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

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Mean Field Spin Glasses and Neural Networks

A Bovier, Weierstrass Institute for Applied Analysis

and Stochastics, Berlin, Germany

Introduction and Models

Rarely has a paper with a simple title as ‘‘A solvable

model of a spin glass’’ had such a tremendous impact

on both physics and mathematics as the seminal

paper of 1972 by Sherrington and Kirkpatrick,

which introduced what is now known as the

Sherrington–Kirkpatrick (SK) mean-field spin glass

model. As solvable as it might have appeared to the

authors, it was soon found that the heuristic

solution, based on the so-called replica method,

was physically unacceptable. The reason was a tacit

assumption, now known as replica symmetry, that

proved unfounded. Several years later, Giorgio

Parisi provided an ingenious way out through his

continuous replica symmetry-breaking scheme, that

presented a solution that, through its complexity

and intrinsic beauty, both stunned and fascinated

the community. Unraveling the mysteries involved in

this solution has presented a challenge and driving

force for the last three decades of mathematical

statistical mechanics, while the use of the method in

theoretical physics opened the path to solving a wide

variety of problems not only in the theory of

disordered magnets, but also in neural networks

and combinatorial optimization. In this article the

focus is on the mathematical results obtained in the

study of this and a number of related models.

Mean-Field Models

Mean-field models have played an important role in

statistical mechanics by providing simple, solvable

models in which some of the complex phenomena,

such as phase transitions, could be studied and under-

stood. For example, the Curie–Weiss model of a

ferromagnet describes N spin variables 

(taking values

1) in interaction. The simplifying assumption com-

pared to more realistic models, such as the Ising model,

is to ignore the spatial structure of the model and allow

all spins to interact with each other with equal strength.

This yields to a Hamiltonian function of the form

ðÞ¼

i;j¼1



þ h

i¼1



½1

where J is a coupling constant and h a magnetic

field. This from of the interaction implies that the

Hamiltonian is in fact just a function of the

empirical magnetization m

() = N

1

i = 1



, and

this allows one to use methods from the theory of

large deviations to analyze rather easily the corre-

sponding Gibbs measures



;N

ðÞ

H

ðÞ

;N

½2

The SK Model

This model was a straightforward attempt to

introduce a mean-field version of models with

randomly interacting spins. The interest in such

models arose from the discovery of certain alloys of

ferromagnets and cond uctors (e.g., AuFe and

CuMn) that had been found to exhibit very unusual

magnetic properties. Ruderman and co-workers had

proposed that in these models the magnetic ions

with magnetic moments S

and S

located at the

points x

and x

would interact via an exchange

interaction of the form

cosðk

ðx

 x

ÞÞ

 x

 S

Since the positions of the magnetic ions in the alloy

are random, the signs of their interaction would be

oscillatory. Anderson proposed a simplified model,

in the spirit of the Ising model, where spins taking

values 1 located on a regular lattice would interact

via nearest-neighbor couplings J

modeled as i.i.d.

random variables uniformly distributed on an inter-

val [ J, J]. In the spirit of the Curie–Weiss model,

Sherrington and Kirkpatrick then proposed the

mean-field model where any two spins would

interact via i.i.d. Gaussian random variables J

mean zero and variance one. The SK Hamiltonian is

thus given by

ðÞ

ﬃﬃﬃﬃﬃ

1i<jN



þ h

i¼1



½3

where the normalization is chosen to ensure that the

variance of H

is an extens ive quantity. Although

the two Hamiltonians superficially look similar, the

main feature that allows one to solve the Curie–

Weiss model is absent in the SK model: there is no

way to write the Hamiltonian as a function of

macroscopic variable(s) such as the magnetization.

This implies that all methods known to solve the

Curie–Weiss model fail here. The approach used

systematically in the physics literature to overcome

Mean Field Spin Glasses and Neural Networks 407

this difficulty is to try to compute the mean free

energy f

, N

(1=N), E ln Z

, N

using the formal

identity ln x = lim

q #0

1

 1). For q 2 N, one

easily sees that (putting h = 0)

;N



;...;

exp



a;b¼1

i<j



This expression looks already more like the parti-

tion function of an ordinary mean-field model, and

the computation with standard methods seemed

feasible. However, passage to the limit q#0remains

a highly r isky enterprise, and it took the genius of

Parisi to develop an approach that provided at least

a physically meaningful and convincing answer.

The replica method being dealt with e lsewher e in

this encyclopedia, this approach is not explained

any further here, although we will explain the

nature of the result in the light of recent rigorous

work later on.

Site Disordered Models

The difficulties encountered with the random-bond

interactions led readily to proposals of mean-field

models that were closer to the Curie–Weiss model –

from the point of view that they allowe d the

Hamiltonian to be written as a function of macro-

scopic variables. The most important of these

models was introduced by Figotin and Pastur. Here

the disorder was introduced as an M-dimensional

vector 

for each site i. The components of this

vector are usually taken as i.i.d. random variables 



taking values 1 with equ al probability. One can

then introduce M-dimensional vectors as macro-

scopic variables that generalize the magnetization

with components



ðÞN

1

i¼1







The Hamiltonian can then be written as

ðÞ¼N

¼1



ðÞ



¼

i;j¼1



¼1









These models were indeed found to be solvable with

tools similar to those used intheCurie–Weisscase;

however, they proved disappointing i n that the

solution did not show the characteristic features

expected in a spin glass. In fact, it turns out the

these m odels behave very much like a mean-field

ferromagnet, except that as they display not just

two e quilibrium states at low temperatures, but 2M

of them, concentrated on spin configurations  for

which m

() takes values close to one of the values

m



()e



,wheree



is the -unit vector in R

and



() s olves the equation m = tanh (m) known

from the Curie–Weiss model. This model might

have been forgotten, had it not been rediscovered in

1982 by Hopfield in the context of neural net-

works. Hopfield realized that if 

are interpreted as

the activation states (‘‘firing’’ and ‘‘not firing’’) of

neurons in the brain, the form of the i nteraction in

this model is exactly the one proposed earlier by

Hebb for synaptic interaction between neurons

having ‘‘learned’’ the M ‘‘patterns’’ 



in the past.

He went on to interpret H

() as the Lyapounov

function of the retrieval algorithm by which the

brain would recognize the lear ned patt ern. Natu-

rally, the f act the the configurations 



are minima

of H

then implies the functioning of the algorithm.

The important observation of Hopfield was that,

based on numerical experiments, the algorithm

failed when M became too large. In fact, he

observed a breakdown of the memory if M 

0.14N. This m eant that the interesting asymptotics

in this model required to consider M as an

increasing function of N. This r egime was not

covered by large-deviation-type results and an

intensive program to investigate this model was

initiated. Again, the replica method could be

employed and yielded a very rich structure of the

model, including an explanation of t he findings of

Hopfield. These models also turned out to be an

important starting point for the rigorous analysis.

Gaussian Processes and Derrida’s Models

While the models discussed so far were motivated

from the point of view of randomly inte racting

spins, Derrida had the consequential idea to view

the Hamiltonian of such a model simply as a

random process indexed by the set of all spin

configuration. In the case of the SK model, this

process was, moreover, a Gaussian process and thus

characterized entirely by its mean and variance. For

h = 0 we see that

ðÞH

ð

Þ¼

; 

ðÞðÞ



ð; 

where r

(, 

)  N

1



is usually called the

overlap. This opened the view to a much larger

class of models. In particular, the simplest model

from this perspective corresponds to taki ng H

()as

a process of i.i.d. random variables. Derrida called

this the random-energy model (REM). He also noted

408 Mean Field Spin Glasses and Neural Networks

that it could be seen as the limit if a sequence of the

so-called p-spin SK models corresponding to the

covariance of the Hamiltonian being N(r

(, 

))

On the other hand, Derrida observed that another

class of models could be defined that were easier to

analyze while exhibiting much of the complex

properties of the SK model. These are obtained by

choosing the covariance not as a function of the

overlap (resp. the Hamming distance), but of a

ultra-metric distance related to d

(, 

)  N

1

( inf

{i : 

6¼ 

}  1). These models, called generalized

Random-Energy Models (GREM) were analyzed by

Derrida and Gardner in the 1980s and are now the

only models where the full predictions of the Parisi

theory can be rigorously justified. This is discussed

in some detail later.

Further Models and Applications

There is a wealth of problems that can be

interpreted in terms of disordered mean-field

models, and which may be analyz ed using methods

developed here. Some of the most notable ones

that have received more attention lately include:

the perceptron, a feed-forward neural network

was analyzed first by Gardner using the replica

method. Very recently, Shcherbina and Tirozzi gave

a rigorous justification of this result. The

p-satisfiability problem is an important problem in

computer science that also can be analyzed with the

replica method. Rigorous results are still very

limited. The number partitioning problem can be

formulated as a rand om-energy model . Also, the

most famous problem in combinatorial optimiza-

tion, the traveling salesman problem, can be solved

heuristically with the replica method. Another

emerging field are applications to coding theory.

Formulation of the Problem

Given a model, that is, a Hamiltonian function

defined as a random process, the ultimate goal is

to describe the asymptotic propertie s of the

corresponding Gibbs measure, ideally identifying

a (random or deterministic) limiting measure, as a

function of the temperature, 

1

, and other

parameters, such as the magnetic field h.

The first steps in this direction concerns global

properties:



Does the ground-state energy density,

lim

N "1

max

2S

ðÞ

converge (in what sense?) and what is the limit?



What is the limit of the free energy

;N



1

N

ln Z

; N

It has been noted in the mid-1990s that such

quantities are usually self-averaging, for example,

in the sense that

lim

N "1

;N

 Ef

;N



¼ 0; a:s :

due to the concentration of measure phenomenon.

However, until very recently, the existence of the

limits was considered an open problem in most of

the models described above. Guerra and Toninelli

(2002) discovered that a clever use of comparison

inequalities for convex functions of Gaussian

processes allows one to prove a priori the existence

of limits at least in the case of models based on

Gaussian processes (SK, GREM). The main task is

the computation of the values of the limit.

If the free energy is known as a function of

sufficiently many parameters, one can frequently

compute a number of correlation functions that

characterize the limiting measure as well. What one

should compute is somewhat mod el dependent.

Geometry of Gibbs Measures

and Multi-Overlap Distributions

The problem of satisfactorly describing the asymp-

totic geometric properties of random Gibbs

measures on { 1, 1} is rendered difficult as the

symmetries of the problem make the use of local

topologies seem unattractive. A reasonable way of

solving this problem is as follows. Let D

be a

distance on S

normalized so that max

,  2S

(, ) = 1. Then consider the mass distribution

around any fixed point ,



ðxÞ

; N

ðD

ð; 

ÞxÞ

and construct the biased empirical average

;N



2S



; N

ðÞ



ðÞ

The set of distributions of these random measures

is compact (with respect to the weak topology)

and thus we can expect to construct limits. The

law of K

, N

is fully determined by t he family of

averaged distributions of the distances between n

independent copies of  drawn from the Gibbs

measures,

E

n

; N

ðD

ð

;

Þ; ...; D

ð

n1

;

ÞÞ

Mean Field Spin Glasses and Neural Networks 409

In the SK models, one chooses

ð; Þ¼1 





so that these quantities can be expressed as distribu-

tions of the overlaps (1=N)





, between n

‘‘replica’’ spin variables. In the GREM models, it is

natural to chose as distance the lexicographic distance

used in the construction of the models. In this case, the

limits of K

, N

can be constructed explicitly and it was

shown that they can be expressed in terms of the size-

biased empirical family size distribution of a certain

continuous state branching process via a model-

dependent time change. Since this plays a key roˆle

not only in the GREMs but in other models as well, we

will go into some detail to elucidate this structure.

Neveu’s Process and Random

Genealogies

The random structure of the limiting Gibbs

measures of the GREM models (and presumably

also the SK models, even though this is not proven)

can be traced to a continuous-state branching

process introduced by Neveu, and an induced

associated random genealogy on the unit interval.

Let Z

be a time-homogeneous continuous-time

Markov process with state space R

characterized

by the Laplace transform of its transition kernel

Eðe

Z

¼ aÞ¼exp ða

t

Based on this process, construct a two-parameter

process Z(t, a) with the property that, for any a, b >

0, the processes Z(  , a) and Z(  , a þ b)  Z(  , a)

are independent and have the same laws as Z

with

initial conditions a, resp. b. It follows that Z(t,)isa

stable subordinator with exponent e

t

. Now let



(a)  Z(t, a)=Z(t, 1), as a function on [0, 1], 

being a random probability distribution function (of

pure point type). Any such family 

of dist ributions

defines in a natural way a genealogical structure on

[0, 1]. Define the ancestor of  2 [0, 1] at time t < 1

to be a

()  

(

1

()), where 

1

is the right-

continuous inverse of the nondecreasing function .

We say that, for , 

2 [0, 1], q(, 

) = t if and

only if t = sup(s : a

() = a

(

)). It is easy to see that

1  q defines an ultra-metric distance. We can

associate with this the distribution size of the offspring

of an ancestor at time t, m



(t) = j

: q(, 

)  tj,and

its size-biased empirical distribution

K

d



ðÞ

In the GREM models, it can be shown that the

quantity K

, N

converges (weakly in law) to the

corresponding K obtained from a time change of

the family of measures 

, namely





 

ln mðtÞln mð0Þ

where m is a nondecreasing function that can be

computed explicitly. Namely, if EX





A(d

(, )), and a

denotes the right-derivative

of the concave hull of A, then

mðxÞ¼min 

1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ln2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃ



aðxÞ

; 1



As explained below, similar results are expected in

the SK models.

Interpolation Methods and Guerra’s

Integral Representation

Among the very important tools for the analysis of

Gaussian models in particular have been the inter-

polation methods that allow one to compare

functions of processes with different covariance.

While these methods go back to early work on

Gaussian processes (Slepian, Kahane), they have

been employed with remarkable success in the

present context. Mostly, they consist in introducing

an interpolating Hamiltonian H

() 

ﬃﬃ

H() þ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  t

K(), where K is a reference process that has

certain desired properties. Given any function F of

the process (e.g., the free energy of the model), one

then represents

FðHÞ¼FðKÞþ

FðH

Often the derivative on the right-hand side can be

controlled rather well, for example, because of some

obvious positivity properties.

Example 1 (Guerra and Tonin elli). Choose

KðÞ¼

ﬃﬃﬃﬃﬃ

i<j¼1



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

N  M

i<j¼Mþ1



and consider the free energy F( H

) = f

, N

. Then, first

F(H

) = F(H

) þ F(H

NM

). On the other hand,



¼



;N

i<j¼1



i<j¼1



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1 tÞM

i<j¼Mþ1



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ð1 tÞðN MÞ

410 Mean Field Spin Glasses and Neural Networks

A key tool to be used at this stage is the so-called

Gaussian integration by parts formula, Egf (g) =

(g). Applied here, this gives







t;2

;N

i¼1



i¼1



N  M

i¼Mþ1



 0

This proves superadditivity of NEf

, N

NEf

;N

 MEf

;M

þðN  MÞEf

;NM

which, in turn, implies convergence of Ef

, N

a limit Ef



. Moreover, standard concentration

of measure estimates show then that f

, N

also

converges almost surely.

Example 2 (Guerra, Aizenman–Sims–Starr). A

more complicated application of the interpolation

method allows one to relate the free energy to

Parisi’s solution. This was first found by Guerra

(2003), but a different, and in some sense more

intuitive formulation, was given later by Aizenman

et al. (2003). It is based on the following construc-

tion. We consider a centered Gaussian process H

()

on S

with covariance given by Ng(R

(, 

)) for

some even convex function g :[ 1, 1] ! [0, 1]. Let

us take F(H

) = ln E



H

()

(the a priori expecta-

tion E



need not be symmetric, but may incorporate

a magnetic field). Before using comparison, we now

want to go to a larger space. For this, introduce some set

A equipped with some positive-definite quadratic form

q, normalized such that q

, 

= 1, and jq

, 

j1,

, 

.LetP



denote some probability measure

on A. Now introduce a centered Gaussian process





on A, independent of H

, whose covariance is

given by E







= r(q

, 

)  q

, 

)  g(q

, 

Define

GðH

ﬃﬃﬃﬃﬃ

Þ¼ln E





ðH

ðÞþ

ﬃﬃﬃ







Obviously, G(H

, ) = F(H

) þ

F( ), where

F() =

ln(E





ﬃﬃﬃ





). The amazing idea is now to

compare the process ( H

þ ) with another process



, 

whose covariance is a linear function of R

()

(this is in some sense a Slepian’s process), and that

otherwise is smaller than the covariance of (H

); to wit

E

;





;

¼ R

ð; 

Þg

ðq

; 

By these choices of covariances, one has that for x 2

[ 1, 1], y 2 [0, 1], since g is even and co nvex,

gðxÞþyg

ðyÞgðyÞxg

ðyÞ

It is an immediate consequence of Kahane’s theo-

rem, respectively the same interpolation argument

given above, that

EGðH

þ ÞEGðÞ

which translates into

EFðH

ÞEGðÞE

FðÞ

It is clear that we can optimize this bound by

choosing A, q, and P



. Of course, the difficulty

would be to find such a minimum. A first

simplification of this optimization problem is to

consider instead of the deterministic structure of P

and q random-probability measures on the space of

probability measures and quadratic forms on A,to

average over the preceding equation with respect to

their laws, and then take the infimum over all such

random structures. This gives a (still incalculable)

bound that Aizenman et al. (2003) have shown to be

asymptotically sharp, that is, they sh owed that

lim

N"1

EFðH

Þ¼lim

N"1

inf

A;



ðEGðÞE

FðÞÞ

where  is short for all probability measures on the

space of (P



, q

, 

)onA (called ‘‘random overlap

structures’’(rosts) in Aizenman et al. (2003)). Guerra’s

bound consists in restricting the infimum to a class of

rosts where the bound is calculable ‘explicitly’.

Maybe unsurprisingly, this is exactly the class of

asymptotic models that have already arisen in the

GREMs. In fact, we set A= [0, 1], M{m : [0, 1] !

[0, 1], non-decreasing}, let q be the random genealo-

gical distance associated to the family of measures 

and let P



be the probability measure on A whose

distribution function is 

(). Then Guerra’s bound

states that

lim

N"1

EFðH

Þlim

N"1

inf

m2M

EGðÞE

FðÞ

where the expectations relate to all random quan-

tities involved. By self-averaging, the same result

holds almost surely. The right-hand side of this

equation is known as (a particular formulation of)

the famous Parisi solution. In fact, define the

function f (q, y) as the solution of the nonlinear

partial differential equation

f þ

f þ m ðqÞð@

f Þ



¼ 0

with final conditions

f ð1; qÞ¼ln cosh y

Mean Field Spin Glasses and Neural Networks 411

These equations can be solved by elementary means

in the case when m is a step function. It turns out

that, for given m,

EGðÞE

FðÞÞ ¼ f ð0; h; m;Þ



qmðqÞdrð qÞ

where h = 

1

cosh

1



). This solution was origi-

nally obtained using the replica method. The preceding

construction gives, at the least, a clear mathematical

meaning to the objects involved. In particular, the

notion of ‘‘ultra-metric zero-dimensional matrices,’’

appears now to be equivalent to ultra-metric structures

on the unit interval.

In a recent pa per, Talagrand (2003) has proven

that converse inequality is also true in the preceding

equation, confirming that Parisi’s solution y ields the

correct free energy in a large class of models of the

SK type.

Ghirlanda–Guerra Relations

The appearance of a universal probabilistic structure

in the asymptotics of these models may appear

surprising. A partial explanation can be found in a

set of remarkable identities between multi-overlap

distributions that has been discovered first by

Ghirlanda and Guerra (1998) in the context of SK

models. If 

n

, N

denotes the n-fold product Gibbs

measure, the Ghirlanda–Guerra relations assert a

recursion relation of the form

E

nþ1

;N

ð

nþ1

;

ÞtjB



‘6¼k

E

n

;N

ð

‘

;

ÞtjB



E

2

;N

ð

;

ÞtjB



þ oð1Þ

These relations hold generically for Gaussian mean-

field models, with D

being the distance through

which the covariance is defined. The proof of these

relations is based on Gaussian integration-by-parts

formulas, and concentration of measure inequalities.

In the case of the GREM models, where D

is ultra-

metric, these recursions are sufficient to determine all

n-replica overlap distributions in terms of the 2-replica

distribution. On the other hand, the set of n-replica

overlap distributions determines the law of the process

K and thus the geometry of the Gibbs measure. In

particular, they leave time changes of Neveu’s process

as the only candidates for limit processes. In the case of

the SK models, the same does not hold a priori,since

the Hamming distance is not an ultra-metric. How-

ever, since the Parisi solution is correct, this suggests

very strongly that asymptotically the overlap distances

are almost surely (with respect to the Gibbs measure)

ultra-metric. Then, the Ghirlanda–Guerra identities

also imply that the geometry of the Gibbs measures is

described by the same structure.

From Mean-Field to Lattice Models

One of the widely discussed issues in the theory of spin

glasses is to what extent the results of mean-field

theory are relevant for lattice models. This issue has

been addressed elsewhere in this encyclopedia by

Newman and Stein. Here, we will only mention a

recent result of Franz and Toninelli (2004) that shows

that the free energy of the SK model can be represented

as the limit of the free energy of lattice models when

the range of the interaction tends to zero while their

strength tends to zero in an appropriate way (the so-

called Kac models). This still leaves open many finer

questions, but hints to the fact that mean-field theory

bears at least some relevance for realistic spin glasses.

See also: Short-Range Spin Glasses: The Metastate

Approach; Spin Glasses.

Further Reading

Aizenman M, Sims R, and Starr SL (2003) An extended

variational principle for the SK spin-glass model. Physical

Review B 6821: 4403.

Amit DJ (1989) Modelling Brain Function. Cambridge:

Cambridge University Press.

Bovier A (2001) Statistical Mechanics of Disordered Systems,

MaphySto Lecture Notes, Aarhus University, Aarhus.

Bovier A and Picco P (eds.) (1998) Mathematical Aspects of Spin

Glasses and Neural Networks, vol. 41. Boston: Birkha¨ user.

Ghirlanda S and Guerra F (1998) General properties of overlap

probability distributions in disordered spin systems. Towards

Parisi ultrametricity. Journal of Physics A 31: 9149–9155.

Guerra F (2002) Broken replica symmetry bounds in the mean

field spin glass model. Communications in Mathematical

Physics 233: 1–12.

Guerra F and Toninelli FL (2002) The thermodynamic limit in

mean field spin glass models. Communications in Mathema-

tical Physics 23: 71–79.

Me´zard, Parisi G, and Virasoro MA (1988) Spin Glass Theory

and Beyond. Singapore: World Scientific.

Ruelle D (1987) A mathematical reformulation of Derrida’s REM

and GREM. Communications in Mathematical Physics 108

(suppl. 2): 225–239.

Sherrington D and Kirkpatrick S (1972) Solvable model of a spin

glass. Physics Review Letters 35: 1792–1796.

Talagrand M (2003) Spin Glasses: A Challenge for Mathemati-

cians. Berlin: Springer.

Talagrand M (2003) The Parisi formula. Annals of Mathematics

(in press – 2005).

Franz S and Toninelli FL (2004) Finite-range spin glasses in the

Kac limit: free energy and local observables. Journal of Physics

A 37: 7433–7446.

412 Mean Field Spin Glasses and Neural Networks

Measure on Loop Spaces

H Airault, Universite

de Picardie, Amiens, France

Introduction

Loop spaces have been considered for their geo-

metric interest ( Freed Daniel 1988) where the space

of based loops on a compact Lie group is endowed

with a Ka¨ hlerian structure; see also the survey by

L Gross (1988). The harmonic analysis on loop

groups, developed by Pressley and Segal, is

reviewed by Hsu (1997). Loop groups have also

an impact in string theory (Bowick and Rajeev

1987). They are related to Yang–Mills theory (Levy

2003). A presentation of the history of measure on

infinite-dimensional spaces has been given by

P Malliavin (see Malliavin (1992) and references

therein). The main problem is the construction of

measures on the loop space which have quasi-

invariance property. This has implications in

representation the ory (Neretin 1994, Jones 1995).

Here we mainly concentrate on the nonlinear

stochastic point of view and its interference with

geometry. The geometrical study of the space of

closed curves over a compact Riemannian manifold

M, that is, the loop space over M, was ini tiated by

Marston Morse in 1932. The loop space is itself a

manifold where one can def ine a Laplace–Beltrami

operator. A diffusion process can be considered on

this manifold. Wiener defined the Brownian loop

by the Fourier series

uðÞ¼

k 1

sin k

½1

where the G

are independent normal variables.

The time evolution of the Wiener loop and the

extension of the theory to the case of a compact

Riemannian manifold of finite dimension has been

considered by Airault and Malliavin (1996,and

references therein). The Brownian loop evolutes in

thetimeparametert as a Brownian sheet where

the independent random variables G

are function

of t.

Starting from the zero loop, one obtains at time t,

a random loop, and the law of this loop gives a

measure on the loop space. A construction of this

measure with functional analysis on infinite-

dimensional manifold was done by Gaveau and

Mazet (1979). The tools of stochastic analysis are

important to the subje ct. The loop space of

continuous maps from the circle to the multi-

plicative group of complex numbers has a group

structure, hence the term ‘‘loop group.’’ On the loop

group, we consider the multiplicative Brownian

motion starting at one point of the circle and

conditioned to come back at this point at time s.It

defines a probability measure on the loop group.

One can also consider the set of continuous maps

from the circle to the set of complex numbers of

modulus equal to 1. The loop group is the space of

continuous closed paths on a Lie group. More

generally, on a Riemannian manifold M, the

Brownian motion on M defines a Wiener measure

on the loops over M. To go from the path space to

the loop space, an important tool is the quasisure

analysis in infinite dimension. The quasisure analysis

was developed by Airault and Malliavin (1996, and

references therein) to obtain disintegrations of the

Wiener measure and they have used this tool in

1992 to construct measures on the loop group. The

main problems are:

1. The construction of heat kernel measures and the

existence of a Brownian motion on the loop

space, the existenc e of pinned Wiener measures

obtained as the law of Brownian motions condi-

tioned on the loops.

2. The quasi-invariance of these transition prob-

ability measures under translation, or multi-

plication if we have a multiplicative structure, o r

under the infinitesimal action o f suitable vector

fields. For the path space over the n-dimensional

Euclidean space R

, the Cameron–Martin theo-

rem (1944) ensures the existence of a density

which shows the quasi-invariance of the Wiener

measure under translations. For the quasi-

invariance, an important fact is the choice of

the m etri c on the Cameron–Martin space. In the

case of the Wiener measure, one considers the

paths of finite energy,

(s)j

ds < þ1.This

corresponds to the metric ‘‘1.’’ P Malliavin

(1989, and references therein) discussed the

case of metrics  with 1=2 <<1.

3. To define the ‘‘good’’ Cameron subspace, that is,

find the vector fields that yield integration-

by-parts formulas. The question occurs whether

the Cameron–Martin space depends on time. For

the loop space, it has been proved by Driver

(2003) that it is not the case. A time evolution of

the tangent Cameron–Ma rtin space could appear

eventually.

4. The determination of the support of the measures

(e.g., the Wiener measure) is carried by the set of

Ho¨ lder functions of order 1=2  .

5. The absolute continuity of the measures with

respect to each other.

Measure on Loop Spaces 413

The Construction of Heat Measures

on the Loop Space and Their

Quasi-Invariance

The construction of measures giving a solution to

the infinite-dimensional heat equation as well as the

study of the quasi-invariance of the Wiener measure

on the path space was started extensively in the

work by Bismut, followed by Gross (1998), then by

Aida and Elw orthy (1995) where the loop group is a

suitable manifold to extend to infinite-dimensional

manifolds the log-Sobolev inequalities, by Malliavin

and Malliavin (1992, and references therein) where

the measures on the path space and the path group

have been studied. Consider a compact Lie group G

with unit e and let G be its Lie algebra. From the

G-valued Brownian motion, one can construct a

family of measures (

)

t 0

on the path space. These

measures 

are the images of the Wiener measure

on G through the Ito map

ðÞ¼

ﬃﬃ

ðÞdxðÞ with g

ð0Þ¼e ½2

The convolution of two measures 

and 

is equal

to 

tþt

. By choosing the initial value of the path

randomly distributed according to the Haar measure

on G, it defines a family of measures (

)

t 0

on the

path space with

f ðÞ

ðdÞ¼

f ðgÞ

ðdÞ

The Laplacian on the path group is defined by

ð

f ÞðgÞ¼lim

!0



f ðgÞ



ðdÞf ðgÞ



The heat equation is valid for the measures (

)

t 0

on the paths,

f ðgÞ

ðdgÞ¼

ð

f ÞðgÞ

ðdgÞ

Moreover, there is a quasi-invaria nce density k

(g)

defined on the path group (g

and g are paths with

values in G) such that



ðg

AÞ¼

ðgÞ

ðdgÞ

where g

A is the translated on the left of the subset

A in the path space over G. This is a generalization

to the path space of the classical Cameron–Martin

theorem. Then, one can consider the loop space. The

free loop space is the set of continuous maps g from

[0, 1] to G such that g(0) = g(1), and the loop space

with a base point is the set of maps such that

g(0) = g(1) = m is fixed. One can define the pinned

Brownian motion on the group G to obtain the

pinned Wiener measures (

)

t 0

on the loop group

(Malliavin and Malliavin 1992, Driver and

Srimurthy 2001). Denote by p

(g) the solution of

the heat equation on the group G.Letg be a map

from [0, 1] to the finite-dimensional Lie group G.For



, 

, ..., 

2 [0, 1], consider the evaluations of the

map g, g



, g



, ..., g



2 G,Letf be a real function

defined on G and denote by dg the Haar measure on

G. The measure 

on the loop group is given by

f ðg



; g



; ...; g



Þd

ðgÞ

f ðg

; g

; ...; g

Þp

t

ðg

Þp

tð



ðg

1

Þ

 p

tð



n1

ðg

1

n1

Þp

tð1

ðg

Þdg

 dg

From 

, one defines a measure 

on the free

loops by taking the mean over G as

f ðÞ

ðdÞ¼

f ðgÞ

ðdÞ

The qua si-invariance property for the pinned Wiener

measure was proved by Malliavin and Malliavin

(1992).

When the measures (

)

t 0

are obtained by

conditioning and quasisure analysis, we have heat

kernel measures. The case of heat kernel measures

defined on the loop group has been studied by

Airault and Malliavin by disinte grating the measures

on the path space and using the quasisure analysis.

The Laplacian on the loop group is defined as it has

been for the Laplacian on the path space,

ð

f ÞðgÞ¼lim

!0



f ðgg

Þ



ðdg

Þf ðgÞ



but now the heat equation has a Kac’s potential 

defined on the loops. On the loop group, the heat

equation is

f ðlÞ

ðdlÞ¼

ð

f ÞðlÞþ

ðlÞf ðlÞ½

ðdlÞ½3

where



ðlÞ¼

dlðsÞlðsÞ

1



2

log p

ðeÞ



dim G

The case of the circle, G = R=2Z, is interesting.

The law of the functional

dlðsÞlðsÞ

1

is given in Airault and Malliavin (1996,and

references therein). Moreover, the study of the heat

414 Measure on Loop Spaces

measures over the loop group of R=2Z brings new

identities on the classical Jacobi theta function

ðÞ¼1 þ 2

n 1

cosðnÞe

n

t=2

at  ¼ 0

Let

¼2

log p

ð0Þ

The following system of differential equations is

given by Airault–Malliavin (1996, and references

therein):

¼

n2Z

ðtÞ

ðtÞ¼

ðtÞþ

2

ðtÞ

To pass from path space to loop space, it is

convenient to use the ‘‘tubular chart’’ introduced

by Gross and the quasisure analysis developed by

Airault–Malliavin. Let  :  !(1)(0)

1

from the

path space to the group G; then the free loop

space over G is 

1

(e). There exists a neighbor-

hood V of the neutral of G such that 

1

(V)is

diffeomorphic to V  L(G), the product of V with

theloopspaceoverG. W ith this diffeomorphism,

one can disintegrate the measures on the path

space and obtain the measures on the loop space.

The Cameron–Martin formula on the path space

of the group G is obtained from the Cameron–

Martin formula for the Wiener space and the Ito’s

map. Let  be a differentiable path with finite

energy on G,thatis,

ðsÞ

1

ðsÞ



< þ1

it holds

f ðgÞ

ðdgÞ¼

f ðgÞk



ðgÞ

ðdgÞ

Let us denote by (j)

the Euclidean scalar product on

the Lie algebra G; then the density is given by



ðgÞ¼exp

ðsÞ

1

ðsÞjdgðsÞgðsÞ

1





ðsÞ

1

ðsÞ



The previous approach relies on the heat equation

on the loop space. Thus, the metric on the

Cameron–Martin loop or path sp ace is important.

The problem of quasi-invariance for metrics  with

1=2 <<1 relates to the random series



ðÞ¼

k 1

sin k



½4

where the G

are independent normal variables.

Driver (2003) solved the problem for 1=2 <<1

by Riemannian geometry in infinite dimension.

The Ricci curvature appears in the integration-by-

parts formulas on the loop space. The case of the

metric 1=2 is out of reach. Fang (1999) calculated

the Ricci curvature of the loop manifold for

metrics >1=2 and showed that when  !1=2,

these Ricci curvatures tend to a limit. Another

presentation of the problem is that of Pickrell

(1987), where he obtains a family of quasi-

invariant measures on Grassmannians.

Given a family of measures (

)

t 0

on the path

space of a Riemannian manifold, one defines a heat

operator as a family (L

)

t 0

of operators depending

upon t 2 [0, þ1[ such that

F d

½5

where F is a function defined on the path space. The

heat equation with a potential as [3] gives an

example of a heat operator. Heat operators have

been constructed for the path space over R

Airault–Malliavin, obtaining, after an integration by

parts on the path space, a heat operator of first

order. This introduces the notion of dilatation vector

fields on the path space. In the case of the flat

Wiener space, to each point x in the path space is

associated the dilatation vector field Y such that

(Yf )(x) = (xj(grad f )(x)). This gives a rescaling of the

Wiener measure under dilatations. This idea has

been exploited by Mancino (1999), who extended

the method to free loop groups.

Integration-by-Parts Formulas

The Cameron–Martin space plays the role of the

tangent space to the Wiener space. The integration-

by-parts formulas are an infinitesimal version of the

Cameron–Martin quasi-invariance property. Let G

be a compact Lie group or any product of R

by a

compact Lie group. For a vector field z, the

differentiation on the right @

right

and differentiation

on the left @

left

are given by

left

FðpÞ¼lim

!0

FðexpðzÞpÞFðpÞ



Measure on Loop Spaces 415

and

right

FðpÞ¼lim

!0

Fðp expðzÞÞ  FðpÞ



The operator @

right

commutes with the translation on

the left, for a translation 

left

, then @

right

(Fo

left

) =

right

F)o 

left

and vice versa for @

left

For the measures on the path space or loop space,

the problem is to prove the integration-by-parts

formulas. On the path spaces on G, let 

be the

Wiener measure on the set of paths starting from e,

there exists a density k

such that E[ exp (ck

)] is

finite and

ðGÞ

left

Fðg Þd

ðgÞ¼

ðGÞ

Fðg Þk

d

ðgÞ

The density k

is defined on the path space by

ðgÞ¼

<g ðtÞz

ðtÞgðtÞ

1

; d!ðtÞ>

This was proved by a number of authors (see, e.g.,

Pickrell (1987) and, in a geometrical context,

Cruzeiro and Malliavin (1996) ).

The existence of a density for the differentiation

on the left is valid for any Lie group. This is not true

for the differentiation on the right. If G is

noncompact or is not the product of R

by a

compact Lie group, the existence of k

is not proved

on the right. This comes from the fact that the map

Ad defined on the path group as a parallel trans port

does not preserve the Cameron–Martin subspace. In

the case where G is not a product of a flat space by

a compact Lie group, the Cameron space, which is a

kind of ‘‘tangent space’’ to the infinite-dimensional

loop manifold, is not closed under the Lie bracket of

vector fields.

The integration-by -parts formulas are obtained

with the stocha stic calculus of variation. On a group

G, consider Y

, Y

, ..., Y

, p independent left-

invariant vector fields. Let G be the Lie algebra of

G. The second-order differential operator 4=

j = 1

defines a left-invariant diffusion g

(t)on

the group G with the stochastic equation

(t) = g

(t)

)

o d!



where (!

) are inde-

pendent Brownian motions on the Euclidean space

G. In the work by Malliavin and Malliavin (1992,

and references therein), the stochastic calculus of

variation is done with the right-invariant connection

on the Lie group by setting



right

d

j¼0

ðg

!þh

Þog

1

where h is a differentiable function of t with

values in the Lie algebra G, with finite energy

(s)j

ds < þ1. By taking the derivative with

respect to  in the Stratonovitch equation



ðtÞ

1

o dg



ðtÞ¼d!ðtÞþh

ðtÞdt

and letting  = 0, it turns out that 

right

is a differenti-

able function of t and its derivative is given by



right

ðtÞ¼g

ðtÞh

ðtÞg

ðtÞ

1

The situation is not the same for



left

d

j¼0

1

oðg

!þh

where d

left

(t) is a stochastic differential. This

generalizes to an arbitrary Riemannian manifold

using a coupling of connections (see Airault and

Malliavin (1996), and references therein). The

construction of the appropriate Cameron subspace,

that is, the choice of the infinitesimal action of

vector fields on the measure, is of im portance. In the

commutative case of the path space over R

, the

classical Cameron–Martin subspace of paths h such

that

(s)j

ds < þ1 is time invariant. To define

the vector fields acting on the path (or loop) space

over M , it is necessary to consider the geometry of

the manifold M. The infinitesimal transformations

which preserve the Riemannian metric are called

Riemannian connections. In the case where M is a

group, the natural connections are those defined by

the parallelism on the group. For a Riemannian

manifold, Driver proved the existence of integration-

by-parts formulas for the measures on the path

space of M when M is endowed with a torsion skew-

symmetric connection. The Levi-Civita connection,

since it is torsionless, is of course a Driver (2003)

connection. If the connection is not skew-symmetric,

then two coupled connections permit study of the

-variation or ‘‘reduced variation’’ of a path, and one

obtains a Cameron–Martin formula on the path and

on the loop space of the Riemannian manifold M

(Fang 1999). The method of reduced variation can be

used to obtain the integration-by-parts formulas over

path and loop spaces. Another approach to the quasi-

invariance problem, using two-parameter processes,

has been provided by Norris (1995).

The Support of the Measures and

Absolute Continuity with Respect

to Each Other

Given a Riemannian manifold M,let(

)

be the heat

kernel measures on the path space of M and let (

)

be heat kernel measures on the loop space of M;the

question arises whether 

is absolutely continuous

416 Measure on Loop Spaces