Probabilistic Methods for Discrete Nonlinear Schrödinger Equations

Probabilistic Methods for Discrete NonlinearSchrödinger Equations

SOURAV CHATTERJEENew York University

KAY KIRKPATRICKNew York University

Abstract

We show that the thermodynamics of the focusing cubic discrete nonlinear Schrö-

dinger equation are exactly solvable in dimension 3 and higher. A number of

explicit formulas are derived. © 2011 Wiley Periodicals, Inc.

1 IntroductionA complex-valued function u of two variables x and t , where x 2 Rd is the

space variable and t 2 R is the time variable, is said to satisfy a d -dimensional

nonlinear Schrödinger equation (NLS) if

i@tu D ��uC �jujp�1u;

where � is the Laplacian operator in Rd , p is the nonlinearity parameter, and � is

a parameter that is either C1 or �1. The case of interest in this article is p D 3 and

� D �1, called the “focusing cubic NLS.” The focusing cubic NLS is an equation

of interest in nonlinear optics, condensed matter physics, and a number of other

areas [2, 13, 14, 15, 19, 31, 34, 35].

The questions of local and global well-posedness of nonlinear Schrödinger equa-

tions are still not completely understood. Local existence results under restrictive

conditions on the initial data have been established in low dimensions [16, 18],

while ill-posedness results are known in higher dimensions [11]. For a survey of

the literature and further references, see the recent monograph of Raphaël [28].

One approach to this problem is via the method of invariant measures for the

NLS flow, initiated by Lebowitz, Rose, and Speer [21] and developed by McKean

and Vaninsky [23, 24, 25] and Bourgain [3, 4, 5, 6]. Invariant measures coupled

with Bourgain’s development of the so-called Xs;b spaces (“Bourgain spaces”) for

constructing global solutions has led to important advances in this field. A striking

recent development is the work of Tzvetkov [33], who used invariant measures and

Bourgain’s method to construct global solutions of certain nonlinear Schrödinger

Communications on Pure and Applied Mathematics, Vol. LXV, 0727–0757 (2012)

© 2011 Wiley Periodicals, Inc.

728 S. CHATTERJEE AND K. KIRKPATRICK

equations for random initial data. The technique was further developed by Burq

and Tzvetkov [9, 10] and Oh et al. [12, 27].

Most of the above works use invariant measures as a tool for proving local or

global well-posedness for various classes of initial data. However, the nature of the

invariant measures themselves have not been so well studied. Studying the nature

of the invariant measures may yield important information about the long-term be-

havior of these systems. The only results we know in this direction are those of

Brydges and Slade [7] in d D 2 and Rider [29, 30] in d D 1. Some progress for

invariant measures of the KdV equation has been made recently in [26]. In this arti-

cle we investigate the case of the discrete NLS in three and higher dimensions. Two

problems that immediately arise are: (a) the construction of the invariant measure

for the cubic NLS due to Lebowitz, Rose, and Speer [21] does not give a meaning-

ful probability distribution when d � 3, and (b) local and global well-posedness

are not well-understood in d � 3. To overcome these (very difficult) hurdles,

we drastically simplify the situation by discretizing space and considering the so-

called discrete NLS. Several well-posedness results under general conditions on

the initial data are known for the discrete system [32, 34], and the existence of the

natural invariant Gibbs measure is straightforward.

In return for this simplification of the problem, we give a large amount of refined

information about the nature of the invariant measure. In particular:

� We “solve” the system “exactly” in the sense of statistical mechanics by

computing the limit of the log partition function.

� Analysis of the partition function yields a first-order phase transition; we

identify the exact point of transition.

� We prove the existence of so-called localized modes (also called discrete

breathers [15, 22, 34]) in functions drawn from the invariant measures and

compute the size of these modes.

� Additionally, we show that the localized mode persists at one site for an

exponentially long time.

The results are stated in Section 2, and the proofs are presented in later sections.

The proofs involve elementary probabilistic arguments.

Incidentally, we do not know how to take our results for the discrete NLS to

some kind of a “continuum limit” as the grid size goes to 0. Neither do we know

how to derive conclusions about the long-term behavior of solutions of the discrete

NLS from our results about the nature of the invariant measures. As of now, these

are open problems that may well be unsolvable.

2 Setup and Statements of Main ResultsIt is of interest to study the nonlinear Schrödinger equation on graphs [1, 8], and

this is the setting of our results.

Let G D .V;E/ be a finite, undirected graph without self-loops. Let D be the

maximum degree ofG, and let n D jV j be the size of the graph. Let h be a positive

PROBABILISTIC METHODS FOR NLS 729

real number denoting the distance between two neighboring vertices inG. WhenG

is a part of a lattice, h is called the lattice spacing (e.g., in [34]).

For example, when G is a discrete approximation of the d -dimensional unit

torus Œ0; 1�d represented as f0; 1=L; 2=L; : : : ; .L � 1/=Lgd , then n D Ld and

h D 1=L. In particular, n ! 1 as L ! 1; moreover, if d � 3, nh2 also tends

to 1 as L ! 1. This last condition will be crucial for us. There is nothing

special about the torus. The condition nh2 ! 1 should hold for any nice enough

compact manifold of dimension � 3. The condition is also satisfied if instead of

the torus we take a cube whose width is increasing to infinity as the grid size 1=L

tends to 0 (since h D 1=L and n � Ld in this scenario), which can be viewed as

a discrete approximation to the whole of Rd .

The condition nh2 ! 1 is satisfied in one and two dimensions only if the

domain that is approximated also tends to an infinite size. For example, in one

dimension, if the grid size is h D 1=L for some large L (tending to infinity), then

the condition nh2 ! 1 holds on an interval Œ�K;K� only if K grows to infinity

with L so fast that K.L/=L ! 1.

The discrete nearest-neighbor Laplacian on G is defined as

z�fx WD 1

h2

Xy�x

.fy � fx/;

where y � x denotes the sum over all neighbors of x and f D .fx/x2V is any

map from V into C. Note that the scaling by h2 is meant to ensure, at least in the

case of the d -dimensional torus, that the discrete Laplacian converges to the true

Laplacian as the grid size goes to 0.

The discrete cubic NLS on G with the discrete nearest-neighbor Laplacian z� is

a family of coupled ODEs with fx D fx.t/:

id

dtfx D � z�fx C �jfxj2fx; x 2 V;

where � may be C1 or �1. These are known as the focusing and defocusing

equations, respectively. The object of interest in this paper, as in Lebowitz, Rose,

and Speer [21] and Bourgain [3], is the focusing NLS:

(2.1) id

dtfx D � z�fx � jfxj2fx :

The discrete Hamiltonian associated with the focusing NLS (2.1) is

(2.2) H.f / WD 2

n

X.x;y/2E

ˇˇfx � fy

h

ˇˇ2

� 1

n

Xx2V

jfxj4:

Up to scaling by a constant, this is simply the discrete analogue of the continuous

Hamiltonian considered in [3, 21, 23].

Let the power be N.f / WD Px2V jfxj2; the mass is n�1N.f /, but we will

use these terms interchangeably. Then we have the conservation of mass under the


dynamics, using standard finite-difference techniques [20]:

d

dtN.f / D 0;

as well as conservation of energy:

d

dtH.f / D 0:

Hence by the Liouville theorem, the measure d� WD e�ˇH.f /Q

x2V dfx is invari-

ant under the dynamics of the discrete NLS (2.1) for any real ˇ. However, this

measure has infinite mass if ˇ > 0.

The problem is easily solved by a mass cutoff as in [3, 21] (allowed due to

conservation of mass) and normalization. The resulting probability measure

(2.3) d z� WD Z�1e�ˇH.f /1fN.f /�BngYx2V

dfx

continues to be invariant under the NLS dynamics. Here B is an arbitrary positive

cutoff, and Z is the normalizing constant (partition function). Of course, both z�and Z depend on the pair .ˇ; B/. Let be a random element of CV with law z�.

That is, is a random function on V such that for each A � CV ,

P . 2 A/ D Z�1

ZA

e�ˇH.f /1fN.f /�Bngdf;

where df D Qx dfx denotes the Lebesgue differential element on CV . Our ob-

jective is to understand the behavior of the random map . The first step is to

understand the partition function Z. The first theorem below shows that if we have

a sequence of graphs with n and nh2 both tending to infinity, the limit of n�1 logZ

can be exactly computed for any positive ˇ and B .

The result can be roughly stated as follows. Letm W Œ2;1/ ! R be the function

(2.4) m.�/ WD �

2� 1

2C �

2

r1 � 2

�C log

�1

2� 1

2

r1 � 2

�

�:

It may be easily verified that m is strictly increasing in Œ2;1/, m.2/ < 0, and

m.3/ > 0. Thus, m has a unique real zero that we call �c . Numerically, �c �2:455407. Let

(2.5) F.ˇ;B/ WD(

log.B�e/ if ˇB2 � �c ;

log.B�e/Cm.ˇB2/ if ˇB2 > �c :

(Figure 2.1 shows a graph of F versus ˇ when B D 1.) Theorem 2.1 below asserts

that if ˇ > 0 and we have a sequence of graphs such that n ! 1 and nh2 ! 1(as in the d -dimensional torus for d � 3), all other parameters remaining fixed,

then

limn!1

logZ

nD F.ˇ;B/:


F

β

2.144...

c =2.455...θ

FIGURE 2.1. The free energy is constant for small inverse temperature

and starts increasing at the critical threshold. Here mass is normalized,

B D 1.

The theorem also gives an explicit rate of convergence.

THEOREM 2.1. Suppose ˇ � 0. Take any � 2 .0; 15/. There exists a positive

constant C depending only on �, ˇ, B , h, and D such that if n > C , then

logZ

n� F.ˇ;B/ � �Cn�1=5 � C.nh2/�1

and

logZ

n� F.ˇ;B/ �

(Cn�1=5C� C Cn�4�=5 if ˇB2 � �c ;

Cn�1=5C� if ˇB2 > �c :

The behavior of the random map is the subject of the next theorem. It turns

out that the behavior is quite different in the two regimes ˇB2 < �c and ˇB2 > �c .

To roughly describe this phase transition, letM1. / andM2. / denote the largest

and second-largest components of the vector .j xj2/x2V . It turns out that when

ˇB2 > �c , there is high probability that M1. / � an and M2. / D o.n/, where

(2.6) a D a.ˇ;B/ WD B

2C B

2

s1 � 2

ˇB2:

In other words, when ˇB2 > �c , there is a single x where x takes an abnormally

large value and is relatively small at all other locations. Moreover, N. / � Bn

with high probability. A consequence is that the largest component carries more

than half of the total mass:

maxx

j xj2Py j y j2 � a

B>1

2:


β

cθ

1

a

FIGURE 2.2. The fraction of mass at the heaviest site jumps from

roughly 0 for small inverse temperature, to roughly :71 at the critical

threshold. (Here B D 1.)

On the other hand, when ˇB2 < �c , then M1. / D o.n/, but still N. / � Bn.

Consequently,

maxx

j xj2Py j y j2 � 0:

(Figure 2.2 shows the graph of the fraction of mass a at the heaviest site versus

ˇ when B D 1.) When ˇB2 > �c , the energy density H. /=n is strictly nega-

tive and approximately equals �a2, whereas in the regime ˇB2 < �c , the energy

density is close to 0. The formula for a shows that a does not tend to 0 as ˇB2

approaches �c from above (in fact, it stays bigger than B=2), demonstrating a first-

order phase transition. These results are detailed in the following theorem:

THEOREM 2.2. Suppose h D n�p for some p 2 .0; 12/. Let a D a.ˇ;B/ be de-

fined as in (2.6), and let M1. / and M2. / be the largest and second-largestcomponents of .j xj2/x2V . First, suppose ˇB2 > �c . Take any q such thatmaxf2p; 4

5g < q < 1. Then there is a constant C depending only on ˇ, B , D,

p, and q such that if n > C , then with probability � 1 � e�nq=C ,

(2.7)

ˇˇH. /n

C a2

ˇˇ � Cn�.1�q/=4;

ˇˇN. /n

� Bˇˇ � Cn�.1�q/=2;ˇ

ˇM1. /

n� a

ˇˇ � Cn�.1�q/=4;

M2. /

n� Cn�.1�q/:

Next, suppose ˇB2 < �c . Take any q satisfying maxf.1 C 2p/=2; 17=18g <q < 1. Then there is a constant C depending only on ˇ, B , D, p, and q such that


whenever n > C , with probability � 1 � e�nq=C ,

(2.8)

ˇˇH. /n

ˇˇ � 2n�2.1�q/;

ˇˇN. /n

� Bˇˇ � n�.1�q/;

andM1. /

n� n�.1�q/:

Finally, if ˇB2 D �c and q is any number satisfying maxf.1 C 2p/=2; 17=18g <q < 1, then there is a constant C depending only on ˇ, B , D, p, and q such thatwhenever n > C , with probability � 1 � e�nq=C , either (2.7) or (2.8) holds.

An obvious shortcoming of Theorem 2.2 is that it does not give a precise de-

scription of the critical case ˇB2 D �c . It is important to know whether (2.7)

or (2.8) is more likely, and how much. Another deficiency of both Theorem 2.1

and Theorem 2.2 is that they say nothing about the thermodynamics of the two-

dimensional cubic NLS. A substantial amount of information is known about the

two-dimensional continuous system (see, e.g., [28] and references therein), but

precise calculations along the lines of Theorems 2.1 and 2.2 would be desirable.

Additionally, it would be nice to be able to extend the theory to other nonlinearities

than cubic.

Theorem 2.2 says that when ˇB2 > �c , there is a single site x 2 V which bears

a sizable fraction of the total mass of the random wave function . This fraction

is nearly deterministic, given by the ratio a=B . Theorem 2.2 also implies that this

exceptional site bears nearly all of the energy of the system. This is because the

total energyH. / is approximately �a2n, while the energy at x is, summing over

just the neighbors y of x,

1

nh2

Xy�x

jfx � fy j2 � jfxj4n

� �n�1M1. /2 CO.h�2/

D �a2nC o.n/;

the equality by Theorem 2.2. Such a site is sometimes called a localized mode.

It easily follows as a corollary of this theorem that typical discrete wave func-

tions above the critical threshold have divergent discrete H 1 norm:

kf k2zH 1

WD 1

n

Xx2V

jfxj2 C 1

n

X.x;y/2E

ˇˇfx � fy

h

ˇˇ2

:

However, it is not so clear that the discrete H 1 norm diverges even if ˇB2 � �c .

The following theorem shows that the divergence happens on the discrete torus in

dimensions � 3 for all values of ˇ and B .

THEOREM 2.3. Suppose the context of Theorem 2.2 holds, with h D n�p for somep 2 .0; 1

2/ and n > C . Let D . x/x2V be a discrete wave function picked

randomly from the invariant probability measure z� defined in (2.3). If ˇB2 > �c ,then there is a positive constant c depending only on ˇ, B , D, and p such thatP .k k zH 1 � cnp/ � e�nc

whenever n � 1=c. On the other hand, if ˇB2 � �c ,


then the same result holds with a small modification: P .k k zH 1 � cpınp/ �

e�ınc

, where ı is the average vertex degree.

Since the measure z� of (2.3) is invariant for the discrete NLS (2.1), one may

expect from the above discussion that if the initial data comes from z�, localized

modes will continue to exist as time progresses. The question is whether the same

site continues to be a mode for a long time (in which case we have a “standing”

or “stationary” wave with a localized mode, sometimes called a discrete breather),

or not. The following theorem shows that indeed, the same site continues to be

the localized mode for an exponentially long period of time. This is an example

of a dynamical result deduced from a theorem about the statistical equilibrium. Of

course, we need to use the NLS equation (2.1) for some basic dynamical informa-

tion at one point in the proof. (As a side note, let us mention that global-in-time

solutions of (2.1) are known to exist [32, 34].)

THEOREM 2.4. Suppose h D n�p for some p 2 .0; 12/ and ˇB2 > �c . Let a

be defined as in (2.6). Let .t/ D . x.t//x2V be a discrete wave functionevolving according to (2.1), where the initial data .0/ is picked randomly fromthe invariant probability measure z� defined in (2.3). Choose any q such thatmaxf2p; 4

5g < q < 1. Then there is a constant C depending only on ˇ, B , D,

p, and q such that if n > C , then with probability � 1 � e�nq=C the inequalities(2.7) hold for .t/ for all 0 � t � enq=C , and moreover there is a single x 2 V

such that the maximum of j y.t/j is attained at y D x for all 0 � t � enq=C . Inparticular, .t/ is approximately a standing wave with localized mode at x for anexponentially long time.

The above theorem proves, in particular, the existence and typicality of solutions

of (2.1) that have unique stable localized modes for exponentially long times if the

initial energy or mass are above a threshold. One key difference between this

theorem and earlier results about existence of discrete breathers (e.g., [34]) is that

the earlier results could prove the existence of localized modes only if the mass

was very large, i.e., tending to infinity, while Theorem 2.4 proves it under finite

mass and energy. Another difference is that it shows the typicality, rather than

mere existence, of a breather solution.

Our final theorem investigates the probability distribution of the individual co-

ordinates of a random map picked from the measure z�ˇ;B . It turns out that it’s

possible to give a rather precise description of the distribution for small collec-

tions of coordinates. If ˇB2 < �c , then for any x1; : : : ; xk 2 V , under a certain

symmetry assumption on G, the joint distribution of B�1=2. x1; : : : ; xk

/ is ap-

proximately that of a standard complex Gaussian vector provided k is sufficiently

small compared to n. When ˇB2 > �c , the same result holds, but for the vector

.B � a/1=2. x1; : : : ; xk

/ where a is defined in (2.6).

The symmetry assumption on G is as follows. Assume that there exists a group

† of automorphisms of G such that:


(1) j†j D n.

(2) No element of † except the identity has any fixed point.

When these conditions hold, we say that G is translatable by the group †. For

example, the discrete torus is translatable by the group of translations. Note that a

translatable graph is necessarily transitive.

THEOREM 2.5. Suppose the graph G is translatable by some group of automor-phisms according to the above definition. Suppose h D n�p for some p 2 .0; 1

2/,

and let be a random wave function picked according to the measure z�. Take anyk distinct points x1; : : : ; xk 2 V . Let � D .�1; : : : ; �k/ be a vector of i.i.d. stan-dard complex Gaussian random variables. If ˇB2 < �c , then there is a constantC > 0 depending only on ˇ, B , D, and p such that if n > C , then for all Borelsets U � Ck ,

jP .B�1=2. x1; : : : ; xk

/ 2 U/ � P .� 2 U/j � kn�1=C :

If ˇB2 > �c , the result holds after B�1=2 is replaced with .B � a/�1=2 wherea D a.ˇ;B/ is defined in (2.6), and the error bound is changed to k3n�1=C .

Obviously, it would be nice to have a similar result for ˇB2 D �c , but our

current methods do not yield such a result.

When ˇB2 > �c , the reader may wonder how x can behave like a complex

Gaussian variable with second moment B � a, when Theorem 2.2 and symmetry

among the coordinates seem to imply Ej xj2 � B . It is exactly the peaked nature

of the field in the case ˇB2 > �c that allows for a convergence in law without that

of the second moment.

The rest of the paper is devoted to the proofs of the theorems of this section.

Some preliminary lemmas are proved in Section 3. Theorem 2.1 is proved in

Section 4, Theorem 2.2 in Section 5, Theorem 2.3 in Section 6, Theorem 2.4 in

Section 7, and Theorem 2.5 in Section 8.

3 Preliminary LemmasFor the rest of this article, C will denote any positive function of .�; ˇ; B; h;D/

whose explicit form is suppressed for the sake of brevity. The value of C may

change from line to line, and usually C will be called a constant instead of a func-

tion. When h D n�p, C will depend on p instead of h.

Some further conventions: Sums without delimiters will stand for sums over all

x 2 V . For each f 2 CV and each k � 2, let Sk.f / WD P jfxjk . In particular,

N.f / D S2.f /. Define

Z0 WDZ

CV

eˇn�1S4.f /1fS2.f /�Bngdf


If S2.f / � Bn, then

0 �X

.x;y/2E

jfx � fy j2 �X

.x;y/2E

.2jfxj2 C 2jfy j2/ � 4BDn:

Thus, we have the important bounds

(3.1) e�C h�2

Z0 � Z � Z0;which precipitate the irrelevance of the kinetic term in the Hamiltonian (although

this is not obvious a priori). For each 0 < a < b, define

a;b WD ˚f 2 CV W a2n2.1 � n�1=5/ � S4.f / � a2n2.1C n�1=5/;

bn.1 � n�1=5/ � S2.f / � bn.1C n�1=5/�:

Define the function

(3.2) L.a; b/ WD ˇa2 C log.b � a/C log� C 1:

It will be shown in Lemma 3.6 that, in fact, F.ˇ;B/ D sup0�a<b�B L.a; b/.

LEMMA 3.1. For any 0 < a < b � B=.1C n�1=5/,

Z � a exp

�nL.a; b/ � Cn4=5

b � a � C

h2

�:

PROOF. Fix 0 < a < b � B=.1C n�1=5/. Let � D .�x/x2V be a collection of

i.i.d. complex Gaussian random variables with probability density function

1

�.b � a/ e�j´j2=.b�a/:

Note that � can also be viewed as a random mapping from V into C. Now,

P .� 2 a;b/ DZ

�a;b

1

�n.b � a/n e� S2.f /

b�a df

� 1

�n.b � a/n exp

�� bn

b � a.1 � n�1=5/

�Vol.a;b/:

Consequently,

(3.3) Vol.a;b/ � exp

�bn

b � a.1 � n�1=5/C n log�.b � a/�

P .� 2 a;b/:

Fix an element o 2 V . Define the sets

E1 WD ff W an.1 � n�1=4/ � jfoj2 � an.1C n�1=4/g;E2 WD ff W max

x¤ojfxj4 � a2n3=4g;

E3 WDnf W .b � a/n.1 � n�1=4/ �

Xx¤o

jfxj2 � .b � a/n.1C n�1=4/o:


Suppose f 2 E1 \E2 \E3. Since f 2 E1 and f 2 E3,

bn.1 � n�1=4/ � S2.f / � bn.1C n�1=4/:

Again since f 2 E1 and f 2 E2, if n � C ,

S4.f / � a2n2.1C n�1=4/2 C a2n7=4 � a2n2.1C n�1=5/;

and similarly

S4.f / � a2n2.1 � n�1=5/:

Thus if n � C , then E1 \E2 \E3 � a;b .

For any x 2 V , the real and imaginary parts of �x are i.i.d. Gaussian with

mean 0 and variance .b � a/=2. Thus, j�xj2 is an exponential random variable

with mean b � a. The inequality e�u � e�v � .v � u/e�v that holds for v � u

gives

P .� 2 E1/ � 2an3=4

b � a exp

��an.1C n�1=4/

b � a�:

Further, note that by a simple union bound

P .� 62 E2/ � .n � 1/e�an3=8=.b�a/;

and by Chebyshev’s inequality

P .� 62 E3/ � .b � a/2.n � 1/..b � a/.n3=4 � 1//2 � Cn�1=2:

Thus, if n � C , then P .� 2 E2\E3/ � 12

. Lastly, observe that the event f� 2 E1gis independent of f� 2 E2 \E3g. Combining these observations, we see that when

n � C ,

P .� 2 a;b/ � P .� 2 E1/P .� 2 E2 \E3/

� a exp

�� an

b � a � Cn3=4

b � a�:

(3.4)

This inequality, together with (3.3), gives that if n � C ,

Vol.a;b/ � a.e�.b � a//ne�C n4=5=.b�a/:

Since b � B=.1C n�1=5/, a;b � ff W S2.f / � Bng. Thus, if n � C ,

Z0 �Z

�a;b

eˇn�1S4.f /df

� eˇa2n.1�n�1=5/ Vol.a;b/

� a.e1Cˇa2

�.b � a//ne�C n4=5=.b�a/ D aenL.a;b/�C n4=5=.b�a/:

This lower bound and (3.1) complete the proof of the lemma. �


LEMMA 3.2. For any n�2�=5 < a < b � B ,

Z0a;b WD

Z�a;b

eˇn�1S4.f /df � exp

�nL.a; b/C Cn4=5C�

b � a�:

PROOF. Fix n�2�=5 < a < b � B . Let � be the random vector defined in the

proof of Lemma 3.1. Let

0a;b WD

[o2V

˚f 2 a;b W an.1 � 3n�1=5C�/ � jfoj2 � an.1C n�1=5C�/;

maxx¤o

jfxj2 � 4an4=5C��:

Recall that for any o, j�oj2 is an exponential random variable with mean b � a.

Thus,

P .� 2 0a;b/ �

Xo2V

P .j�oj2 � an.1 � 3n�1=5C�//

D n exp

�� an

b � a C Cn4=5C�

b � a�:

(3.5)

Next, define

A1 WD ˚f W jfx W jfxj > n.1��/=5gj > n.4C2�/=5

�;

A2 WDnf W 9U � V; jU j � n.4C2�/=5;

Xx2U

jfxj2 � an.1C n�1=5C�/o:

We claim that if n > a�5=2� (which is true by assumption), then

a;b � 0a;b [ A1 [ A2:(3.6)

To see this, take any f 2 Ac1 \ Ac

2 \ a;b . Let

U WD fx W jfxj > n.1��/=5g:Since f 2 Ac

1, jU j � n.4C2�/=5. Therefore, since f 2 Ac2,

(3.7)Xx2U

jfxj2 < an.1C n�1=5C�/:

Again since f 2 Ac2, we also have

(3.8) maxx

jfxj2 < an.1C n�1=5C�/:

Note that since n > a�5=2�,Xx 62U

jfxj4 DX

x W jfx j�n.1��/=5

jfxj4 � n1C4.1��/=5 < a2n9=5:

Since f 2 a;b , Xx

jfxj4 � a2n2.1 � n�1=5/:


Thus, Xx2U

jfxj4 DX

x

jfxj4 �Xx 62U

jfxj4

� a2n2.1 � n�1=5 � n�1=5/ D a2n2.1 � 2n�1=5/:

On the other hand by (3.7),Xx2U

jfxj4 � .maxx

jfxj2/Xx2U

jfxj2

� .maxx

jfxj2/an.1C n�1=5C�/:

Combining the last two displays implies that

maxx

jfxj2 � a2n2.1 � 2n�1=5/

an.1C n�1=5C�/

� an.1 � 3n�1=5C�/:

Together with (3.8), this shows that

(3.9) an.1 � 3n�1=5C�/ � maxx

jfxj2 � an.1C n�1=5C�/:

Next, let o be a vertex at which f attains its maximum modulus. Let x be any

other vertex. If x 62 U , then since n > a�5=2�, jfxj2 � n2.1��/=5 < an2=5 �an4=5C�. If x 2 U , then by (3.7),

jfoj2 C jfxj2 �Xy2U

jfy j2 � an.1C n�1=5C�/;

and therefore by (3.9),

jfxj2 � an.1C n�1=5C�/ � an.1 � 3n�1=5C�/ D 4an4=5C�:

Hence f 2 0a;b

from the above display and (3.9), and the claim (3.6) follows.

Consequently, if n > a�5=2�,

(3.10) P .� 2 a;b/ � P .� 2 0a;b/C P .� 2 A1/C P .� 2 A2/:

If � 2 A1, then there is a set U � V such that jU j D dn.4C2�/=5e and j�xj >n.1��/=5 for all x 2 U . Therefore, a union bound over all possible U gives

P .� 2 A1/ ��

n

dn.4C2�/=5e� �e�n2.1��/=5=.b�a/

�n.4C2�/=5

� exp

�Cn.4C2�/=5 logn � n6=5

b � a�:

(3.11)


Take any U � V and let j WD jU j. Then .b � a/�1P

x2U j�xj2 is the sum of j

i.i.d. exponential random variables with mean 1. Thus, for any t > 2,

P�X

x2U

j�xj2 � .b � a/t�

DZ 1

t

xj �1

.j � 1/Š e�x dx

D e�t

Z 1

0

.x C t /j �1


� e�t

Z 1

0

2j �2.xj �1 C tj �1/


� e�t .2j �2 C tj �1/ � tj e�t :

On the other hand, for 0 � t � 2, the bound Ce�t works. If � 2 A2, then there

exists U � V with jU j D bn.4C2�/=5c andP

x2U jfxj2 � an.1 C n�1=5C�/.

Thus, with t D .b � a/�1an.1 C n�1=5C�/ and j D bn.4C2�/=5c, the above

inequality, the assumption that a > n�2�=5, and a union bound over all possible U

show that if n > C ,

P .� 2 A2/ ��

n

bn.4C2�/=5c� �

Cn

b � a�n.4C2�/=5

exp

��an.1C n�1=5C�/

b � a�

� exp

�Cn.4C2�/=5 log

n

b � a � an.1C n�1=5C�/

b � a�

� exp

�Cn.4C2�/=5 log

n

b � a � an

b � a � n.4C3�/=5

b � a�

� exp

��n

.4C3�/=5

2.b � a/ � an

b � a�:(3.12)

Together with (3.5), (3.10), and (3.11), this shows that if n > C (and n > a�5=2�),

then

P .� 2 a;b/ � exp

�Cn4=5C�

b � a � an

b � a�:

Now, similar to the beginning of Lemma 3.1,

P .� 2 a;b/ DZ

�a;b

1

�n.b � a/n e� S2.f /

b�a df

� 1

�n.b � a/n e� bn

b�a.1Cn�1=5/ Vol.a;b/:


Therefore, if n > a�5=2�,

Vol.a;b/ � exp

�bn

b � a.1C n�1=5/C n log�.b � a/�

P .� 2 a;b/

� .e�.b � a//neC n4=5C�=.b�a/:

(3.13)

Finally, observe that

Z0a;b D

Z�a;b

eˇn�1S4.f /df � eˇa2n.1Cn�1=5/ Vol.a;b/:

This completes the proof of the lemma. �

LEMMA 3.3. For each n � 1 and r > 0, let

Un;r WD ff 2 Cn W S2.f / � rng:Then Vol.Un;r/ � .r�e/n D enL.0;r/.

PROOF. Note that

1 DZ

Cn

e�S2.f /=r

.�r/ndf

�Z

Un;r

e�S2.f /=r

.�r/ndf � 1

.r�e/nVol.Un;r/:

(The last inequality holds because S2.f / � rn on Un;r .) �

LEMMA 3.4. For each f 2 CV , let a.f / WD pS4.f /=n and b.f / WD S2.f /=n.

Note that a � b. For each c > 0 let

Wc WD ff 2 CV W b.f / � B; b.f / � a.f / � cg:Then Z

Wc

eˇn�1S4.f / df � e.C Clog c/n:

PROOF. Take any f 2 Wc . Let a D a.f / and b D b.f /. Then

maxx

jfxj2 �P

x jfxj4Px jfxj2 D a2n

b:

Thus, if o is a vertex at which jf j is maximized, then

Xx¤o

jfxj2 � b2 � a2

bn � 2.b � a/n:


Let c0 WD cn=.n � 1/. The above inequality and Lemma 3.3 show that

Vol.Wc/ �Xo2V

Vol�nf W jfoj2 � Bn;

Xx¤o

jfxj2 � 2cno�

D B�n2 Vol.Un�1;2c0/

� B�n2.2c0�e/n�1:

Since ZWc

eˇn�1S4.f /df � eˇB2n Vol.Wc/;

this completes the proof. �

LEMMA 3.5. For each a > 0, let

Va WD ff W b.f / � B; a.f / � ag:Then Z

Va

eˇn�1S4.f /df � eˇa2nCL.0;B/n:

PROOF. Simply note thatZVa

eˇn�1S4.f /df � eˇa2n Vol.Un;B/;

and apply Lemma 3.3. �

LEMMA 3.6. For each � � 0, let f� W Œ0; 1/ ! R be the function

f� .x/ WD �x2 C log.1 � x/:Then there is a �c > 0 such that if � < �c , f� has a unique maximum at x D 0,whereas if � > �c , then f� has a unique maximum at the point

x�.�/ WD 1

2C 1

2

r1 � 2

�> 0:

When � D �c , f� attains its maximum at two points, one at 0 and the otherat x�.�c/. The number �c is the unique real solution of

�

2� 1

2C �

2

r1 � 2

�C log

�1

2� 1

2

r1 � 2

�

�D 0:

Numerically, �c � 2:455407. Lastly, if F.ˇ;B/ is the constant defined in (2.5),then

F.ˇ;B/ D sup0�a<b�B

L.a; b/ D sup0�a 0

for sufficiently large � . Moreover, M� is an nondecreasing function of � . Let

�c WD inff� W M� > 0g D supf� W M� D 0g:If � < �c , then we claim that f� has a unique maximum at 0. It is clear that 0 is

a point of maximum. To show that it is unique, suppose not. Then there is some

x > 0 where f� .x/ D 0. Then for a � 0 2 .�; �c/, f� 0.x/ > 0, contradicting the

definition of �c .

When � > �c , it is clear that the maximum must be attained at a nonzero point.

To show that it is unique, observe that

f 0� .x/ D 2�x � 1

1 � x ; f 00� .x/ D 2� � 1

.1 � x/2 ;

and therefore there may be at most two points in .0; 1/ where f 0�

vanishes, and

exactly one of them can be a maximum. Solving the quadratic equation shows

that the maximum is attained at x�.�/ (and also shows that �c � 2). From the

formula, it is clear that when � ! �c , the limit of x�.�/ exists and is positive.

By uniform convergence on compact intervals, it follows that f�c.x�.�c// D 0.

The equation f�c.x�.�c// D 0 is equivalent to the defining equation for �c . It is a

unique solution of the equation because the left-hand side can easily be shown to

be a strictly increasing function of � in Œ2;1/.

Finally, note that for any 0 � a �c . �

4 Proof of Theorem 2.1 (Partition Function)Let f� be defined as in Lemma 3.6. Recall equation (3.14), which implies that

L.a;B/ is maximized at Bx�.a=B/. Let us call this point a�. (The point a� is

called a.ˇ;B/ in the statement of Theorem 2.2.) Note that L.a; b/ is maximized

at .a�; B/, since L.a; b/ � L.a;B/ for any 0 � a 0 when ˇB2 > �c .

Moreover, when ˇB2 ¤ �c , a� is the unique point of maximum.

4.1 Supercritical Partition FunctionFirst, consider the case ˇB2 > �c , so that a� > 0. By Lemma 3.1, it follows

that if n > C ,

logZ

n� L.a�; B=.1C n�1=5// � Cn�1=5 � C

nh2:


But L has bounded derivatives in a neighborhood of .a�; B/. Thus, if n > C ,

logZ

n� L.a�; B/ � Cn�1=5 � C

nh2:(4.1)

Since L.a�; B/ > L.0; B/, there exists a0 D a0.ˇ; B/ small enough such that

ˇa20 C L.0;B/ < L.a�; B/ � ı;

where

ı WD 1

2.L.a�; B/ � L.0;B//:

Therefore by Lemma 3.5,ZVa0

eˇn�1S4.f /df � en.L.a�;B/�ı/:(4.2)

Since log c ! �1 as c ! 0, there exists c0 D c0.ˇ; B/ > 0 such that

C C log c0 < L.a�; B/ � ı;

where C is the constant in Lemma 3.4. Therefore by Lemma 3.4,ZWc0

eˇn�1S4.f /df � en.L.a�;B/�ı/:(4.3)

Now take any f such that b.f / � B . (Recall the definitions of a.f / and b.f /

from the statement of Lemma 3.4.) Let A be a finite collection of .a; b/ such that[.a;b/2A

a;b ff W b.f / � B; a.f / > a0; b.f / � a.f / > c0g:

It is easy to see that if n > C , then A can be chosen such that jAj � Cn2=5, and for

each .a; b/ 2 A, a > a0=2 and b � a > c0=2. Let us choose such a collection A.

Then by Lemma 3.2, for each .a; b/ 2 A,

Z0a;b � enL.a;b/CC n4=5C�

:(4.4)

Now, if f 62 Va0and f 62 Wc0

, then a.f / > a0 and b.f /�a.f / > c0. Therefore

by (4.2), (4.3), and (4.4), it follows that if n > C ,

Z0 �Z

Va0[Wc0

eˇn�1S4.f /df CX

.a;b/2AZ0

a;b

� 2en.L.a�;B/�ı/ C jAj max.a;b/2A

en.L.a;b/CC n�1=5C�/

� Cn2=5en.L.a�;B/CC n�1=5C�/:

A combination of the above inequality, (4.1), (3.1), and Lemma 3.6 proves the

conclusion of Theorem 2.1 when ˇB2 > �c .


4.2 Subcritical Partition FunctionNext, consider the case ˇB2 � �c , when L.a; b/ is maximized at .0; B/. (The

point of maximum is not unique when ˇB2 D �c , but that will not matter in the

proof.) Taking a D n�1=5 and b D B=.1C n�1=5/ in Lemma 3.1, it follows that

logZ

n� L.n�1=5; B=.1C n�1=5// � Cn�1=5 � C

nh2� C logn

n:

Since L has bounded derivatives in a neighborhood of .0; B/, this shows that if

n > C ,

logZ

n� L.0;B/ � Cn�1=5 � C

nh2:(4.5)

By Lemma 3.4, a constant c1 D c1.�; ˇ; B/ > 0 can be chosen so small that when

n > C , ZWc1

eˇn�1S4.f /df � e12

nL.0;B/:(4.6)

Lemma 3.2 shows that whenever n�2�=5 < a c1,

logZ0a;b

n� L.a; b/C Cn�1=5C� � L.0;B/C Cn�1=5C�:(4.7)

Let a1 WD n�2�=5. Let A be a set of .a; b/ such that[.a;b/2A

a;b ff W b.f / � B; b.f / � a.f / > c1; a.f / > 2a1g:

From the definition of a;b , it should be clear that A can be chosen such that

jAj � nC , and for each .a; b/ 2 A, a > a1 and b � a > c1=2. Therefore, (4.7)

holds for every .a; b/ 2 A. Thus, from (4.6), (4.7), and Lemma 3.5, we get

Z0 �Z

V2a1[Wc1

eˇn�1S4.f /df CX

.a;b/2AZ0

a;b

� e4ˇa21nCL.0;B/n C e

12

L.0;B/n C jAjeL.0;B/nCC n4=5C�

:

Since jAj � nC and a21n D n1�4�=5, the proof for the case ˇB2 � �c is complete.

5 Proof of Theorem 2.2 (Gibbs Measure)Recall that h D n�p in this theorem, where p 2 .0; 1

2/. In this proof, whenever

we say “for all .a; b/ satisfying . . . ,” it will mean “for all .a; b/, where 0 � a �c . Choose q such that maxf2p; 4

5g < q < 1.

Choose � satisfying 45

C � D q. Note that � 2 .0; 15/, as required. Let r WD

.1 C q/=2. Let a�, a0, c0, ı, a.f /, and b.f / be as in the proof of Theorem 2.1.

Let

A1 WD ff W b.f / 2n�.1�r/=2g \ V ca0

\W cc0:

Let A1 be a collection of .a; b/ such that 0 � a a0=2, b � a > c0=2, and either b n�.1�r/=2

(or both). With these properties of A1, two conclusions can be drawn. First, by

Lemma 3.2, one can conclude that for each .a; b/ 2 A1,

Z0a;b � enL.a;b/CC n4=5C�

:(5.1)

Next, observe that by equation (3.14) and Lemma 3.6, at the point .a; b/ D .a�; B/,@L=@a D 0 and @2L=@a2 < 0. Also observe that at all points .a; B/ such that

B � a > c0=2, @L=@b is uniformly bounded away from 0 and positive. Moreover,

the derivatives of L are continuous, L is increasing in b, and L.a; b/ is uniquely

maximized at .a�; B/.Combining all of this, it follows that if n > C , then for all .a; b/ such that

ja � a�j > n�.1�r/=2 and b � a > c0=2,

L.a; b/ � L.a;B/ � L.a�; B/ � n�.1�r/

C;

and for all .a; b/ such that b c0=2,

L.a; b/ � L.a;B/ � n�.1�r/

C� L.a�; B/ � n�.1�r/

C:

Thus, if n > C , then for all .a; b/ 2 A1,

L.a; b/ � L.a�; B/ � n�.1�r/

C:

From the above display, inequality (5.1), and the fact that 45

C� < r , it follows that

if n > C , then for all .a; b/ 2 A1,

Z0a;b � enL.a�;B/�nr =C :(5.2)


Let A WD Va0[Wc0

[A1 [A2. Then by (5.2), (4.2), (4.3), (4.1), (3.1), and the

observation that r > maxf4=5C �; 2pg, we get that if n > C ,

P . 2 A/� Z�1

� ZVa0

[Wc0

eˇn�1S4.f /df C jA1j max.a;b/2A1

Z0a;b

�

� e�nL.a�;B/CC n4=5CC n2p �2en.L.a�;B/�ı/ C Cn2=5enL.a�;B/�nr =C

�

� e�nr =C :

Thus, with A0 WD ff W b.f / � B; f 62 Ag, we have that for n > C ,

(5.3) P . 2 A0/ � 1 � e�nr =C :

Note that if f 2 A0, then

(5.4) jn�1N.f / � Bj D jb.f / � Bj � 2n�.1�r/

and

jn�1H.f /C a�2j � Cn�.1�2p/ C jn�2S4.f / � a�2j� Cn�.1�2p/ C ja.f / � a�jja.f /C a�j� Cn�.1�2p/ C 2Bja.f / � a�j� Cn�.1�r/=2:

(5.5)

Let A2 be a collection of .a; b/ such that

(5.6)[

.a;b/2A2

a;b A0:

From the definition of A0, it should be clear that if n > C , then A2 can be chosen

such that jA2j � Cn2=5, and for each .a; b/ 2 A2, b � a > c0=2, a > a0=2,

(5.7) jb � Bj � 4n�.1�r/;

and

(5.8) ja � a�j � 4n�.1�r/=2:

Take any .a; b/ 2 A2. Let 0a;b

and � be defined as in the proof of Lemma 3.2.

Then from (3.6), (3.11), and (3.12), it follows that if n > C ,

P .� 2 a;b n 0a;b/ � exp

�� an

b � a � n4=5C�

C

�:


(The exponent .4C 3�/=5 in (3.12) improves to 45

C � since a > a0=2 here. This

is easy to verify in the derivation of (3.12).) Therefore, as in (3.13) and from the

fact that b � a > c0=2,

Vol.a;b n 0a;b/

� exp

�bn

b � a.1C n�1=5/C n log�.b � a/�

P .� 2 a;b n 0a;b/

� en.L.a;b/�ˇa2/�n4=5C�=C :

Thus, by (4.1) and (3.1) and the observation that 45

C � D q > 2p,

P . 2 a;b n 0a;b/ � Z�1

Z�a;bn� 0

a;b

eˇn�1S4.f /df

� Z�1eˇna2.1Cn�1=5/ Vol.a;b n 0a;b/

� e�n4=5C�=C :(5.9)

Let

Q WD[

.a;b/2A2

0a;b:

By (5.6),

A0 � Q [� [

.a;b/2A2

.a;b n 0a;b/

�:

Therefore by the fact that jA2j � Cn2=5 and (5.9),

P . 2 A0/ � P . 2 Q/C jA2j max.a;b/2A2

P . 2 a;b n 0a;b/

� P . 2 Q/C Cn2=5e�n4=5C�=C :

(5.10)

By (5.10) and (5.3) we see that for n > C ,

P . 2 A0 \Q/ � P . 2 A0/C P . 2 Q/ � 1� 2P . 2 A0/ � e�nq=C � 1� 1 � e�nr =C � e�nq=C � 1 � e�nq=C :

Recall that M1.f / and M2.f / denote the largest and second-largest components

of the vector .jfxj2/x2V . Note that if f 2 Q, then f 2 0a;b

for some .a; b/ 2 A2.

The definition of 0a;b

implies that

jM1.f / � anj � Cn4=5C�; M2.f / � Cn4=5C�:

Combining this with (5.8) and the fact that 45

C � D q gives

jM1.f / � a�nj � Cn.3Cq/=4:


The last two displays and the inequalities (5.4) and (5.5) complete the proof of the

theorem in the case ˇB2 > �c .

5.2 Subcritical Gibbs MeasureNext, consider the case ˇB2 < �c . Let q satisfy

maxf.1C 2p/=2; 17=18g < q < 1:Choose � such that

1 � 2�

5D q:

Since q > 1718

, it follows that � < 17

and therefore 1 � 2�5> 4

5C �, a fact that will

be needed below.

If M1. / > n1�2�=5, then S4. / > n2�4�=5 and therefore 2 a;b for some

n�2�=5 < a C , then for any .a; b/ satisfying 2n�2�=5 < a c1=2,

P . 2 a;b/ � Z�1Z0a;b

� e�nL.0;B/CC n4=5CC n2pCnL.a;b/CC n4=5C�

� e�nL.0;B/CC n4=5CC n2pCnL.a;B/CC n4=5C�

� e�C �1n1�2�=5CC n4=5C�CC n2p

:

By our choice of �, it follows that for any such .a; b/, if n > C then

(5.11) P . 2 a;b/ � e�nq=C :

As usual, we can choose a set A3 of size � nC such that[.a;b/2A3

a;b f.a; b/ W 2n�2�=5 < a c1g:

Moreover, we can ensure that b � a > c1=2 and a > n�2�=5 for all .a; b/ 2 A3.

Therefore by (5.11), (4.5), and our choice of c1, we see that if n > C , then

P .M1. / > n1�2�=5/ � jA3j max

.a;b/2A3

P . 2 a;b/C P . 2 Wc1/

� e�nq=C CZ�1e12

nL.0;B/

� e�nq=C :(5.12)

Define the sets

E1 WD ff W S2.f / > Bn � nqg;E2 WD ff W jH.f /j � 2n2q�1g;E3 WD ff W M1.f / � nqg:


First, observe that if n > C , then E3 � E2; if M1.f / � n1�2�=5 D nq and

n > C , then

jH.f /j � Cn2p C n�1M1.f /2

� Cn2p C n2q�1

� 2n2q�1;

since 2q � 1 > 2p. In particular, if n > C ,

P . 2 E2/ D P .jH. /j � 2n2q�1/ � 1 � e�nq=C :

Next, observe that by (4.5) and Lemma 3.3, the probability of belonging to

Ec1 \E2 is

P .S2. / � Bn � nq; jH. /j � 2n2q�1/

D Z�1

Zff WS2.f /�Bn�nq ; jH.f /j�2n2q�1g

e�ˇH.f /1fS2.f /�Bngdf

� Z�1e2ˇn2q�1

Vol.Un;Bn�nq /

� exp.�n.L.0; B/ � L.0;B � n�.1�q///C Cn4=5 C Cn2p C Cn2q�1/

� e�nq=C CC n4=5CC n2pCC n2q�1 � e�nq=C :

Combining the last two displays with (5.12) gives that for n > C ,

P . 2 E1 \E2 \E3/ D P . 2 E2 \E3/ � P . 2 Ec1 \E2 \E3/

D P . 2 E3/ � P . 2 Ec1 \E2 \E3/

� 1 � e�nq=C � P . 2 Ec1 \E2/

� 1 � e�nq=C :

This completes the proof in the case ˇB2 < �c .

5.3 Critical Gibbs MeasureFinally, when ˇB2 D �c , L.a; b/ is maximized at exactly two points, .0; B/

and .a�; B/. As before, this implies that with high probability, cannot belong to

any a;b where .a; b/ is away from both of these optimal points. It can be deduced

from this, exactly as before, that with high probability either (2.7) or (2.8) must

hold. We omit the details. This completes the proof of Theorem 2.2.

6 Proof of Theorem 2.3 (Blowup of H 1 Norm)The conclusion for the supercritical case ˇB2 > �c is almost immediate from

(2.7). To see this, note that if x is the mode, thenP

y�x j x � y j2 � an � o.n/

with high probability.


Consider the subcritical case ˇB2 < �c . Define E2 WD ff W jH.f /j �2n2q�1g. Then by the lower bound on Z from Theorem 2.1 and the subcritical

part of Theorem 2.2, if n > C , then for any A � CV ,

P . 2 A/ � P . 62 E2/C P . 2 A \E2/

� e�nq=C CZ�1

ZA\E2


� e�nq=C C .B�e/�neC n4=5CC n2pCC n2q�1

ZA\E2

1fS2.f /�Bngdf:

Observe that if � is a random vector whose components are i.i.d. standard complex

Gaussian, then for any U � CV ,

.B�e/�n

ZU

1fS2.f /�Bngdf � .B�e/�n

ZU

en�S2.f /=B1fS2.f /�Bngdf

� .B�/�n

ZU

e�S2.f /=Bdf

D P .pB� 2 U/:

Since the graph G has maximum degree D, it is easy to see (e.g., by a greedy

algorithm) that there is a subset of edges E 0 � E such that jE 0j � jEj=D, and any

two edges inE 0 are vertex disjoint. Therefore, for a sufficiently largeC (depending

on the usual parameters), using the fact that j�x � �y j2 is an exponential random

variable with mean 2,

E.e� P.x;y/2E j�x��y j2/ � E.e� P

.x;y/2E0 j�x��y j2/

DY

.x;y/2E 0

E.e�j�x��y j2/ D 3�jE 0j � e�jE j=C :

Thus, there is a constant C sufficiently large such that

P

� X.x;y/2E

j�x � �y j2 � jEjC

�� e�jE j=C :

Combining this with the two earlier observations and the observation that ı D2jEj=n completes the proof in the case ˇB2 < �c .

Finally, for the critical case, when ˇB2 D �c , either (2.7) or (2.8) must hold

by Theorem 2.2. If (2.7) holds, the result is trivial. For functions satisfying (2.8),

an argument similar to the above has to be made with small modifications. (For

example, instead of a single set E2, we have to consider two sets E2 and E 02 such

that E2 [E 02 has high probability.) We omit the details.


7 Proof of Theorem 2.4 (Standing Wave)As usual C denotes any constant that depends only on ˇ, B , D, p, and q. Let

C0 denote the constant C from the first part of Theorem 2.2, so as not to confuse it

with other C ’s in this proof. Let A denote the interval Œ0; enq=2C0 �. Define

T WD ft 2 A W .t/ violates (2.7) for C D C0g:Then by Theorem 2.2 and the invariance of the flow, if n > C0,

EjT j DZ

t2A

P .t 2 T /dt � e�nq=C0enq=2C0 D e�nq=2C0 ;

where jT j denote the Lebesgue measure of the set T . Therefore,

P .jT j � e�nq=4C0/ � EjT je�nq=4C0

� e�nq=4C0 :

Also, by Theorem 2.2, P .0 2 T / � e�nq=C0 . Thus,

P .jT j < e�nq=4C0 and 0 62 T / � 1 � e�nq=4C0 � e�nq=C0 :

Suppose this event happens, that is, jT j < e�nq=4C0 and 0 62 T . We claim that

under this circumstance, the discrete wave function cannot have modes at two

different locations for two distinct times t; s 2 A; moreover, .t/ satisfies (2.7) for

all t 2 A. This would complete the proof.

By (2.7), we know that if t 62 T , then for each y 2 V , j y.t/j2 must either

belong to the interval Œan � C0n.3Cq/=4; an C C0n

.3Cq/=4� (such a point will be

called a type I point) or is � C0nq (such a point will be called a type II point). Say

that a point y is type III at time t if 2C0nq � j y.t/j2 � an � 2C0n

.3Cq/=4, and

type IV if j y.t/j2 � anC 2C0n.3Cq/=4. Note that if t 62 T , there cannot be any

type III or type IV points.

Let x be the mode at time 0. Since 0 62 T , x is a type I point at time 0. Sup-

pose x is a type III point at some time in A; let s be the first such time (there is a

first time by the time continuity of the wave function). Let s0 be the last time before

s such that x was type I at time s0. Then by continuity,

an � 2C0n.3Cq/=4 < j x.u/j2 < an � C0n

.3Cq/=4 for all u 2 .s0; s/:

Thus, the interval .s0; s/ is a subset of T . Since jT j � e�nq=4C0 , this implies that

s � s0 � e�nq=4C0 . Moreover, j x.s/j2 � an � 2C0n.3Cq/=4 and j x.s

0/j2 �an � C0n

.3Cq/=4. Thus, there exists u 2 .s0; s/ such that

(7.1)d

dtj x.t/j2

ˇˇtDu

� �C0n.3Cq/=4

e�nq=4C0:


On the other hand by (2.1), for any y 2 V and at any t ,

d

dtj y.t/j2 D y.t/

d

dt y.t/C y.t/

d

dt y.t/

D 2Re

� y.t/

d

dt y.t/

�D 2Re

� y.t/.i z� y.t/C i j y.t/j2 y.t//

�:

Note that N. .t// is the same for all t and is bounded by nC at t D 0 (and hence

for all t ) by the assumption that 0 62 T . Since j y.t/j2 � N. .t// for any y and t

and j2Re.u/j � 2juj, this implies that for any y and t ,ˇˇ ddt j y.t/j2

ˇˇ � 2j y.t/j

h2

X´�y

.j ´.t/j C j y.t/j/C j y.t/j4

� 4n2pDN. .t//CN. .t//2 � nC :

But this is in contradiction with (7.1) if n is sufficiently large. The contradiction is

to the assumption that x is type III at some time in A. Thus, x cannot be type III at

any time in A.

Similar arguments show that if jT j < e�nq=4C0 and 0 62 T , and x is the mode

at time 0, then x cannot be type IV any time in A, and for any y ¤ x, y cannot be

a type III point at any time in A.

These three deductions combined with continuity of the flow and the conser-

vation of mass and energy establish that x must be the mode at all times in A;

moreover, (2.7) must hold for .t/ for all t 2 A (with a C different than C0), thus

completing the proof of the theorem.

8 Proof of Theorem 2.5 (Limiting Distribution)Fix k distinct elements x1; : : : ; xk 2 V . Define an undirected graph structure

on † as follows: for any two distinct ; � 2 †, say that .; �/ is an edge if

fx1; : : : ; xkg \ f�x1; : : : ; �xkg ¤ ¿:

Since† is a group and assumption 2 of translatability on page 735 holds, therefore

for each and each 1 � i � k, there are at most k automorphisms in † that take

xi into the set fx1; : : : ; xkg. In other words, there are at most k possible � 2 †

such that ��1xi 2 fx1; : : : ; xkg. This shows that the maximum degree of the

graph on † is bounded by k2. Consequently, there is an independent subset †0 of

† of size � n=k2.

Let � D .�x/x2V be a vector of i.i.d. standard complex Gaussian random vari-

ables. Fix a Borel set U � Ck , and let

Q.�/ WD jf 2 †0 W .��x1; : : : ; ��xk

/ 2 U gjj†0j :


Since†0 is a set of automorphisms and the HamiltonianH is invariant under graph

automorphisms,

EQ.�/ D P ..�x1; : : : ; �xk

/ 2 U/:By the design of †0, Q.�/ is an average of independent random variables taking

value in f0; 1g; consequently, by Hoeffding’s inequality [17], for any � � 0,

P .jQ.�/ � EQ.�/j > �/ � 2e�j†0j�2=2 � 2e�n�2=2k2

:

8.1 Subcritical Limiting DistributionIf ˇB2 < �c , then as in the proof of Theorem 2.3, for any � > 0,

P .jQ.B�1=2 / � EQ.�/j > �/� e�nq=C C eC n4=5CC n2pCC n2q

P .jQ.�/ � EQ.�/j � �/

� e�nq=C C 2eC n4=5CC n2pCC n2q�n�2=2k2

:

A simple computation using the above bound and the identity

E.X/ DZ 1

0

P .X > t/dt;

which holds for any nonnegative random variable X , shows that there is a constant

C > 0 depending only on the usual model parameters such that if n > C , then

EjQ.B�1=2 / � EQ.�/j � kn�1=C :

By Jensen’s inequality, this gives

EjQ.B�1=2 / � EQ.�/j� jEQ.B�1=2 / � EQ.�/jD jP .B�1=2. x1

; : : : ; xk/ 2 U/ � P ..�x1

; : : : ; �xk/ 2 U/j:

This completes the proof in the case ˇB2 < �c .

8.2 Supercritical Limiting DistributionNext, suppose ˇB2 > �c . Take any q as in the first part of Theorem 2.2, and let

C0 denote the constant C . Let R denote the set of functions satisfying (2.7) with

C D C0. For each x 2 V , let Rx be the subset of R consisting of functions with

mode at x. Note that the Rx’s are disjoint and R D Sx Rx . Let a D a.ˇ;B/ be

defined as in (2.6). Recall that

F.ˇ;B/ D ˇa2 C log..B � a/�e/:Fix � > 0 and let

A WD ff W jQ..B � a/�1=2f / � EQ.�/j > �g:


Then for n > C0,

P . 2 A/ � P . 62 R/C P . 2 A \R/� e�nq=C CZ�1

ZA\R


� e�nq=C C e�nˇa2

..B � a/�e/�neC n4=5CC n2p

ZA\R

enˇa2CC n.3Cq/=4

df

� e�nq=C C ..B � a/�e/�neC n4=5CC n2pCC n.3Cq/=4Xx2V

Vol.A \Rx/:

Fix x 2 V . For each f , let f 0 be the function such that f 0y D fy for all y ¤ x and

f 0x D n�1=2fx . Note that the map f 7! f 0 is linear with determinant n�1 (and

not n�1=2, since we are contracting both the real and imaginary parts of fx).

There are at most k many 2 † such that �1x 2 fx1; : : : ; xkg. Therefore for

any f ,

jQ..B � a/�1=2f / �Q..B � a/�1=2f 0/j � k

j†0j � k3

n:

Consequently, if f 2 A then f 0 2 A0 where

A0 WD ff W jQ..B � a/�1=2f / � EQ.�/j > � � k3=ng:Again if f 2 Rx , then f 0 2 R0 where

R0 WD ff W S2.f / � .B � a/nC Cn.3Cq/=4g:Combining the above observations gives

Vol.A \Rx/ DZ

1ff 2A\Rxgdf

� n

Z1ff 02A0\R0gdf 0

� neC n.3Cq/=4Cn

ZA0

e�S2.f 0/=.B�a/df 0

D n..B � a/�e/neC n.3Cq/=4

P .jQ.�/ � EQ.�/j > � � k3=n/:

The proof is now completed as before.

Acknowledgment. The authors thank Riccardo Adami, Jose Blanchet, Jeremy

Marzuola, Stefano Olla, Jalal Shatah, and Terence Tao for enlightening comments.

SC thanks Persi Diaconis and Julien Barré for bringing the problem to his attention.

Sourav Chatterjee’s research was partially supported by NSF Grant DMS-1005312

and a Sloan Research Fellowship. Kay Kirkpatrick’s research was partially sup-

ported by NSF Grant OISE-0730136.


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SOURAV CHATTERJEE

Courant Institute

251 Mercer St., Rm. 813

New York, NY 10012

E-mail: [email protected]

KAY KIRKPATRICK

Courant Institute

251 Mercer St., Rm. 813

New York, NY 10012

and

Université Paris IX Dauphine

Centre De Recherche en Mathèmatiques

de la Décision

Place du Maréchal

de Lattre de TASSIGNY

F-75775 Paris Cedex 16

FRANCE

Received February 2011.

Documents

Probabilistic Methods for Discrete Nonlinear Schrödinger Equations