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

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complicated. Indeed, d ordinary differential equations

R = v(R, t) generally produce chaotic dynamics (for

d  3 already for steady flows and for d = 2 for time-

dependent flows). The tools for the description of what

is called chaotic advection are similar to those of the

theory of dynamical chaos. The description consistently

exploits two simple ideas: to single out the variables

that can be represented by the sum of a large number of

independent random quantities and to separate vari-

ables that fluctuate on different timescales.

The distance, R

= R

 R

, between two fluid

particles with trajectories R

(t) = R(t; r

) passing at

t = 0 through points r

satisfies the equation

¼ vðR

; tÞvðR

; tÞ½12

If the velocity field can be considered smooth on

the scale R

, then one expands v(R

, t)  v(R

, t) =

(t, R

, introducing the strain matrix  which

can be treated as independent of R

. The distance

thus satisfies locally a linear system of ordinary

differential equ ations (we omit subscripts replacing

by R)

RðtÞ¼ðtÞRðtÞ½13

This equation, with the strain treated as given and

R(0) = r, may be explicitly solved for arbitrar y (t)

only in the 1D case

ln½RðtÞ=r¼ln WðtÞ¼

ðsÞds  X ½14

When t is much larger than the correlation time  of

the strain, the variable X is a sum of N independent

equally distributed random numbers with N = t=

and one can apply [8]. In the multidimensi onal case,

to use the large deviation theory, one introduces the

evolution matrix W such that R(t) = W(t)R (0). The

modulus R is expressed via the positive symmet ric

matrix W

W. In almost every realization of the

strain, the matrix t

1

ln W

W stabilizes at t !1,

that is, its eigenvectors tend to d-fixed orthonormal

eigenvectors f

. To understand that intuitively,

consider some fluid volume, say a sphere, which

evolves into an elongated ellipsoid at later times. As

time increases, the ellipsoid is more and more

elongated and it is less and less likely that the

hierarchy of the ellipsoid axes will change. The

limiting eigenvalues



¼ lim

t!1

1

ln jWf

j½15

are called Lyapunov exponents. The major property

of the Lyapunov exponents is that they are realiza-

tion independent if the flow is ergodic (i.e., spatial

and temporal averages coincide). The relation [15]

states that two fluid particles separated initially by r

pointing into the direction f

will separate (or

converge) asymptotically as exp (

t). The incom-

pressibility constraints det (W) = 1 and



= 0

imply that a positive Lyapunov exponent will exist

whenever at least one of the exponents is nonzero.

Consider indeed

EðnÞ¼lim

t!1

1

lnh½RðtÞ=r

i½16

whose derivative at the origin gives the largest

Lyapunov exponent 

. The function E(n) obviously

vanishes at the origin. Furthermore, E(d) = 0, that

is, incompressibility and isotropy make that hR

d

i is

time independent as t !1. Apart from n = 0, d,

the convex function E(n) cann ot have other zeroes if

it does not vanish identically. It follows that dE=dn

at n = 0, and thus 

, is positive. A simple way to

appreciate intuitively the existence of a positive

Lyapunov exponent is to consider the saddle-point

2D flow v

= x, v

= y with the axes randomly

rotating after time interval T. A vector initially at

the angle  with the x-axis will be stretched after

time T if cos   [1 þ exp (2T)]

1=2

, that is, the

measure of the stretching dire ctions is larger

than 1=2.

A major consequence of the existence of a positive

Lyapunov exponent for any random incompressible

flow is the exponential growth of the interparticle

distance R(t). In a smooth flow, it is also possible to

analyze the statistics of the set of vectors R(t)andto

establish a multidimensional analog of [8].Theideais

to reduce the d-dimensional problem to a set of d

scalar problems for slowly fluctuating stretching

variables excluding the fast fluctuating angular degrees

of freedom. Consider the matrix I(t) = W(t)W

(t),

representing the tensor of inertia of a fluid element

such as the above-mentioned ellipsoid. The matrix is

obtained by averaging R

(t) R

(t)d=‘

over the initial

vectors of length ‘ and I(0) = 1. Introducing the

variables that describe stretching as the lengths of the

ellipsoid axis e

2

, ..., e

2

, one can deduce similarly to

[8] the asymptotic PDF:

Pð

; ...;

; tÞ

/ exp tHð

=t  

; ...;

d1

=t  

d1

Þ½

 ð

 

Þ...ð

d1

 

 ð

þþ

Þ½17

The entropy function H depends on the statistics

of . In the -correlated case, H is everywhere

quadratic:

HðxÞ/d

1

i¼1

;

/ dðd  2i þ 1Þ½18

Lagrangian Dispersion (Passive Scalar) 257

Two-Particle Dispersion in Nonsmooth

Flows

To consider dispersion in the inertial interval of

turbulence, one should assume v(r, t)j/r



, where

generally <1. Rewriting then eqn [12] for the

distance between two particles as

R = v (R, t), we

infer that dR

=dt = 2R  v(R, t) / R

1þ

. It suggests

RðtÞ

1

 Rð0Þ

1

/ t ½19

For large t, R(t) / t

1=(1)

, with the dependence of

the initial separation quickly forgotten. Of course,

for the random process R(t), relation [19] is of the

mean-field type and should pertain (if true) to the

large-time behavior of the averages ðhR(t)

p=(1)

, for p > 0Þ implying their super-diffusive

growth, faster than the diffusive one / t

p=2

. The

power-law scaling may be amplified to the scaling

behavior of the PDF of the interparticle distance,

P(R, t) = P(R, 

1

t). The power-law growth of

the second moment, hR(t)

i/t

, is the celebrated

Richardson dispersion relation, which was the first

quantitative pheno menological prediction in devel-

oped turbulence. It seems to be confirm ed by

experimental data and the numer ical simulations. It

is important to remark that, even assuming the

validity of the Richardson relation, it is impossible

to establish general large-time properties of the PDF

P(R; t) such as those for the single-particle PDF of

the distance between two particles. This is because

the correlation time of the Lagrangian velocity

difference, R=v(R) /hR

1=3

/ t, is comparable

with the total time of the process.

It is instructive to contrast the exponential growth

[16] of the distance between the trajectories with the

power-law growth [19]. In a smooth flow, the closer

two trajectories are initially, the more time is needed

to effectively separate them. In a nonsmooth

turbulent flow, the trajectori es separate in a finite

time independent of their initial distance R(0),

provided that the latter is also in the inertial range.

This explosive separation of trajectories results in a

breakdown of the deterministic Lagran gian flow

since the trajectories cannot be labeled by the

initial conditions. That agrees with the fundamental

theorem stating that the ordinary differential equa-

tion

R = v(R, t) does not have unique solution

if v(r, t) is non-Lipschitz. As shown by the

example of the equation

x = jxj



with two solutions

x = [(1  )t]

1=(1)

and x = 0 both starting at zero,

one should expect multiple Lagrangian trajectories

starting or ending at the same point for velocity

fields with <1. Even though the deterministic

Lagrangian description breaks down, the statistical

description is still possible and one can make

sense of propagators like P(r, s; R, tjv). They are

expected to be weak solutions of the equation

rv(R, t)]P(r, s; R, tjv) = 0 in the nonsmooth

case. According to this assumption, the Lagrangian

trajectories behave stochastically already in a

given velocity field and for negligible molecular

diffusivity – and not only due to a random noise or

to random fluctuations of the velocities.

The general conjecture about the existence and

diffuse na ture of propagators is known to be true for

the Gaussian ensemble of velocities decorrelated in

time (Kraichnan 1968):

ðr; tÞv

ðr

; t

Þi ¼ 2ðt  t

ÞD

ðr  r

Þ½20

Here the Lagrangian velocity v(R, t) has the same

white noise temporal statistics as the Eulerian

velocity v(r, t) for fixed r and the displacement

along a Lagrangian trajectory R(t)  R(0) is a

Brownian motion for all times. To model non-

smooth velocity field of turbulence, we choose

(r) = D



 (1=2)d

(r) and

ðrÞ¼D

½ðd  1 þ Þ



 r

2

½21

Here D

gives the eddy diffusivity of a single fluid

particle (discussed earlier), wher eas d

(r) describes

the statistics of the velocity differences. For 0 <<2,

the Kraichnan ensemble is supported on the velo-

cities that are Ho¨ lder continuous in space with a

fixed exponent  arbitrarily close to =2. It mimics

this way the main property of turbulent velocities.

The rough (distributional) behavior of Kraichnan

velocities in time, although not very physical, is not

expected to modify essentially the qualitative prop-

erties of propagators (it is the spatial regularity, not

the temporal one, of a vector field that is crucial for

the uniqueness of its trajectories).

In ex actly the same way as one derives [6] and [7]

from [4], one gets P(R, t) = j

j

1=2

(4t)

d=2



=4t

where (



1

)

= D

(0) þ 

. In much the same way

one can examine the two-particle PDF. The PDF

(r, s; R, t) of the distance R between two particles

satisfies the equation

ð@

 M

ÞP

ðr; s; R; tÞ¼ðt  sÞðr  RÞ½22

where M

= D

(d  1)r

1d

d1þ

and [22] can

be readily solved:

lim

r!0

ðr; s; R; tÞ/

d1

jt  sj

d=ð2Þ

 exp const:

2

jt  sj



½23

258 Lagrangian Dispersion (Passive Scalar)

That confirms the diffusive character of the limiting

process describing the Lagrangian trajectories in

fixed non-Lipschitz velocities: the endpoints of the

process stay at finite distance when the initial points

converge. The PDF [23] changes from Gaussian to

log–normal when  changes from 0 to 2. The

Richardson dispersion hR

(t) i/t

is reproduced

for  = 4=3.

Multiparticle Propagators

In studying multiparticle statistics, an important

question is what memory of the initial configuration

remains when final distances far exceed initial

ones. To answer this question, one must analyze

the conservation laws of turbulent diffusion.

Many-particle evolution in nonsmooth velocities

exhibits nontrivial statistical integrals of motion

(martingales) that are proportional to the positive

powers of the distances. The integrals involve

geometry in such a way that the distance growth is

balanced by the decrease of the shape fluctuations.

The existence of multiparticle conservation laws

indicates the presence of a long-time memory and is

a reflection of the coupling among the particles due

to the simple fact that they are all in the same

velocity field. The conserved quantities may be easily

built for the limiting cases. Already for a smooth

velocity, the d-volume 

...i

...R

is indeed

preserved for ðd þ 1Þ Lagrangian trajectories. In the

opposite case of a very irregular velocity, the fluid

particles undergo a Brownian motion. The distances

between the Brownian particles grow according to

(t) i= R

(0) þ Dt. The statistical integrals

of motion are hR

 R

i, h2(d þ 2)R



d(R

þ R

)i, and an infinity of similarly built

harmonic polynomials (zero modes of Laplacian).

The statistics of the relative motion of N particles

is described by the joint PDF averaged over rigid

translations: P

rel

(r, s ; R, t) =

(s, r; R þ , t)d.

For smooth velocities,

rel

ðr;0; R;tÞ¼

n¼1

ðR

þ WðtÞr

d ½ 24

Such PDF depends only on the statistics of the

evolution matrix W(t) discussed earlier. Under the

evolution governed by W(t), all distances between

points grow exponentially for large times while their

ratios R

tend to a constant. For whatever initial

positions, asymptotically in time, the points tend to be

situated on the line. Note that the existence of

deterministic trajectories leads to the collapse property

lim

N1

rel

(r; R; t ) = P

rel

N1

; R

; t) (R

N1

 R

where

= (R

, ..., R

N1

The long-time asymptotics of the propagators in

the nonsmooth case can be found explicitly for the

Kraichnan ensemble of velocities:

ð@

þ M

ÞP

rel

ðr; s; R; t Þ¼ðt  sÞðR  rÞ½25

n<m

ðr

Þr

½26

When initial points get close or final points far

apart and time gets large, the multiparticle PDF is

factorized:

lim

!0

rel

ðr; 0; R; tÞ¼









ðrÞg



ðR; tÞ½27

where f



must be taken as zero modes of M

and its

powers while @



= M



. The remarkable fea-

ture of the zero modes of M

is that they are

conserved in mean by the Lagrangian evolution:

f ðRðtÞÞ

f ðRÞM

rel

ðr; 0; R; tÞdR

rel

ðr; 0; R; tÞM

f ðRÞdR

¼ 0

The scaling exponents of the zero modes depend, in

a nontrivial way, on the number of particles N. For

  1 and d  1, one finds



ð2  Þ

NðN  2Þ

2ðd þ 2Þ

 ½28

Passive Scalar

For practical applications, for example, in the

diffusion of pollution, the most relevant quantity is

the average h(r, t)i which can be expressed via the

single-particle propagator. As discussed earlier, for

times longer than the Lagrangian correlation time,

the particle diffuses and hi obeys the effective heat

equation

ðr; tÞ

¼ D

þ 



ðr; tÞ

½29

with the eddy diffusivity D

given by [11]. The

simplest decay problem is that of a uniform scalar

spot of size L release d in the fluid. Its averaged

spatial distribution at later times is given by the

solution of [11] with the appropriate initial condi-

tion. On the other hand, the decay of the scalar in

the spot is governed by the multipoint Lagrangian

propagators. Taking the point of measurement

inside the spot, consider the single-point moment

h

i(t) described by [2]. If there is no molecular

diffusion and the trajectories are unique (spatially

smooth velocity), particles that end at the same

Lagrangian Dispersion (Passive Scalar) 259

point remained together throughout the evolution

and all the moments are preserved. On the contrary,

when velocity is nonsmooth and the propagator is

diffusive, we expect the decay even at the limit  ! 0.

This is an example of the so-called dissipative

anomaly: the symmetry t !t remains broken

even when the symmetry-breaking factor  goes

to zero. Consider a spherical spot of  released in

a spatially smooth incompressible 3D flow with



>

> 0 >

. During the time less than

= j

1

ln (L=r

), diffusion is unimportant and 

inside the spot does not change. At larger time, the

dimensions of the spot with negative Lyapunov

exponents are frozen at r

, while the rest keep

growing exponentially, resulting in an exponential

growth of the total volume exp (

þ 

). That leads

to an ex ponential decay of scalar moments averag ed

over velocity statistics: h[(t)]

i/exp (

t ). The

decay rates 

can be expressed via the PDF [18] of

stretching variables 

. Since  decays as the inverse

volume,

h½ðt Þ

d

exp tHð

=t  

;

=t  

Þ½

 Nð

þ 

Þ ½30

At large t, the integral is determined by the saddle

point. At small N, the saddle point lies within the

parabolic domain of H so 

increases with N

quadratically. At large N, the main contribution is

due to the realization with smallest possible spot of

size L so 

saturates.

For the decay in incompressible nonsmooth flow,

using the Kraichnan model one gets

h

ðtÞi ¼

0; R; 1ðÞC

1=ð2Þ

R;0



dR ½31

When J

(r, t)dr 6¼ 0, the function t

d=(2)

1=(2)

r, 0) tends to J

(r) in the long-time limit

and [31] is reduced to

h

ðtÞi  ð2n  1Þ!! J

nd=ð2Þ



0; R

; R

; ...; R

; R

; 1ðÞdR ½32

The decay is self-similar: P(t, ) = t

d=2(2)

Q(t

d=2(2)

). That means that the PDF of =

ﬃﬃ





is asymptotically time indepe ndent, with



(t ) =

h(r)

i being time-dependent (decreasing) dissipa-

tion rate. This should be contrasted with the lack of

self-similarity for the smooth case.

One can also consider steady state of  pumped by

a source (r, t ):

 þðv rÞ þ  ¼  ½33

Assuming that pumping is white Gaussian with a

zero mean and variance

(r

, t

)(r

, t

) = (r

)

 t

), r

= r

 r

, one can express the correlation

functions via the multiparticle propagators. For

example, assuming zero conditions at the distant

past and space homogeneity, one gets

ðr; tÞ¼

1

PðR; r; t

ÞðRÞdR ½34

The function (R) is nonz ero within the correlation

scale L of the pumping which restricts integration to

R(t) < L. For smooth velocity, this gives

(r) = j

1

(0) ln (L=r)atr < L. For nonsmooth

velocity, the statistics of scalar fluctuations at

small scales is described by the set of structure

functions S

(r) h[(r)  (0)]

i/r



with the

scaling exponents determined by the zero

modes (see Falkovich et al. (2001)). Therefore,

existence of Lagrangian statistical invariants

explains the anomalous scaling of passive scalar

(here, anomaly means that scale invariance broken

by pumping is not restored even when the pumping

scale goes to infinity).

See also: Anomalies; Intermittency in Turbulence; Large

Deviations in Equilibrium Statistical Mechanics; Lyapunov

Exponents and Strange Attractors; Random Walks in

Random Environments; Stochastic Differential Equations;

Turbulence Theories.

Further Reading

Ellis R (1995) Entropy, Large Deviations, and Statistical

Mechanics. Berlin: Springer.

Falkovich G, Gawedzki K, and Vergassola M (2001) Particles and

fields in fluid turbulence. Reviews of Modern Physics 73:

913–975.

Kraichnan RH (1968) Small-scale structure of a scalar field

convected by turbulence. Physics of Fluids 11: 945–963.

Majda A and Kramer PR (1999) Simplified models for turbulent

diffusion: theory, numerical modelling, and physical phenom-

ena. Physics Reports 314: 237–574.

Monin A and Yaglom A (1979) Statistical Fluid Mechanics.

Boston: MIT Press.

Pope SB (1994) Lagrangian PDF methods for turbulent flows.

Annual Review of Fluid Mechanics 26: 23–63.

Zinn-Justin J (1989) Quantum Field Theory and Critical

Phenomena. Oxford: Science Publishing.

260 Lagrangian Dispersion (Passive Scalar)

Large Deviations in Equilibrium Statistical Mechanics

S Shlosman, Universite

de Marseille, Marseille,

France

Introduction

Large deviation theory (LDT) deals with the study

of probabilities of extremely rare events. As an

example, consider the case of independent identi-

cally distributed random variables 

, ..., 

with

the mean value E(

) = m. Then the typical devia-

tions of the sum M

= 

þþ

from its mean

value Nm are of the order of

ﬃﬃﬃﬃﬃ

, while in LDT we

study the probabilities of the deviations which are

linear in N. In ‘‘good’’ cases we know that for b > 0

PrfM

 Nm  bNgexpfIðbÞNg

as N !1

½1

where I() > 0 is the ‘‘rate’’ function.

Questions of LDT are very natural in statistical

mechanics, and they have deep physical meaning,

notwithstanding the fact that the corresponding

events are rare. One reason is that (some) rare

events in the grand canonical ensemble become

typical events in the canonical ensemble.

An interesting feature of LDT in statistical

mechanics is that the behavior [1] of LD is not

universal, and sometimes is replaced by a nonclassi-

cal one:

PrfM

 Nm  bNgexp 

IðbÞN





½2

with <1. That usually happens in the ‘‘phase

transition’’ regime, and then the quantity

I(b), as

well as the exponent , have very much to do with

the geometry of a droplet of one phase formed inside

the other.

Below, we will illustrate all these features on the

example of the Ising model.

The Ising Model in the Finite Box

Our random variables 

will take values 1, with

x 2 Z

. They are called spins . For every finite box

  Z

, we will define Gibbs states in . To do this

we need the Hamiltonians

;

ðÞ¼

x;y n:n:

x;y2





x;y n:n:

x2;y62





Here,  is some spin configuration on Z

, which is

called ‘‘boundary condition,’’ while  2 



is any

spin configuration in .

The ‘‘grand canonical Gibbs measure’’ 

, , T

in 

with boundary condition  at inverse temperature

 = T

1

is given by



;;T

ðÞ¼Z

1

;;T

expðH

;

ðÞÞ ½3

where

;;T

2



expðH

;

ðÞÞ

is called ‘‘partition function’’; it makes the measure

[3] to be a probability distribution.

The boundary condition  þ1(1) will be

denoted by þ(). For every value of T, the Gibbs

measures 

(l),, T

with ()-boundary condition in

the cubic box (l) of size l converge, as l !1,to

the probability measures that we will denote by



, T

. If the two happen to be different, then 

þ, T

called the (þ)-phase, and 

, T

the ()-phase. That

happens to be the case iff the temperature T is lower

than the critical temperature T

= T

(d ). The critical

temperature depends on dimension; T

(1) = 0, while

(d ) > 0 for d  2. The expect ation



þ;T

ð

ÞmðÞ

is called spontaneous magnetization; m() > 0 iff

>T

1

LD Properties of the Gibbs States 

(l), , T

In what follows, we will discuss the LD properties of

the sum M



= 

þþ

jj

, where the spins



, x 2 , are distributed according to the Gibbs

state 

, , T

. Note that E



, T

(

) =  m().

Classical Case

If we look on the LDs of the sum M



when the

temperature T is high enough (in which case the

limiting states 

þ, T

and 

, T

coincide), or else if the

temperature is low, and the deviations are negative –

that is, we consider the events M



þ jjm(T

1

)  b jj

with b < 0 – then their probabilities behave classically:

There exists a (high) temperature T

such that if

T > T

, then

lim

!Z

jj

Pr M



þ 

1



 b 



¼I

ðbÞ for b  0 ½4

lim

!Z

jj

Pr M



þ 

1



 b 



¼I

ðbÞ for b  0 ½5

Large Deviations in Equilibrium Statistical Mechanics 261

where the function I

(b)  0 is strictly concave on

the segment (m(T

1

)  1, m(T

1

) þ 1). It vanishes

at only one point b = 0.

There exists a (low) temperat ure T

such that if

T < T

, then the relation [4] holds with the function

(b) > 0 strictly concave on the segment (m(T

1

) 

1, 0). The limit [5] also does exist, but it can vanish

once we are in the phase transition region. In order

to see some nontrivia l behavior, we have to change

the normalization 1= 

in [5].

Nonclassical Case

The proper normalization happens to be the surface

term, 1= 

(d1)=d

There exists a temperature T

such that if T < T

then

lim

!Z

jj

ðd1Þ=d

Pr M



þ 

1



 b 



¼W

bðÞ for b > 0 ½6

The function W

(b) obeys W

(b) = b

(d1)=d

, with

> 0, provided the value b > 0 is not too large:

b  b(d), where b(d) is some constant, depending on

the dimension and temperature; one can show that

b(d )  1=2

. For larger b’s the dependence is more

complex.

The key object here is the constant w

. To obtain

it, one has to solve the following variational

problem. Let 

(),  2 S

d1

be the surface tension

between the (þ)-phase and the ()-phase of the Ising

model at the temperature T. Then, for every closed

compact (hyper)surface M

d1

 R

, we define its

surface energy as

MðÞ¼





ðÞds

where 

is the normal vector to M at s 2 M. The

functional W

(M) has the meaning of the energy of

the M-shaped droplet of the (þ)-phase floating in the

()-phase. It is called the ‘‘Wulff functional.’’ Let

be the surface which minimizes W

() over all

the surfaces enclosing the unit volume. Such a

minimizer does exist and is unique up to translation.

It is called the ‘‘Wulff shape.’’ The value w

is just

the surface energy of the Wulff shape:

¼W

ðÞ

The value b(d) is defined as the maximal value of

b’s, for which the dilatation b

1=d

can fit into the

unit cube. For higher values of b, the shape of the

(þ)-phase droplet in the cube with ()-boundary

condition is deformed by its walls, so its surface

energy is given by a more complicated variational

problem.

Moderate Deviations and the Droplet

Condensation

The reason behind the different order of the

probabilities of the events M



þ 

m(T

1

) 

b 

, b < 0, and M



þ 

m(T

1

)  b 

, b > 0, at

low temperatures is the following. A typical config-

uration contributing to the first event contains many

small droplets of ()-spins, of size  ln 

, floating

in the sea of (þ)-spins. On the contrary, in the case

of the second event a typical configuration con-

tains, in addition to small droplets, one large

droplet of the size of . It ha s a random shape,

but in the limit  ! Z

that shape converges to a

nonrandom one, which happens to be the Wulff

shape W

. (The precise meaning of that statement

depends on dimension; in case d = 2 the conver-

gence holds in the Hausdorff metrics, while in

higher dimensions it is known only in L

sense.)

That statement makes the following question

natural: consider the event



 E M



ðÞ



; 0 <<1

For which  should we expect, in addition to

microscopic (þ)-droplets of size  ln 

, the forma-

tion of a large droplet, of volume  



,ina

corresponding typical configuration? In other

words, how many extra (þ)-spins should we pump

into our systems in order for the microscopic

droplets to condense into a macroscopic one? (In

the formulation of this question, we have to use the

expectation E(M



) instead of the asymptotically

equivalent quantity  

m(T

1

). The difference,

E(M



) þ 

m(T

1

)  O( @

), being irrelevant in

the LD case, becomes significant here.)

The answer is the following:



if <d=(d þ 1), then a typical configuration

contains only microscopic droplets;



if >d=(d þ 1), then any typical configuration

contains, in addition to microscopic droplets, one

large droplet of volume  



Therefore, the condensation happens at the value

 = d=(d þ 1). This picture has its counterpart in the

behavior of the probabilities of ‘ ‘moderate deviations’ ’

(MD), that is, events when M



þ 

m(T

1

)  





if <d=ðd þ 1Þ, then the deviation is due to

independent fluctuations of sizes of many small

droplets, and the usual Gaussian behavior holds:

Pr M



 E M



ðÞ



 exp 





ðÞ

2Var M



ðÞ

()

¼ exp c 

21

262 Large Deviations in Equilibrium Statistical Mechanics



if >d=ðd þ 1Þ, then the deviation is due to the

formation of a large droplet, and so

Pr M



 E M



ðÞ



 exp c



ððd1Þ/dÞ

Note that the two estimates match at  = d=(d þ 1).

Further Reading

Biscup M, Chayes L, and Kotecky R (2003) Critical region for

droplet formation in the two-dimensional Ising model.

Communications in Mathematical Physics 242: 137–183.

Dobrushin RL and Shlosman SB (1994) Large and moderate

deviations in the Ising model. In: Dobrushin RL (ed.)

Probability Contributions to Statistical Mechanics, Advances

in Soviet Mathematics, vol. 18, pp. 91–220. Providence, RI:

American Mathematical Society.

Dobrushin RL, Kotecky R, and Shlosman SB (1992) Wulff

Construction: A Global Shape from Local Interaction. AMS

Translations Series. Providence, RI: American Mathematical

Society.

Large-N and Topological Strings

R Gopakumar, Harish-Chandra Research Institute,

Allahabad, India

Introduction

Topological strings have been well studied since

they were introduced in the early 1990s. Essen-

tially, they are simplified string theories that

capture the information about a sector of the full

(or ‘‘physical’’) string theory. Thus, while sharing

many of the structural features of usual string

theory, they hold out the possibility of being

amenable to explicit calculations. This is especially

true with regard to stringy quantum corrections

(the higher genus contributions from the point of

view of the string world sheet), which are normally

rather intractable in the full physical string theory.

This has allowed them to play a useful role in

enhancing the understanding of string theory and

many of its mysterious quantum properties, such as

the various dualities.

In particular, in the last several years, topological

strings have served as an important laboratory for

testing and understanding the connection between the

large-N expansion of gauge theories and closed-

string theories. In this article we will sketch how

this connection is illustrated in a duality between

large-N Chern–Simons gauge theory and closed

topological string theories. We will survey the origin

and current status of these developments and

indicated some of its remarkable mathematical

ramifications.

Background

In order to appreciate the conjecture relating the

Chern–Simons theory and topological string the-

ories, we need to go back to the seminal work of

’t Hooft, who pointed to the connection between the

large-N expansion of gauge field theories and string

theories.

The starting point is a gauge field theory (with,

say, gauge group U(N)), where we take the limit of

the rank N of the gauge group to infinity (see Brezin

Large-N and Topological Strings 263

and Wadia (1993) for a collection of papers on the

topic). The idea is then to make an expansion in

inverse powers of N for various observables such as

the free energy and correlation functions. For

definiteness, let us take a gauge theory containing

only gauge fields A in the adjoint representation of

U(N). The quantum theory is (schematically) defined

by the path integral

Z ¼

½DAe

iSðAÞ

½1

For now, the action S(A) for the gauge fields is left

unspecified. It could be either the usual Yang–Mills

functional or of the Chern–Simons form which we

describe below. S(A) is normalized in such a

way that the gauge coupling constant, denoted

by , only appears via an overall multiplicative

factor of 1=.

Then the expression, for instance, for the free

energy F = ln Z has an expansion in a power series

in , whose individual terms are given by the usual

Feynman diagrammatic r ules. Namely, we have is

a sum over connected vacuum diagrams (those

without any external legs) formed from the

vertices determined by the action S(A). Even

without going into the details of the action, we

can write down the dependence on N and 

comingfromadiagramwithh faces, V vertices,

and E edges. Every edge is associated with a

propagator (arising from the inverse of the quad-

ratic term in S(A)) and thus comes with a weight

of . Every vertex, coming f rom the cubic and

higher-order terms in S(A), comes with a factor of



1

.ThereisafactorofN coming from summing

over the color indices that circulate in every loop

(face). We thus get a weight of N



EV

and so the

total contribution t o the free energy can be

organized as

F ¼

g¼0;h¼1

g;h



2g2þh

g¼0;h¼1

g;h

22g



2g2þh

½2

Here we have defined   N , the ’t Hooft

coupling, as the combination that will be kept

fixed when taking the limit of large N.Wehave

also used the fact that V  E þ h = 2  2g,whereg

is the number of handles of the closed two-

dimensional surface one can associate with the

Feynman diagram. (It is bes t to visualize the

Feynman diagram as a ‘‘fatgraph’’ which forms

the s keleton of a closed Riemann surface.) The

coefficients C

g,h

represent the sum of the

contributions from all genus g diagrams with h

boundaries and depend on the details of the

theory.

We note that the reorganization of the contribu-

tions to the free energy is reminiscent of the genus

expansion in a string theory. In fact, eqn [2] as it

stands looks like an open-string expansion on world

sheets with g handles and h boundaries. Indeed, in

many cases the gauge theory arises as a limit of an

open-string theory. (Recall that a massless nonabe-

lian gauge boson is one of the low-lying excitations

of an open-string theory.) So the double expansion

in terms of g and h is not too surprising.

However, the interesting conjecture of ’t Hooft

is in the relation to closed-string theory. Note

that the expansion in inverse powers of N depends

only on the number of handles g.Infact,1=N

seems t o play the role of closed-string coupling in

that it suppresses higher genus diagrams. The total

contribution to a given genus g comes from

summing over all the holes h in eqn [2],for

example,

F ¼

g¼0

22g

ðÞ½3

The conjecture is to identify this with a closed-string

expansion in which F

() is a closed-string ampli-

tude on a genus g Riemann surface. (In carrying out

the sum over the holes, we have assumed the

existence of a radius of convergence. This is

plausible since the number of planar diagrams

(g = 0), for instance, grows only exponentially with

the number of holes.) The question, since ’t Hooft,

has been: what is this closed-string theory? In other

words, what is the background on which the closed

string propagates?

A breakthrough came from Maldacena’s identi-

fication of the background for the particular case of

U(N) N = 4 supersymmetric Yang–Mills theory.

His conjecture was that t his theory is dual to type

IIB closed-string theory on AdS

 S

with a

curvature scale set by  and w ith closed-string

coupling / = N. This proposal passed a number of

nontrivial checks and is widely held to be true. It

also stimulated the search for closed-string duals to

other large-N gauge theories.

In what follows, we explain how the conjecture of

’t Hooft has a nice realization in the case of three-

dimensional U(N) Chern–Simons gauge theory on

. The dual closed-string theory, obtai ned by

summing ov er the holes , turns out to be the

A-model topological string on the (six-dimensional)

resolved conifold background . The parameter 

maps into a Kahler parameter in the closed-string

264 Large-N and Topological Strings

geometry and once again the closed-string coupling

is /=N.

The Large-N Expansion of Chern–Simons

Theory

Nonabelian Chern–Simons theory is based on the

following action functional for the U(N) gauge

connection A:

ðAÞ¼

4

tr A ^ dA þ

A ^ A ^ A



½4

Here M is a three-dimensional manifold. k is called the

level and is integer quantized for the path-integral

equation [1] to be single valued. Note that, classically,

 as defined earlier is proportional to 1=k. One of the

nice properties of S

(A) is that it is independent of the

metric on M, unlike the Yang–Mills functional. Thus,

it is a prototype of a topological field theory. In fact,

the observables in this theory capture topological

information about the 3-manifold M.

Witten succeeded in quantizing the Chern–

Simons theory by relating its Hilbert space to the

space of conformal blocks in the two-dimensional

U(N) WZW theory. (for more details on the

quantization, see Chern-Simons Models: Rigorous

Results). Here, merely the answers for various

observables in the theory will be quoted. In

particular, the free energy for the theory on S

can be written in a completely explicit form:

ZðS

; N; kÞ¼exp FðS

; N; kÞ

ðN þ kÞ

N=2

N 1

j¼1

2 sin

j

N þ k



Nj

½5

One of the features one observes in the quantization

is the shift (‘‘finite renormalization’’) of the effective

level from k to k þ N. This can also be seen in

perturbation theory. Consequently, while taking the

large-N limit, the natural quantity to be held fixed

as the ’t Hooft coupling is  = 2N=(k þ N).

We can then carry out the ’t Hooft expansion in

powers of  and 1=N, of expressions, for example,

for the free energy in eqn [5]:

F ¼

log  





log N þ 

ð1Þ

g¼2

2g2

2gð2g  2Þ

g¼0

h¼2

g;h



2g2þh

½6

The coefficents F

g,h

are nonzero only for even h and

are given by

0;h

¼

2ð h  2Þ

ð2Þ

h2

ðh  2Þhðh  1Þ

1;h

ðhÞ

6ð2Þ

g;h

2ð2g  2 þ hÞ

ð2Þ

2g2þh

2g  3 þ h





2gð2g  2Þ

½7

where the last line is for g > 1. B

are the Bernoulli

numbers. The first few terms in eqn [6] are

nonperturbative contributions which do not have a

Feynman-diagram interpretation. The power series in

 is, on the other hand, of the same form as eqn [2].

In fact, there is an open-string interpretation for these

terms which will be considered later.

Given the explicit form of the answer, we can

carry out the summation over the holes h. Using

some resummation techniques, we find

F ¼

g¼0

i



2g2

ðtÞ½8

with t  i and

ðtÞ¼

ð1Þ

2g2

2gð2g  2Þð2g  2Þ !

2gð2g  2Þ!

n¼1

2g3

nt

½9

(This expression is for g > 1. There are very similar

expressions for genus 0 and 1 as well.) With the

identification of the string coupling g

=  it=N, the

(t) actually turn out to be the genus g amplitudes

of a closed topological string, in line with the

general expectation of the previous section. This is

explained in the following.

Topological Strings

Physical strings are defined in terms of a two-

dimensional sigma model (the theory on the world

sheet) made reparametrization invariant by coupling

to two-dimensional gravity. Topological strings are

simpler versions of this, where the world-sheet

theory is a two-dimensional topological sigma

model. The latter is defined in terms of a sigma

model (usually with N = 2 superconformal symme-

try) with an additional twist which drastically cuts

down the physical states to a subset of the low-lying

modes. There are actually two inequivalent twists

Large-N and Topological Strings 265

denoted by A and B, respectively, but we will

restrict to the A twist in this article. One of the

simplifications of the A twisted sigma model is that

the path integral localizes to contributions from only

holomorphic maps from the world sheet to the

target spa ce (which will be taken to be a Calabi–Yau

3-fold). Also, all the observables in the theory

depend only on the Kahler parameters of the target

space and not the complex structure parameters (see

Topological Sigma Models as well as the book by

Hori et al. (2003) for more details).

The topological string theory is defined by an

appropriate integration of the observa bles of the

topological sigma model over the moduli space of

the world-sheet Riemann surface. For instance, the

free energy of the string theory at genus g is given by

top

ðtÞ¼

6g6

i¼1

ðb;

Þ >

½10

Here b is one of the reparametrization ghost fields

on the world sheet and 

are Beltrami differentials.

The averaging is with respect to the world-sheet

sigma model for the Calabi–Yau target X, as the

subscript indicates. We have also shown the depen-

dence of F

on the Kahler parameters of X,

collectively denoted by t. The localization to the

holomorphic maps in the path integral implies that

top

(t) takes the generic form

top

ðtÞ¼



g;





½11

Here q

= e

t

and n

are the integer coefficents

labeling the element  2 H

(X). This is in the same

basis of two cycles of H

(X) in terms of which the

complex Kahler parameters t

are expressed. (Recall

that in string theory the Kahler parameters are

complexified because of the presence of an addi-

tional 2-form field.) The N

g,

are the Gromov–

Witten invariants for X and are in general rational

numbers. For nonzero , the corresponding terms

are often called world-sheet instanton contributions

since they correspond to topologically nontrivial

maps from the world sheet to 2-cycles in the target

space. The all-genus free energy of the topological

string is also defined to be

top

ðt; g

Þ¼

g¼0

2g2

top

ðtÞ½12

with g

being the string coupling.

Since topological strings are related to physical

strings by a twist on the world sheet, it is natural

that topological string computations are related to

computations in the physical string theory. In fact,

as shown by Antoniadis, Gava , Narain, and Taylor

as well as Bershadsky, Cecotti, Ooguri, and Vafa,

observables such as F

top

(t) are related to special

superpotential terms in the type II string compacti-

fication on the Calabi–Yau X. Using duality to

M-theory, these answers were reinterpreted by

Gopakumar and Vafa in terms of contributions

coming from BPS states of wrapped D-branes. This

gives a completely different perspective on topolog-

ical strings. For instance, the all-genus free energy

can naturally be reorganized as

top

ðt; g

g¼0



d¼1



2 sin



2g2

d

½13

where the n



are integer invariants (Gopakumar–

Vafa) since they count the number of BPS states.

This will prove to be useful in extracting all-genus

answers for topological string amplitudes, which is

normally quite difficult using the perturbative

definition given earlier.

The Large-N Dual to Chern–Simons

Theory

We are now in a position to state the duality

(Gopakumar and Vafa 1999) between large-N

Chern–Simons theory and topological strings in a

precise way. The conjecture is that the closed

topological string theory on the S

resolved conifold

geometry is exactly dual to the U( N) Chern–Simons

theory on S

. The resolved conifold geometry is a

noncompact Calabi–Yau 3-fold described by the

equation

xy  zw ¼ 0 ½14

where the singularity is resolved by a 2-sphere

x = z, w = y. The resulting space can thus be

characterized as an O(1) þ O(1) bundle over P

It has a single Kahler param eter t for the nontrivial

2-cycle of the S

. In addition, the string theory is

characterized by the string coupling g

. These

parameters map on the gauge theory side to the

’t Hooft parameter  and N via the dictionary

t ¼ i; g



½15

This conjecture can be checked by comparing

various exact calculations in the Chern–Simons

theory with corresponding calculations in the topo-

logical string on this conifold background. The use

of the duality to M-theory enables us to make exact

computations on this side as well. One of the

266 Large-N and Topological Strings