensions, d !0, the kinetic term vanishes

and the remaining ergodic term yields precisely the

unfolded two-level correlations of the Gaussian

ensembles. Thus, RMT can be viewed as the zero-

dimensional limit of field theory for disordered

systems. For d > 0, there is a competition between

the two terms. The diffusion constant D and the

system size determine an energy scale, the Thouless

Figure 5 The Sinai billiard.

Random Matrix Theory in Physics 345

energy E

, within which the spectral statistics is of

the Gaussian ensemble type and beyond which it

approaches the Poisson limit. In Figure 6, this is

schematically shown for the level number variance



(L), which bends from Gaussian ensemble to

Poisson behavior when L > E

. This relates to the

crossover transitions in RMT. Gaussian ensemble

statistics means that the electron states extend over

the probe, while Poisson statistics implies their

spatial localization. Hence, the Thouless energy is

directly the dimensionless conductance.

A large number of issues in disordered and

mesoscopic systems have been studied with the

supersymmetric nonlinear  model. Most results

have been derived for quasi-one-dimensional sys-

tems. Through a proper discretization, a link is

established to models involving chains of random

matrices. As the conductance can be formulated in

terms of the scattering matrix, the experience with

RMT for scat tering systems can be applied and

indeed leads to numerous new results.

Quantum Chromodynamics

Quarks interact by exchanging gluons. In quantum

chromodynamics, the gluons are described by gauge

fields. Relativistic quantum mechanics has to be

used. Analytical calculations are only possible after

some drastic assumptions and one must resort to

lattice gauge theory, that is, to demandi ng numerics,

to study the full problem.

The massless Dirac operator has chiral symmetry,

implying that all nonzero eigenvalues come in pairs

(

, þ

) symmetrically around zero. In chiral

RMT, the Dirac operator is replaced with block

off-diagonal matrices

W ¼

0 W



½31

where W

is a random matrix without further

symmetries. By construction, W has chiral symmetry.

The assumption underlying chiral RMT is that the

gauge fields effectively randomize the motion of the

quark. Indeed, this simple schematic model correctly

reproduces low-energy sum rules and spectral statis-

tics of lattice gauge calculations. Near the center of

the spectrum, there is a direct connection to the

partition function of quantum chromodynamics.

Furthermore, a similarity to disordered systems exists

and an analog of the Thouless energy could be found.

Other Fields

Of the wealth of further investigations, we can

mention but a few. RMT is in general useful for

wave phenomena of all kinds, including classical

ones. This has been shown for elastomechanical and

electromagnetic resonances.

An important field of application is quantum

gravity and matrix model aspects of string theory.

We decided not to go into this, because the reason

for the emergence of RMT concepts there is very

different from everything else discussed above.

RMT is also succe ssful beyond physics. Not

surprisingly, it always received interest in mathema-

tical statistics, but, as already said, it also relates to

harmonic analysis. A connection to number theory

exists as well. The high-lying zeros of the Riemann 

function follow the GUE predictions over certain

interval lengths. Unfortunately, a deeper under-

standing is still lacking.

As the interest in statistical concepts grow s, RMT

keeps finding new applications. Recently, one even

started using RMT for risk management in finance.

See also: Arithmetic Quantum Chaos; Chaos and

Attractors; Determinantal Random Fields; Free Probability

Theory; Growth Processes in Random Matrix Theory;

Hyperbolic Billiards; Integrable Systems in Random

Matrix Theory; Integrable Systems: Overview; Number

Theory in Physics; Ordinary Special Functions; Quantum

Chromodynamics; Quantum Mechanical Scattering

Theory; Random Partitions; Random Walks in Random

Environments; Semi-Classical Spectra and Closed

Orbits; Supermanifolds; Supersymmetry Methods in

Random Matrix Theory; Symmetry Classes in Random

Matrix Theory.

Further Reading

Beenakker CWJ (1997) Random-matrix theory of quantum

Transport. Reviews of Modern Physics 69: 731–808.

Bohigas O (1984) Chaotic motion and random matrix theory. In:

Dehesa JS, Gomez JMG, and Polls A (eds.) Mathematical and

Computational Methods in Physics, Lecture Notes in Physics,

vol. 209. Berlin: Springer.

0.0

0.5

1.0

1.5

2.0

(L)

0 20406080

Figure 6 Level number variance 

(L). In this example, the

Thouless energy is E

 10 on the unfolded scale. The

Gaussian ensemble behavior is dashed.

346 Random Matrix Theory in Physics

Brody TA, Flores J, French JB, Mello PA, Pandey A, and Wong

SSM (1981) Random-matrix physics: spectrum and strength

fluctuations. Reviews of Modern Physics 53: 385–479.

Efetov K (1997) Supersymmetry in Disorder and Chaos. Cambridge:

Cambridge University Press.

Forrester PJ, Snaith NC, and Verbaarschot JJM (eds.) (2003)

Special issue: random matrix theory. Journal of Physics A 36.

Guhr T, Mu¨ ller-Groeling A, and Weidenmu¨ ller HA (1998)

Random-matrix theories in quantum physics: common con-

cepts. Physics Reports 299: 189–428.

Haake F (2001) Quantum Signatures of Chaos, 2nd edn. Berlin:

Springer.

Mehta ML (2004) Random Matrices, 3rd edn. New York:

Academic Press.

Sto¨ ckmann HJ (1999) Quantum Chaos: An Introduction. Cambridge:

Cambridge University Press.

Verbaarschot JJM and Wettig T (2000) Random matrix theory

and chiral symmetry in QCD. Annual Review of Nuclear and

Particle Science 50: 343–410.

Random Partitions

A Okounkov, Princeton University, Princeton, NJ,

USA

ª 2006 A Okounkov. Published by Elsevier Ltd.

Partitions

A partition of n is a monotone sequence of non-

negative integers,

 ¼ð

 

0Þ

with sum n. The number n is also denoted by jj and

is called the size of n. The number of nonzero terms

in  is called the length of  and often denoted by

‘(). It is convenient to make the sequence  infinite

by adding a string of zeros at the end.

A geometric object associated to partition is its

diagram. The diagram of  = (4, 2, 2, 1) is shown in

Figure 1. A larger diagram, flipped and rotated by 135



can be seen in Figure 2. Flipping the diagram introduces

an involution on the set of partitions of n known as

transposition. The transposed partition is denoted by 

Partitions serve as natural combinatorial labels for

many basic objects in mathematics and physics. For

example, partitions of n index both conjugacy classes

and irreducible representations of the symmetric

group S(n). Partitions  with ‘()  n index irredu-

cible polynomial representations of the general linear

group GL(n). More generally, the highest weight of a

rational representation of GL(n) can be naturally

viewed as two partitions of total length  n.

For an even more basic example, partitions  with



 m and ‘()  n are the same as upright lattice

paths making n steps up and m steps to the right

(just follow the boundary of ). In particular, there

are

nþm



of such. By a variation on this theme,

partitions label the standard basis of fermionic Fock

space (Miwa et al. 2000). They also label a standard

basis of the bosonic Fock space.

In most instances, partitions naturally occur

together with some weight function. For example,

the dimension, dim , of an irreducible representation

of S(n), or some power of it, is what always appears in

harmonic analysis on S(n). By a theorem of Burnside,

Planch

ðÞ¼

ðdim Þ

½1

is a probability measure on the set of partitions of n;it

is known as the Plancherel measure. Besides harmonic

analysis, there are many other contexts in which

it appears, for example, by a theorem of

Schensted (see Sagan (2001) and Stanley (1999)), the

distribution of the first part 

of a Plancherel random

partition  is the same as the distribution of the longest

increasing subsequence in a uniformly random permu-

tation of {1, 2, ..., n}.

Figure 1 Diagram of a partition.

1.5

0.5

–2 –1 21

Figure 2 A Plancherel-random partition of 1000 and the limit

shape.

Random Partitions 347

Partitions of n being just a finite set, one is often

interested in letting n !1. Even if the original

problem was not of a probabilistic origin, one can

still often benefit from adopting a probabilistic

viewpoint because of the intuition and techniques

that it brings. This is best illustrated by concrete

examples, which is what we now turn to. These

examples are not meant to be a panorama of

random partitions. This is an old and still rapidly

growing field and a simple list of all major

contributions will take more space than is allowed.

The books Kerov (2003), Pitman (n.d.) Sagan

(2001),andStanley (1999) offer much more

information on the topics discussed below.

Plancherel Measure

Dimension of a Diagram

There are sever al formulas and interpretations

for the number dim  in [1];seeSagan (2001) and

Stanley (1999). The one that often appears in the

context of growth processes is the following:

dim  is the n umber of ways to grow the diagram

 from the empty diagram ; by adding a square

at a time. That is, dim  is number of chains of

the form

;¼

ð0Þ

 

ð1Þ



ðn1Þ

 

ðnÞ

¼ 

where j

(k)

j= k and    means inclusion of

diagrams.

From the classical formula

dim  ¼

jj!

ð

þ k  iÞ!

ijk

ð

 

þ j  iÞ½2

where k is any number such that 

kþ1

= 0, one sees

that the Planche rel measure is a discrete analog of

the eigenvalue density

ð1/2Þ

i<j

ðx

 x

of a GUE random matrix (Mehta 1991). Indeed,

the first f actor in [2], which looks like a multi-

nomial coefficient, is the analog of the Gaussian

weight. Kerov (2003) and Johansson were a mong

the first to recognize the analogy between Plan-

cherel measure and GUE. One comes across many

partition sums that are discrete analogs o f r andom

matrix integrals.

The most compact formula for dim  is the hook

formula

dim 

jj!

&2

hð&Þ

1

½3

Here the product is over all squares & in the

diagram of  and

hð&Þ¼1 þ að&Þþlð&Þ

where a(&) and l( & ) is the number of squares to the

right of the square & and below it, respe ctively.

(These are known as arm-length and leg-length.)

Limit Shape and Edge Scaling

When the diagram of  is very large, the logarithm

of the hook product approximates a double

integral. The analysis of the corresponding integral

plays the central role, (see Kerov (2003), chapter 3)

in the proof of the following law of large numbers

for the Plancherel measure.

Take the diagram of , flip and rotate it as in Figure 1

and rescale by a factor of

ﬃﬃﬃ

so that it has unit area. In

this way one obtains a measure on continuous and, in

fact, Lipschitz functions. By a result of Logan and Shepp

and, independently, Vershik and Kerov these measures

converge as n !1 to the -measure on a single

function (x). This limit shape for the Plancherel

measure, is also plotted in Figure 2. Explicitly,

ðxÞ¼



x arcsinðx=2Þþ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4  x



; jxj2

jxj; jxj > 2

This is an analog of Wigne r’s semicircle law (Mehta

1991) for spectra of random matrices. The Gaussian

correction to the limit shape was also found by

Kerov (2003).

The limit shape result can be refined to show

that 

ﬃﬃﬃ

! 2 in probability. Together with

Schensted’s theorem, this answers the question

posed by Ulam about the l ongest increasing

subsequence in a random permutation. Further

progress came in the work of Baik, Deift, and

Johansson (see Deift (2000)), who conjectured

(and proved for i = 1 and 2) that as n !1 the

joint distribution



 2

ﬃﬃﬃ

1=6

; i ¼ 1; 2; ...

becomes exactly the same as the distribution of

largest eigenvalues of a GUE random matrix. In

particular, the longest increasing subsequence,

suitably scaled, i s distributed exactly like the

largest eigenvalue. The distribution of the l atter is

known as the Tracy–Widom distribution; it is

given in terms of a particular solution of the

Painleve´ II equation. For more information about

the proof of the full conjecture, see Aldous and

Diaconis (1999), Deift (2000),andOkounkov

(2002).

348 Random Partitions

Correlation Functions

One way to prove the full BDJ conjecture is to use the

following exact formula first obtained in a more general

setting by Borodin and Olshanski (see Olshanski (2003),

and Okounkov (2002) for further generalizations).

Look at the downsteps of the zig-zag curve in Figure 2.

The x-coordinates of their midpoints are the numbers

SðÞ¼ 

 i þ



 Z þ

½4

The map  7!S() makes a random partition a random

subset of Z þ

, that is, a random point field on a lattice.

These random points should be treated like eigenvalues

of a random matrix. In particular, it is natural to consider

their correlations, that is, the probability that X  S()

for some fixed X  Z þ

Many formulas work better if we replace the

Plancherel measures M

Planch, n

on partitions of a

fixed number n by their Poisson average,



¼ e



n0



Planch;n

Here >0 is a parameter. It equals the expected

size of . For any finite set X, we have

Prob



X  SðÞðÞ¼det K

Bessel

ðx

; x

; Þ



½5

where K

Bessel

is the discrete Bessel kernel given by

Bessel

ðx;y;Þ

ﬃﬃﬃ



x1=2

ð2

ﬃﬃﬃ



ÞJ

yþ1=2

ð2

ﬃﬃﬃ



ÞJ

xþ1=2

ð2

ﬃﬃﬃ



ÞJ

y1=2

ð2

ﬃﬃﬃ



x y

Note that only Bessel function of integral order

enter this formula.

For large argument , J

ﬃﬃﬃ



) has sine asymptotics

if n  2

ﬃﬃﬃ



and Airy function asymptotics if n 

ﬃﬃﬃ



. Consequently, one gets the random matrix

behavior near the edge of the limit shape and

discrete sine kernel asymptotics of correlations in

the bulk of the limit shape.

Permutation Enumeration

A basic combinatorial problem is to count per-

mutations 

, ..., 

2 S(n) of given cycle types



(1)

, ..., 

(p)

such that





¼ 1 ½6

A geometric interpretation of this problem is to count

covers of the sphere S

= CP

branched over p given

points with monodromy 

(1)

, ..., 

(p)

. Elementary

character theory of S(n)gives(Jones 1998)

#f

2 C



ðiÞ

;



¼ 1g¼



ðiÞ

Planch

½7

where C



is the conjugacy class with cycle type

 and



ðÞ¼jC









dim 

is the central charac ter of the irreducible representa-

tion . Here 





is the character of any  2 C



in the

representation .

Let  be of the form (



,1,1,...) with



 fixed.

By a result of Kerov and Olshanski,





j



1



(), is a

polynomial in  of degree j



j. See [11] for the

simplest example



 = (2), that is, for the central

character of a transposition. We thus recognize in

[7] a discrete analog of the GUE expectation of a

polynomial in traces of a random matrix. This

analogy becomes even clearer in the Gromov–Witten

theory of CP

, which can be viewed as taking into

account contributions of certain degenerate covers,

see Okounkov (2002).

There is a generalization, due to Burnside, of [7]

to counting branched covers of surfaces of any

genus g; see Jones (1998). The only modification

required is that a representation  is now counted

with the weight ( dim )

22g

. For example, covers of

the torus correspond to a uniform measure on

partitions. In particular, the probability that two

random perm utation from S( n) commute is p(n)/n!,

where p(n) is the number of partitions of n.

Generalizations of Plancherel Measure

Schur Functions and Cauchy Identity

Schur functions s



, ..., x

), where  is a parti-

tion with at most n parts, form a distinguished

linear basis of the algebra of symmetric polyno-

mials in x

, ..., x

. Various definitio ns and many

remarkable properties of these function are dis-

cussed in, for example, Sagan (2001) and Sta nley

(1999). One of them is that s



(x)isthetraceofa

matrix with eigenvalues {x

}inanirreducible

GL(n) module with highest weight . The follow-

ing stability of s



ðx

; ...; x

; 0Þ¼s



ðx

; ...; x

Þ;‘ðÞn

allows one to define Schur functions in infinitely

many variables. The formulas













; s











zðÞ



where

zðÞ¼

jj!



¼jAutðÞj



Random Partitions 349

establish the transition between the basis of Schur

function and the basis of power sum functions



; p

In particular, the dimension function dim  is the

following specialization of the Schur function:

dim 

jj!

¼ s

 p

¼1; p

¼p

¼¼0



We will discuss other important specializations of

Schur functions later.

A typical situation in which a random matrix

integral can be reduced to a sum over partition is

when one uses the Cauchy identity

ð1  x

¼ exp

ðxÞp

ðyÞ





ðxÞs



ðyÞ½8

to expand the integrand in Schur function and

integrate term by term using, for example, the

orthogonality of characters or the identity

UðnÞ



ðAgBg

1

Þdg ¼

dim



ðAÞs



ðBÞ½9

Here s



(A) denotes the Schur function in eigenvalues

of a matrix A,dg is the normalized Haar measure on

the unitary group U(n), and

dim

 ¼ s



ð1; ...; 1

|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ}

n times

is the dimension of irredu cible GL(n) module V



with highest weight . The meaning of [9] is that

normalized characters are algebra homomorphisms

from the cente r of the group algebra of U(n)to

numbers. This method of converting a random

matrix problem to a random partition problem is

known as character expansion (see, e.g., Kazakov

(2001)).

Inspired by the Cauchy identity, one can general-

ize Plancherel measure to

Schur

ð1  x

Þs



ðxÞs



ðyÞ

where x and y, or, equivalently, p

(x) and p

(y), are

viewed as parameters. This is known as the Schur

measure. If p

(x) = p

(y) =

ﬃﬃﬃ



and all other p

’s

vanish, we get M

Schur

= M



. Many properties of the

Plancherel measure can be generalized to Schur

measure, in particular, exact formulas for correla-

tion functions, description of the limit shape, etc.

(Okounkov 2002).

Dimension Functions

We already met the function dim

. There is a

useful formula

dim

 ¼

&2

n þ cð&Þ

hð&Þ

½10

where c((i, j)) = j  i is the content of the square

ith row and jth column. From [10] it is clear that dim

makes sense for arbitrary complex values of n. The

corresponding specializations of the Schur measure

x ¼

ﬃﬃﬃ



; ...;

ﬃﬃﬃ



|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

z times

; y ¼

ﬃﬃﬃ



; ...;

ﬃﬃﬃ



|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

times

where , z, z

are parameters, are related to

the so-called Z-measures and their theory is much-

developed ( Olshanski 2003). As z, z

, 

1

!1 in

such a way that zz

 ! 

, we get M



in the limit.

The enumerative problems discussed in the section

‘‘Permutation enumeration’’ have analogs for the

unitary groups U(n) and, suitably interpreted, the

answers are the same with the dimension dim



replacing dim . For example, instead of counting the

solutions to [6], one may be interested in the volume

of the set of p-tuples of unitary matrices with given

eigenvalues that multiply to 1. Geometrically, such

data arise as the monodromy of a flat unitary

connection over S

n{p points}, which is a U(n)analog

of a branched cover. The analog of Burnside’s

formula is Witten’s formula for the volumes of

moduli spaces of flat connections on a genus g

surface with given holonomy around p punctures,

(see, e.g., Witten (1991) and Woodward (2004)). It

involves summing normalized characters over all

representations V



, not necessarily polynomial, with

the weight (dim V



)

22g

. If additionally weighted by

a Gaussian of the form exp(A(f

() þ (n=2)jj)),

where

ðÞ¼



 i þ



i þ



&2

cð&Þ½11

this becomes Migdal’s formula for the partition

function of the 2D Yang–Mills theory , the positive

constant A being the area of the surfa ce (see, e.g.,

Witten (1991) and Woodward (2004 )).

A further generalization naturally arising in the

theory of quantum groups is the quantum dimension

dim

n;q

 ¼ s



ðq

1n

; q

3n

; ...; q

n3

; q

n1

&2

nþcð&Þ

 q

ncð&Þ

hð&Þ

 q

hð&Þ

350 Random Partitions

where q is a parameter (it is more common to use

dim

n, q

1=2

instead). Obviously, dim

n, q

! dim

as q ! 1.

The function dim

n, q

is an important building block of,

for example, quantum invariants of knots and 3-folds,

and various related objects (see, e.g., Bakalov and

Kirillov (2001)). The Verlinde formula (Bakalov and

Kirillov 2001) can be viewed as an analog of Burnside’s

formula with weight dim

n, q

.Whenq is a root of unity

the summation over  is naturally truncated to a

finite sum.

The next level of generalization is obtained by

deforming Schur function to Jack and, more generally,

Macdonald symmetric functions (Macdonald 1995).

In particular, the Jack polynomial analog of the

Plancherel measure is

Jack

ðÞ

&2

n!ðt

ððað&Þþ1Þt

þ lð&Þt

Þðað&Þt

þðlð&Þþ1Þt

where t

, t

are parameters, and a(

)andl(

)

denote, as above, the arm- and leg-length of a

square

. This measure depends only on the ratio

which is the usual parameter of Jack poly-

nomials. To continue the analogy with random

matrices, this should be viewed as a general 

analog of the Plancherel measure.

The measure M

Jack

naturally arises in Atiyah–

Bott localization computations on the Hilbert

scheme of n points in C

. By definition, this

Hilbert scheme parametrizes ideals I  C[x, y]of

codimension n as linear spaces. The torus (C



)

acts on it by rescaling x and y and the fixed points

of this action are



¼ Span of x

j1

i1



ði;jÞ=2

where  is a partition of n. The weight of this fixed

point in the Atiyah–Bott formula is proportional to

Jack

(), the parameters t

and t

being the

standard torus weights. Corresponding formulas in

K-theory involve a Macdonald polynomial analog of

dim .

Nekrasov defines the partition functions of N = 2

supersymmetric gauge theories by formally applying

the Atiyah–Bott localization formula to (noncom-

pact) instanton moduli spaces. The resulting ex pres-

sion is a sum over partitions with a weight which is

a generalization of M

Jack

. In this way, random

partitions enter gauge theory. What is more,

statistical properties of these random partitions are

reflected in the dynamics of gauge theories. For

example, the limit shape turns out to be precisely the

Seiberg–Witten curve (see Nekrasov and Okounkov

(2003), Okounkov (2002), and also Nakajima and

Yoshioka (2003)).

Harmonic Functions on Young Graph

Definitions

Partitions form a natural directed graph Y, known

as Young graph , in which there is an edge from  to

 if  is obtained from  by ad ding a square. We

will denote this by % . Let  be a non-negative

function (called multiplicity) on edges of Y.A

function  on the vertices of Y is harmonic if it

satisfies

ðÞ¼

-

ð; ÞðÞ½12

for any . For given edge multiplicities , non-

negative harmonic functions normalized by (;) = 1

form a convex compact (with respect to pointwise

convergence) set, which we will denote by H(). The

extreme points of H() are the indecomposable or

ergodic harmoni c functions. They are the most

important ones. One defines

dim



= ¼

¼

%

%%

jjjj

¼

ð

;

iþ1

and dim



 = dim



=;. For example, if   1 then

dim



 = dim . Any function  2 H() defines a

probability measure on partitions of fixed size

n, n = 0, 1, 2, ...,by

;n

ðÞ¼ðÞdim



; jj¼n ½13

The mean value property [12] implies a certain

coherence of these measures for different values of

n, which, in general, does not hold for measures like

Schur

. Two multiplicity functions  and 

are

gauge equivalent if



ð; Þ¼f ðÞð; Þf ðÞ

1

for some function f.Inthiscase,H()andH(

)are

naturally isomorphic and the measures M



are the same.

First Example: Thoma Theorem

Let F be a central function on the infinite symmetric

group S(1) =

S(n), normalized by F(1) = 1.

Restricted to S(n), F is a linear combination of

irreducible characters

SðnÞ

jj¼n

ðÞ



The branching rule 



S(n1)

 %





implies that

the Fourier coefficients  are harmonic with respect

to   1. They are non-negative if and only if F is a

positive-definite function on S(1), which means that

the matrix (F(g

1

)) is non-negative definite for any

}  S(1). The description of all indecomposable

positive-definite central functions on S(1) was first

Random Partitions 351