A connection between the Ghirlanda--Guerra identities and ultrametricity

0.26 MB·English

by  Dmitry Panchenko

TheAnnalsofProbability
2010,Vol.38,No.1,327–347
DOI:10.1214/09-AOP484
(cid:13)c InstituteofMathematicalStatistics,2010
A CONNECTION BETWEEN THE GHIRLANDA–GUERRA
1
IDENTITIES AND ULTRAMETRICITY
0
1
0 By Dmitry Panchenko
2
Texas A&M University
n
a We consider a symmetric positive definite weakly exchangeable
J
infiniterandom matrixandshowthat,underthetechnicalcondition

7 thatitselementstakeafinitenumberofvalues,theGhirlanda–Guerra
2
identities imply ultrametricity.

]
R
1. Introduction and main result. Let us consider an infinite random
P
matrix R = (R ) which is symmetric, nonnegative definite [in the
l,l′ l,l′≥1
.
h sense that (R ) is nonnegative definite for any n 1] and weakly
l,l′ 1≤l,l′≤n
t ≥
a exchangeable, which means that for any n 1 and any permutation ρ
m of 1,...,n , the matrix (R ) h≥as the same distribution as
ρ(l),ρ(l′) 1≤l,l′≤n
{ }
[ (Rl,l′)1≤l,l′≤n. Following [6], we will call the matrix with such properties
a Gram–de Finetti matrix. We assume that diagonal elements R =1 and
4 l,l
v nondiagonal elements take only a finite number of values,
3
4 (1.1) P(R =q )=m m
1,2 l l+1 l
7 −
0 for 1 l k and for some 1 q <q < <q 1 and 0=m < <
1 2 k 1
0. mk <≤mk+≤1 =1. We say tha−t th≤e matrix R··s·atisfies≤the Ghirlanda–Gu·e·r·ra
1 identities [7]if,foranyn 2,anyboundedmeasurablefunctionsf:Rn(n−1)/2
8 R and ψ:R R, ≥ →
0 →
: n
v 1 1
(1.2) Ef ψ(R )= Ef Eψ(R )+ Ef ψ(R ),
i n 1,n+1 n 1,2 n 1,l
X n n
l=2
X
r
a wheref =f((R ) ).Inotherwords,conditionallyon(R ) ,
n l,l′ 1≤l<l′≤n l,l′ 1≤l<l′≤n
the law of R is given by the mixture n−1 (R )+n−1 n δ . By
1,n+1 L 1,2 l=2 R1,l
the positivity principle proven by Talagrand (Theorem 6.6.2 in [14], see also
P
Received March 2009.
1Supportedin part by NSFgrant DMS-0904565.
AMS 2000 subject classifications. 60K35, 82B44.
Key words and phrases. Sherrington–Kirkpatrick model, Parisi ultrametricity conjec-
ture.
This is an electronic reprint of the original article published by the
Institute of Mathematical Statistics in The Annals of Probability,
2010,Vol. 38, No. 1, 327–347. This reprint differs from the original in pagination
and typographic detail.
1
2 D. PANCHENKO
[9]), the Ghirlanda–Guerra identities imply that R 0 with probability
1,2
≥
1 and, therefore, from now on, we can assume that q 0. However, this a
1
≥
priori assumption is not necessary since it will also be clear from the proof
that q must be nonnegative. The main result of the paper is the following.
1
Theorem 1. Under assumptions (1.1) and (1.2), the matrix R is ultra-
metric, that is,
(1.3) P(R min(R ,R ))=1.
2,3 1,2 1,3
≥
Another way to express the event in (1.3) is to say that
(1.4) R q ,R q = R q for all 1 l k.
1,2 l 1,3 l 2,3 l
≥ ≥ ⇒ ≥ ≤ ≤
This question of ultrametricity originates in the setting of the Sherrington–
Kirkpatrick model [13], where R corresponds to the matrix of the overlaps,
or scalar products, of i.i.d. replicas from a random Gibbs measure. The ul-
trametricity property (1.3) was famously predicted by Parisi in [8] as a part
of complete description of the expected behavior of the model and it still
remains an open mathematical problem.On the other hand,the Ghirlanda–
Guerra identities, which are implicitly contained in the Parisi theory, were
proven rigorously in [7] in some approximate sense; namely, one can slightly
perturb the parameters of the model such that, on average over the pertur-
bation (or for some specific choice of perturbed parameters, see [15]), the
Ghirlanda–Guerra identities hold in the thermodynamic limit. In this pa-
per, we consider an asymptotic distribution of the overlap matrix for which
the Ghirlanda–Guerra identities hold precisely, as in (1.2), and show that
they automatically imply ultrametricity, under a technical condition (1.1).
In some sense, the main idea of the paper is nothing but a reversal of the
proof of the Ghirlanda–Guerra identities, which arise from the information
providedbythe“stochasticstability”ofthesystemwithregardtosmallper-
turbations of the parameters. Our main technical contribution, the invari-
anceprincipleofTheorem4,isaveryspecificformofthestochasticstability
of thesystem implied by theGhirlanda–Guerra identities. TheParisi theory
for the Sherrington–Kirkpatrick model also predicts that the distribution of
the overlap R has a nontrivial continuous component, so the assumption
1,2
(1.1) is rather restrictive. Nevertheless, Theorem 1 provides some hope that
the Ghirlanda–Guerra identities imply ultrametricity in the general case as
well.
This work was motivated by a paper of Arguin and Aizenman [3] and,
in particular, by a beautiful application of the Dovbysh–Sudakov represen-
tation theorem [6] for Gram–de Finetti matrices that will play the same
crucial role here. In [3], the authors prove ultrametricity in a slightly differ-
ent setting as a consequence of what they call the “robustquasi-stationarity
GHIRLANDA–GUERRAIDENTITIES ANDULTRAMETRICITY 3
property.” The main idea in [3] utilizes the robust quasi-stationarity in or-
der to prove “quasi-stationarity under free evolution” at each step of the
inductive argument, which, in turn, implies weak exchangeability and, via
the application of the Dovbysh–Sudakov representation, induces clustering
of the type (1.4). Our proof is based on exactly the same idea. The differ-
ence now is that quasi-stationarity under free evolution—the invariance or
stochastic stability property of Theorem 4 below—will be a consequence of
the Ghirlanda–Guerra identities and, of course, the induction in the proof
of Theorem 1 will be different since it is also based on (1.2). In addition, we
give a new proof in Theorem 3 below that the invariance implies exchange-
ability, which is based on the explicit control of the mixing induced by the
random permutation in the invariance principle.
Simultaneously with the present work, Talagrand developed a different
approach to Theorem 1 in [15] based on the Ghirlanda–Guerra identities
and a form of invariance. Theorem 4 below shows that sufficient invariance
is already contained in the Ghirlanda–Guerra identities and one can now
find a new more direct proof of Theorem 4 in [15]. In addition, [15] clar-
ifies the physicists’ idea of decomposing the system into pure states and
explains how both the Ghirlanda–Guerra identities and invariance arise in
the Sherrington–Kirkpatrick model.
The rest of the paper is organized as follows. In the next section, we
start with the Dovbysh–Sudakov representation result which shows that
any Gram–de Finetti matrix R can begenerated by i.i.d. replicas from some
random Gibbs measure on a separable Hilbert space, which is called the
directing measure ofR,inalmostexactlythesamewayastheoverlapmatrix
is generated by the Gibbs measure in the Sherrington–Kirkpatrick model.
We first study some basic properties of the directing measure which follow
from the Ghirlanda–Guerra identities; namely, that it always concentrates
on a nonrandom sphere of the Hilbert space and is either continuous or
discrete with probability 1. In particular, it is discrete when (1.1) holds.
In Section 3, we formulate the invariance and exchangeability properties of
the configuration of the directing measure and use them to prove Theorem
1 by induction on k. Finally, in Section 4, we show how the Ghirlanda–
Guerra identities imply the invariance of the directing measure and how the
invariance implies weak exchangeability.
2. BasicconsequencesofGGIandexchangeability. Sincealloftheprop-
erties of the matrix R=(R ) considered above—symmetry, positive
l,l′ l,l′≥1
definiteness, weak exchangeability, satisfying the Ghirlanda–Guerra iden-
tities (GGI)—were expressed in terms of its finite-dimensional distribu-
tions, we can think of R as a random element in the product space M =
[ 1,1] with the pointwise convergence topology and the Borel σ-
1≤l,l′ −
algebra . Let denote the set of all probability measures on . Suppose
Q M P M
4 D. PANCHENKO
that P is such that for all A ,
∈P ∈M
(2.1) P(A)= Q(ω,A)dPr(ω),
ZΩ
where Q:Ω [0,1] is a probability kernel from some probability space
×M→
(Ω, ,Pr) to M such that: (a) Q(ω, ) for all ω Ω; (b) Q(,A) is mea-
F · ∈P ∈ ·
surable on for all A . In this case, we will say that P is a mixture of
F ∈M
laws Q(ω, ). Under (2.1), we can write the expectation of any measurable
·
P-integrable function φ:M R as
→
(2.2) Eφ(R)= φ(R)Q(ω,dR) dPr(ω).
ZΩ(cid:18)ZM (cid:19)
We will say that a law Q of a Gram–de Finetti matrix is generated by
∈P
an i.i.d. sample if there exists a probability measure η on H [0, ), where
× ∞
H is a separable Hilbert space, such that Q is the law of
(2.3) (x x +a δ ) ,
l l′ l l,l′ l,l′≥1
·
where(x ,a) is ani.i.d.sequencefrom η and x y denotes thescalar product
l l
·
on H. The analysis of the distribution of R will utilize the following repre-
sentation result for Gram–de Finetti matrices due to Dovbysh and Sudakov
[6].
Proposition 1. A law P of any Gram–de Finetti matrix is a mix-
∈P
ture (2.1) of laws in such that for all ω Ω, Q(ω, ) is generated by an
P ∈ ·
i.i.d. sample.
We will denote by η a probability measure on H [0, ) corresponding
ω
× ∞
to Q(ω, ) and let µ be the marginal of η on H. Following the terminology
ω ω
·
of Aldous [2], we will call µ the directing measure of the matrix (R ).
ω l,l′
The main result of this section shows that the Ghirlanda–Guerra identities
imply the following basic geometric properties of the directing measure.
Theorem 2. Let R be a Gram–de Finetti matrix on M that satisfies the
Ghirlanda–Guerra identities (1.2) and let F be the law of R . If q∗ is the
1,2
largest point of the support of F, then, for Pr-almost all ω: (a) µ ( x 2 =
ω
k k
q∗)=1; (b) µ is continuous if F( q∗ )=0; (c) µ is discrete if F( q∗ )>
ω ω
{ } { }
0, in which case the sequence of weights of µ has the Poisson–Dirichlet
ω
distribution PD(1 F( q∗ )).
− { }
We recall that, given s (0,1), if (u ) is the decreasing enumeration
l l≥1
∈
of a Poisson point process on (0, ) with intensity measure x−1−sdx and
∞
w = u / u , then the distribution of the sequence (w ) is called the
l l j j l
Poisson–Dirichlet distribution PD(s). If s=0, then we define PD(0) to be
P
thetrivial distribution with w =1. Theproofwillbebased on thefollowing
1
consequence of the Ghirlanda–Guerra identities.
GHIRLANDA–GUERRAIDENTITIES ANDULTRAMETRICITY 5
Lemma 1. Consider a measurable set A [ 1,1]. For Pr-almost all ω:
⊂ −
(a) if F(A)>0, then for µ -almost all x , µ (x :x x A)>0;
ω 1 ω 2 1 2
· ∈
(b) if F(A)=0, then for µ -almost all x , µ (x :x x A)=0.
ω 1 ω 2 1 2
· ∈
Proof. (a) Let m=F(Ac)<1. Then, by the Ghirlanda–Guerra iden-
tities (1.2),
P( 2 l n+1,R Ac)=EI( 2 l n,R Ac)I(R Ac)
1,l 1,l 1,n+1
∀ ≤ ≤ ∈ ∀ ≤ ≤ ∈ ∈
n 1+m
= − P( 2 l n,R Ac)
1,l
n ∀ ≤ ≤ ∈
(n 1+m) (1+m)m
(by induction on n)= − ···
n!
m(1+m) m m
= 1+ 1+
n 2 ··· n 1
(cid:18) (cid:19) (cid:18) − (cid:19)
m(1+m) m(1+m)
(using 1+x ex) emlogn= .
≤ ≤ n n1−m
Since m<1, letting n + gives P( 2 l,R Ac)=0. By Proposition
1,l
→ ∞ ∀ ≤ ∈
1, P is a mixture (2.1) of measures generated by an i.i.d. sample from η
ω
and we can write
P( 2 l,R Ac)= µ⊗(l≥1)( 2 l,x x Ac)dPr(ω)=0,
∀ ≤ 1,l∈ ω ∀ ≤ 1· l∈
ZΩ
which implies that for Pr-almost all ω, by Fubini’s theorem,
µ⊗(l≥2)((x ) : l 2,x x Ac)dµ (x )=0.
ω l l≥2 ∀ ≥ 1· 2∈ ω 1
Z
This implies that for µ -almost all x ,
ω 1
µ⊗(l≥2)((x ) : l 2,x x Ac)=µ (x :x x Ac)∞=0,
ω l l≥2 ∀ ≥ 1· 2∈ ω 2 1· 2∈
which means that µ (x :x x Ac)<1 and, therefore, µ (x :x x
ω 2 1 2 ω 2 1 2
· ∈ · ∈
A)>0. To prove part (b), it is enough to express P(R A)=0 using
1,2
Proposition 1. (cid:3) ∈
ProofofTheorem2. (a)SinceF([ 1,q∗])=1andF([q∗ n−1,q∗])>0
− −
for all n 1, Lemma 1 implies that for Pr-almost all ω, for µ -almost all
ω
≥
x ,
1
µ (x :x x q∗)=1, µ (x :x x q∗ n−1)>0
ω 2 1 2 ω 2 1 2
· ≤ · ≥ −
(2.4)
n 1.
∀ ≥
Letusfixanysuchω.First,theequalityin(2.4)impliesthatµ ( x 2 q∗)=1.
ω
k k ≤
Otherwise, there exists h H with h 2 >q∗ such that µ (B (h))>0 for
ω ε
∈ k k
6 D. PANCHENKO
any ε>0, where B (h) is the ball of radius ε>0 centered at h. Taking
ε
ε> 0 small enough so that x x > q∗ for all x ,x B (h) contradicts
1 2 1 2 ε
· ∈
the first equality in (2.4). Next, let us show that µ ( x 2 <q∗)=0. Oth-
ω
k k
erwise, there again exists an open ball B (h) x: x 2 <q∗ such that
ε
⊂ { k k }
µ (B (h))>0. For some δ>0, x 2 <q∗ δ for all x B (h) and, there-
ω ε ε
k k − ∈
fore,forallx B (h)andx x 2 q∗ ,wehavex x < q∗(q∗ δ)
1 ε 2 1 2
∈ ∈{k k ≤ } · − ≤
q∗ n−1forlargeenoughn 1.Sincewehavealreadyproventhatµ ( x 2
ω
− ≥ p k k ≤
q∗)=1, this contradicts the second inequality in (2.4). We have proven that
µ ( x 2=q∗)=1.
ω
k k
(b) If F( q∗ )=0, then, by Lemma 1 for Pr-almost all ω, for µ -almost
ω
{ }
all x , we have µ (x :x x =q∗)=0. Therefore, for Pr-almost all ω for
1 ω 2 1 2
·
which also µ ( x 2=q∗)=1, µ must be continuous.
ω ω
k k
(c) If F( q∗ )=1, then, by Lemma 1 for Pr-almost all ω, for µ -almost
ω
{ }
all x , we have µ (x :x x =q∗)=1. By part (a), µ ( x 2=q∗)=1 and,
1 ω 2 1 2 ω
· k k
therefore,µ mustbeconcentrated ononepoint.IfF( q∗ ) (0,1),thenfor
ω
{ } ∈
Pr-almost all ω, for µ -almost all x , we have µ (x :x x =q∗)>0 and
ω 1 ω 2 1 2
·
sinceµ ( x 2=q∗)=1,wegetthatforµ -almostallx ,µ ( x )>0.This
ω ω 1 ω 1
k k { }
provesthatµ isdiscrete.Let(w )bethesequenceofweightsofµ arranged
ω l ω
in decreasing order [we keep the dependence of (w ) on ω implicit]. The fact
l
that(w )hasPD(1 F( q∗ )) distributionwillfollowfromtheanalogofthe
l
− { }
Ghirlanda–GuerraidentitiesforthePoisson–Dirichletdistributionprovenby
Talagrand in Chapter 1 of [14]. Let us explain how this result applies in our
setting. Let us take any m 1 and n ,...,n 1 and for n=n + +n ,
1 m 1 m
≥ ≥ ···
consider a function f on M which is the indicator of the set
R =q∗:n + +n +1 l=l′ n + +n +n ,0 p m 1 .
l,l′ 1 p 1 p p+1
{ ··· ≤ 6 ≤ ··· ≤ ≤ − }
Let us express the expectation Ef using (2.2) and write the inside integral
in terms of the weights (w ) of the directing measure µ . Since, by part
l ω
(a), µ ( x 2 =q∗)=1 and R =x x =q∗ only if x =x when x 2=
ω l,l′ l l′ 1 2 1
k k · k k
x 2=q∗, by Fubini’s theorem,
2
k k
(2.5) f(R)Q(ω,dR)= wn1 wn2 wnm.
l l ··· l
Z l≥1 l≥1 l≥1
X X X
Similarly, if we take ψ(x)=I(x=q∗), then
(2.6) fψ(R )Q(ω,dR)= wn1+1 wn2 wnm,
1,n+1 l l ··· l
Z l≥1 l≥1 l≥1
X X X
and we have fψ(R )=f for 2 j n and
1,j 1
≤ ≤
(2.7) fψ(R )Q(ω,dR)= wn1+np wnp−1 wnp+1 wnm
1,j l ··· l l ··· l
Z l≥1 l≥1 l≥1 l≥1
X X X X
GHIRLANDA–GUERRAIDENTITIES ANDULTRAMETRICITY 7
for n +1 j n when n + +n +1 j n + +n +n . If we let
1 1 p 1 p p+1
≤ ≤ ··· ≤ ≤ ···
S(n ,...,n )=E wn1 wn2 wnm
1 m l l ··· l
l≥1 l≥1 l≥1
X X X
and s=1 F( q∗ ), then plugging (2.5), (2.6) and (2.7) into (1.2) implies
− { }
that
n s
1
S(n +1,n ,...,n )= − S(n ,...,n )
1 2 m 1 m
n
m
n
p
+ S(n +n ,...,n ,n ,...,n ).
1 p p−1 p+1 m
n
p=2
X
This coincides with equation (1.52) in [14]. It is explained there that this
equation can be used recursively to compute S(n ,...,n ) in terms of s
1 m
∈
(0,1) only and that the set of numbers S(n ,...,n ) uniquely determines
1 m
the distribution of the weights (w ) as the Poisson–Dirichlet distribution
l l≥1
PD(s). (cid:3)
3. Ultrametricity inthediscretecase. Inthecasewhen(1.1)holds,The-
orem 2 implies that for Pr-almost all ω, the directing measure µ is discrete
ω
and concentrated on the sphere of radius √q , that is,
k
(3.1) µ = w δ
ω l ξ(l)
l≥1
X
for some distinct sequence ξ(l) H with ξ(l) 2=q and w w >0.
k 1 2
∈ k k ≥ ≥···
In particular, by (2.3), this implies that a nondiagonal element R =x
l,l′ l
·
x =q if and only if x =x whichproves theultrametricity at the last level
l′ k l l′
k,
(3.2) R q , R q = R q .
1,2 k 1,3 k 2,3 k
≥ ≥ ⇒ ≥
Thisisalreadyaseriousachievement, based,inadditionto(1.1),onlyonthe
weak exchangeability of the matrix R via the application of the Dovbysh–
Sudakov representation. Bringing this idea to light was one of the main
contributions of [3]. The description of the directing measure µ provided
ω
by (3.1) corresponds to the pure states picture in physics, where each point
ξ(l) can be thought of as a “pure state” of the asymptotic Gibbs measure.
One of the crucial results in the alternative approach of Talagrand in [15]
is a very general construction of pure states for measures in Hilbert spaces
which provides a way around the representation results for exchangeable
arrays that we are relying on here.
Since we will hereafter deal with the discrete directing measure (3.1), let
usintroducesomenotation thatwillbemoreconvenient throughouttherest
8 D. PANCHENKO
of the paper. We will keep the dependence of the directing measure µ on
ω
ω implicit and write E for the integration in ω. To describe an i.i.d. sample
from measure µ in (3.1), consider i.i.d. random variables
(3.3) σ1,σ2,... N
∈
that take any value l 1 with probability w , which is the weight corre-
l
≥
sponding to the index l in the directing measure (3.1). Let us denote by
h·i
the expectation in these random indices for a given measure µ, that is, for
any n 1 and a function h:Nn R,
≥ →
(3.4) h = h(σ1,...,σn) = h(l ,...,l )w w .
h i h i 1 n l1··· ln
l1X,...,ln
By the configuration of µ, we will understand the weights and configuration
of its atoms,
(3.5) w=(w ) and =(ξ(l) ξ(l′)) .
l l≥1 R · l,l′≥1
Since ξ(σl) are i.i.d. from distribution µ, Proposition 1 can be rephrased
by saying that the nondiagonal elements of the matrix R can be generated
by first generating a random measure µ [or its configuration (3.5)], then
sampling indices (3.3) and setting
(3.6) R =ξ(σl) ξ(σl′)= .
l,l′ · Rσl,σl′
The Ghirlanda–Guerra identities (1.2) can be rewritten as
n
1 1
(3.7) E f ψ(R ) = E f E ψ(R ) + E f ψ(R )
n 1,n+1 n 1,2 n 1,l
h i n h i h i n h i
l=2
X
and (1.1) can be rewritten as
(3.8) E I(R =q ) =m m for 1 l k.
1,2 l l+1 l
h i − ≤ ≤
Thefactthatin(3.1),all ξ(l) 2=q impliesthatin(2.3),alla =1 q and,
k l k
k k −
therefore, we can safely omit the term a δ in (2.3) and redefinethe matrix
l l,l′
R by R =ξ(σl) ξ(σl′) for all l,l′ 1 so that, from now on, the diagonal
l,l′
· ≥
elements are equal to q . As mentioned in the Introduction, as in [3], the
k
crucial step which will allow us to use the Dovbysh–Sudakov representation
(2.3) in order to make the induction step in the proof of Theorem 1 is the
following.
Theorem3. Thematrix in(3.5)isweaklyexchangeable conditionally
R
on w=(w ).
l
GHIRLANDA–GUERRAIDENTITIES ANDULTRAMETRICITY 9
Of course, this means that is also weakly exchangeable uncondition-
R
ally, which is how it will be used in the proof of Theorem 1, but the proof of
the stronger statement of conditional exchangeability is exactly the same.
The proof of Theorem 3 will be based on a certain invariance property
of the joint distribution of w and —quasi-stationarity under free evolu-
R
tion in the terminology of [3]—which, in our setting, will follow from the
Ghirlanda–Guerra identities. Consider i.i.d. Rademacher random variables
(ε ) independent of the measure µ. Given t 0, consider a new sequence
l l≥1
≥
of weights
w etεl
(3.9) wt= l ,
l p≥1wpetεp
defined by a random change of denPsity proportional to etεl. Of course, these
weights are not necessarily decreasing anymore, so let us denote by (wπ) the
l
weights (wt) arranged in decreasing order and let π:N N bethe permuta-
l →
tion keeping track of where each index came from, wπ =wt . Let us define
l π(l)
by
(3.10) µπ = wπδ and π =(ξ(π(l)) ξ(π(l′)))
l ξ(π(l)) R · l,l′≥1
l≥1
X
theprobabilitymeasureµafterthechangeofdensityproportionaltoetεl and
thematrix rearrangedaccordingtothereorderingofweights.Analogously
R
to (3.4), let us denote by h the average
n π
h i
(3.11) h = h(l ,...,l )wπ wπ
h iπ 1 n l1··· ln
l1X,...,ln
and let E now denote the expectation in the randomness of the measure
µ and the Rademacher sequence (ε ). Theorem 3 is a consequence of the
l
following invariance principle.
Theorem 4. For t<1/2, we have (wπ, π)=D (w, ).
R R
This result can be expressed by saying that the directing measure µ is
stochastically stable under the change of density (3.9) and is a nontrivial
consequence of the Ghirlanda–Guerra identities; in fact, as mentioned in
the Introduction, this is, in some sense, a reversal of the usual derivation of
the Ghirlanda–Guerra identities. There is a very important technical reason
why we use Rademacher instead of the more obvious Gaussian change of
density (as in [3]) that will become clear from the proof. Of course, as will
be shown in the proof of Theorem 3, Theorem 4 for Rademacher change of
density (3.9) implies the same result for Gaussian change of density as well.
10 D. PANCHENKO
Let us now show how Theorem 3 can be used to prove ultrametricity by
induction on k.
ProofofTheorem1. GivenaGram–deFinettimatrixR,weconsider
the configuration matrix of its directing measure defined in (3.5), which
R
is symmetric and nonnegative definite.By (3.1), =q and by (1.1), with
l,l k
R
probability 1, nondiagonal elements
(3.12) q ,...,q .
l,l′ 1 k−1
R ∈{ }
By Theorem 3, is weakly exchangeable and, thus, is a Gram–de Finetti
R
matrix. Therefore, using Proposition 1, there exists a random probability
measure η′ on H [0, ) such that
× ∞
D
(3.13) ( )=(y y +b δ ),
l,l′ l l′ l l,l′
R ·
where (y ,b ) is an i.i.d. sequence from the distribution η′. Let us now
l l
show that (3.12) implies that if µ′ is the marginal of η′ on H, then µ′ =
w′δ for some distinct sequence ξ′(l) H, w′ w′ >0, and,
l≥1 l ξ′(l) ∈ 1 ≥ 2 ≥···
moreover,
P
(3.14) ξ′(l) ξ′(l′), ξ′(l) 2 q ,...,q .
1 k−1
· k k ∈{ }
Indeed,ifapointy belongstothesupportofµ′,inthesensethatµ′(B (y))>
ε
0 for all ε>0, then in the i.i.d. sequence (y ) from this distribution, there
l
will be infinitely many elements from B (y). Since, by (3.12), the scalar
ε
product y y of these elements belongs to q ,...,q with probability
l l′ 1 k−1
· { }
1, letting ε 0 proves that y 2 q ,...,q . If z is another point in
1 k−1
→ k k ∈{ }
the support of µ′ such that z B (y), then z 2 q ,...,q and, there-
ε 1 k−1
∈ k k ∈{ }
fore, z 2= y 2 if ε is small enough. The same argument also proves that
k k k k
y z q ,...,q , which implies that y =z if ε is small enough. This
1 k−1
· ∈{ }
proves that the measure µ′ is discrete and (3.14) holds. [Remark: Note that
(3.14) guarantees that q 0 and by a forthcoming induction, one can
k−1
≥
similarly conclude that all q 0, which means that we did not need to
l
≥
invoke Talagrand’s positivity principle and assume that q 0.]
1
≥
Let us now explain the induction step. If r− = min(r,q ), then the
k−1
truncated matrix R− =(R− ) is symmetric, weakly exchangeable and, ob-
l,l′
viously, automatically satisfies the Ghirlanda–Guerra identities (1.2). Also,
since R = and R− = − , R− is nonnegative definite whenever
l,l′ Rσl,σl′ l,l′ Rσl,σl′
− is, and the fact that − is nonnegative definite can be seen as follows.
R R
By (3.13), (3.14) and since =q by (3.1), we get
l,l k
R
(3.15) ( − )=D (y y +(q y 2)δ ).
Rl,l′ l· l′ k−1−k lk l,l′
Since, by (3.14), y 2 q , the right-hand side is obviously nonnegative
l k−1
k k ≤
definite. This implies that − and, therefore, R− are nonnegative definite
R