3
Ergodic operators
In discussing random operators with homogeneous disorder, which are the
main topic of these notes, it is natural to bring up the ergodicity of such operators under translations. This carries useful implications:
1. certain properties, such as the spectrum, are predictable in the sense that
they almost surely assume non-random values,
2. averages over translations within a given configuration almost surely yield
the same statistics as averages over the random coefficients.
It is worth appreciating that the class of ergodic operators includes also other
examples, such as operators which are almost-periodic under shifts. Disorder
and almost-periodicity are easily told apart, and are associated with rather different phenomena. In this section we shall however focus on a structure which
they share (except for the degenerate periodic case of the latter), and that is
ergodicity.
3.1 Definition and examples
3.1.1 Some basic terminology
A natural setup for the discussion of random operators from the perspective
of ergodicity, is to start by presenting the operators as acting over a transitive
graph. Following is a quick summary of some of the key concepts.
A (vertex) transitive graph is a graph (G) endowed with a group of graph
automorphisms (I) which acts on it transitively. A stochastic processes indexed by the graph G, is a family of random variables, Y(x) : ⌦ ! R,
! 7! Y(x, !), indexed by x 2 G, and defined over a common probability space
(⌦, A, P).
28
3.1 Definition and examples
29
In this language, a random potential, which is one of the components of
the Schrödinger operator in `2 (G), is given by a stochastic process over G. Its
values over the graph are given by V(x, !).
Definition 3.1 The action of a family (T x ) x2I of transformations on a probability space (⌦, A, P) , which are measure preserving, i.e. P(T x 1 A) = P(A) for
all A 2 A, is said to be ergodic if all events A 2 A which are invariant, i.e.,
T x 1 A = A for all x 2 I, are of probability either zero or one.
An equivalent property is that any random variable, Y : ⌦ ! R [ {1},
which is invariant under all transformations , i.e. Y T x = Y for all x 2 I, is
almost surely constant, i.e., there exists c 2 R [ {1} such that
P (Y = c) = 1 .
(3.1)
The terminology is used with some flexibility: in situations where the natural starting points is a symmetry group, the term “ergodic” is often pegged on
the probability measure.
Of particular interest for us is the case where G = Zd and the automorphism group I consists of the lattice shifts . In this (amenable) case one has the
following extension of Birkho↵’s fundamental theorem.
Proposition 3.2 (Birkho↵) Let (Y(x)) x2Zd be an ergodic stochastic process
and suppose that Y(0) 2 L1 (⌦, P). Then P-almost surely:
lim
L!1
1 X
Y(x) = E [Y(0)] ,
|⇤L | ⇠2⇤
(3.2)
L
where ⇤L := {x 2 Zd | |x|1 < L}.
A modern proof can be found in [158], and extensions of the theorem to
non-ameanable groups (including that of graph automorphisms of a regular
tree graph) are discussed in [?].
3.1.2 Ergodic operators – definition and examples
For a broad class of operator-valued functions defined on a probability space
the notion of a random variable is naturally extended as follows.
Definition 3.3
A function defined on a probability space (⌦, A, P) which
Random Operators c M. Aizenman & S. Warzel
DRAFT
30
Ergodic operators
assigns to every ! 2 ⌦ a self-adjoint operator H(!) in a common separable Hilbert space H is said to be weakly measurable if the functions ! 7!
h' , f (H(!)) i are measurable for all f 2 L1 (R) and all ', 2 H.
It suffices to check this property for all f (x) = (x z) 1 with z 2 C\R. We
shall not dwell here on this and on related questions of measurability, on which
further discussion may be found in references [50, 129].
Following are two natural extensions of the notion of ergodicity to operator
valued functions.
Definition 3.4 Let (⌦, A, P) be a probability space endowed with the action of a family of ergodic transformations (T x ) x2I . Any weakly measurable
map ! 7! H(!) into the self-adjoint operators on a separable Hilbert space
H is called a family of ergodic operators if H(T x !) is unitarily equivalent
to H(!).
If H(!) is an ergodic operator then any functional which is invariant under
unitary transformations is almost surely constant with respect to any ergodic
measure. An example of such functions is ! 7! kH(!)k, and – as will be verified below – also the spectrum (with the appropriate definition in case of pure
point spectrum). In case of standard ergodic operators more can be added, e.g.,
concerning the density of states.
Following are three examples of random operators in which both of the
above criteria are met.
Example 3.5 (Random Schrödinger operator on Zd ) The symmetry group on
which we shall focus is that of lattice shifts, which for specificity we denote
as (S x ) x2Zd (i.e., in this case I = Zd ). A natural and convenient choice for
d
the probability space is ⌦ = RZ , with the product topology and the product
-algebra of Borel sets B. Under this explicit construction, the values of the
random potential at sites x 2 Zd are given by the components !(x). The lattice
shifts are implemented by the mappings:
(S x !) (⇠) := !(⇠
x) ,
⇠, x 2 Zd ,
(3.3)
which are assumed to preserve the probability measure on (⌦, B). The class of
ergodic measures for such a system includes all the product measures, for each
of which (!(x)) x2Zd form iid random variables.
In the above setup, the induced action of the shifts on the random Schrödinger
operator H(!) =
+ V(!) in `2 (Zd ) (with V(!) the multiplication operator
3.1 Definition and examples
31
by !(x)) correspond to:
H(S x !) = U x H(!) U x†
(3.4)
with (U x ) x2Zd the unitary transformations on `2 (Zd )
(U x ) (⇠) :=
(⇠
x) .
(3.5)
Consequently, H(!) is an ergodic operator.
Example 3.6 (Random Schrödinger operator with a constant magnetic field)
A quantum particle on Z2 with charge q 2 R in a perpendicular magnetic field,
and subject to a random potential V(!), is described by an operator of the form
H (A) (!) =
(A)
+ V(!)
(3.6)
(A)
with
the magnetic Laplacian, a concept on which more is said here in
Chapter 12. An explicit version of the magnetic Laplacian which incorporates
a constant magnetic field B is (in the symmetric gauge):
⇣
⌘
(A)
(x) = eiBqx2 /2 (x1 + 1, x2 ) + e iBqx2 /2 (x1 1, x2 )
+e
X ⇣
=:
e
|y x|=1
iBqx1 /2
iqA(x,y)
(x1 , x2 + 1) + eiBqx1 /2 (x1 , x2
⌘
(y)
(x) .
1)
4 (x)
(3.7)
Here A(x, y) = A(x, y) |x y|,1 is an antisymmetic function over the oriented
nearest neighbor edges (which represents the line integral of a vector potential). Its relation to the magnetic field is the discrete version of B = Curl A,
which is presented in (12.9).
For the magnetic Schrödinger operator, the simple shift relation (3.4) is replaced by:
H (A) (S x !) = U x(A) H (A) (!) U x(A) † .
(3.8)
where U x(A) are the magnetic shifts, which combine U x with a gauge transformation:
⇣
⌘
U x(A) (⇠) := e i'A (⇠,x) (⇠ x) ,
(3.9)
with 'A (⇠, x) := qB(⇠1 x2 ⇠2 x1 )/2. Due to the twist by the gauge transformation, the magnetic shifts do not form a group, however that in itself is not
detrimental for our purpose.
The next example is rather di↵erent, yet is also fits within the framework
discussed here.
Random Operators c M. Aizenman & S. Warzel
DRAFT
32
Ergodic operators
Example 3.7 (Almost-periodic Schrödinger operator on Z) Let ⌦ = [0, 2⇡)
and P be the uniform distribution on the interval. For a fixed irrational ↵ 2
(0, 1), the corresponding Harper Hamiltonian (dubbed also as the almost-Mathieu
operator [?]) in `2 (Z) is
H↵ (✓) =
+ V↵ (✓)
(3.10)
with the potential
V↵ (x, ✓) := cos(2⇡↵ x + ✓) ,
(3.11)
defined for x 2 Z. In this context it seem natural and is customary to replace
the parameter ! by ✓ 2 [0, 2⇡). However the concepts of ergodicity are equally
applicable. The symmetry group is formed of the rotations, T x : ⌦ 7! ⌦, with
T x ✓ := ✓ + 2⇡↵ x
mod 2⇡ ,
(3.12)
with respect to which H↵ (✓) forms a standard ergodic operator, provided ↵ is
irrational.
In this case there is no disorder: while the values of the potential form an
ergodic process, this process is deterministic: V(0, ✓) is (easily) determined
from the values of {V(n, ✓)}n<0 .
3.2 Deterministic spectra
For ergodic operators certain questions have a predictable answer. Examples
are the spectra and the density of states. The latter will be the topic of the
next chapter. Concerning the spectra, the following result dates back to L. Pastur [197] with extensions by H. Kunz, B. Souillard [163] and W. Kirsch and
F. Martinelli [129].
Theorem 3.8 (Pastur) The spectrum of a family of ergodic operators H(!) is
P-almost surely non-random, i.e., there is ⌃ ⇢ R such that
P ( (H) = ⌃) = 1 .
(3.13)
The same applies to any subset in the Lebesgue-decomposition of the spectrum,
i.e., there are ⌃# ⇢ R, # = ac, sc, pp, such that
⇣
⌘
P # (H) = ⌃# = 1 , # = ac, sc, pp .
The above non-random set ⌃ is called the almost-sure spectrum associated
with H(!) and likewise for the spectral components.
Let us stress that the set of eigenvalues of a family of ergodic operators is
usually heavily dependent on the realization !. Only the closure of this set is
3.2 Deterministic spectra
33
deterministic. Just think of the random multiplication operator corresponding
to iid random variables on Zd .
Proof of Theorem 3.8: For any E1 , E2 2 R, the functions
! 7! XE1 ,E2 (!) := dim Range P(E1 ,E2 ) (H(!)) ,
defined on the underlying probability space (⌦, A, P) are:
1. measurable, i.e., random variables taking values in [0, 1].
This is most easily seen by identifying them as (possibly divergent) nonP
negative series, XE1 ,E2 (!) = 1
k=1 h k , P(E1 ,E2 ) (H(!)) k i, given in terms
of an arbitrary orthonormal basis ( k ) in H.
2. invariant under the action of the ergodic transformations (T x ) x2I ,
dim Range P(E1 ,E2 ) (H(T x !)) = dim Range P(E1 ,E2 ) (H(!)) ,
by unitary equivalence.
Exercise 3.2 then ensures that there are constants cE1 ,E2 2 [0, 1] such that
P XE1 ,E2 = cE1 ,E2 = 1. The characterization (A.18) of the spectrum as the
support of the spectral projections identifies
⌃ = E 2 R | For all E1 , E2 2 Q with E1 < E < E2 : cE1 ,E2 > 0
as a possible candidate for the almost-sure spectrum. The reason for choosing
intervals with rational endpoints becomes apparent from the fact that the event
\
XE1 ,E2 = cE1 ,E2 ⇢ { (H) = ⌃} ,
E1 ,E2 2Q
cE1 ,E2 >0
still has probability one, since it is a countable intersection of such events. ⇤
The same proof idea also applies to the components of the spectrum. The only
subtle point is the measurability of the spectral projections associated to the
restriction of H(!) to the subspaces H # , # = ac, sc, pp. We refer the interested
reader to [129, 163].
Determining the almost-sure spectral components of an ergodic operator is
in general a hard question which much of the later chapters are devoted to. In
contrast, the almost-sure spectrum is much easier to identify. One strategy is
based on the construction of Weyl sequences, i.e., sequences of approximate
normalized eigenfunctions.
Random Operators c M. Aizenman & S. Warzel
DRAFT
34
Ergodic operators
Proposition 3.9 (Weyl criterion)
ator on a Hilbert space H. Then
(A) =
⇢
2 R|9(
n)
Let A : D(A) ! H be a self-adjoint oper-
⇢ D(A), k
nk
= 1 : lim k(A
n!1
)
nk
=0
.
A more general statement and a proof can be found in [233].
Let us illustrate the strategy to determine the spectrum in case of a standard ergodic operator of the Example 3.5. Associated to the random potential
(V(x; !)) x2Zd defined on some probability space (⌦, A, P) are two notions of
the support of the probability distribution. The first is:
supp1 (P) := { 2 R | 8" > 0 : P (|V(x)
| < ") > 0}
(3.14)
which for any ergodic process (V(x)) x2Zd is almost-surely equal to the closure
of the set of realized values:
a.s.
supp1 (P) =
V(x, !) | x 2 Zd .
(3.15)
The proof of this assertion is left as an exercise to the reader.
The notion which is directly relevant for determining the almost-sure spectrum through Weyl sequences is the slightly di↵erent set:
supp2 (P) :=
2 R | 8" > 0 , ⇤ ⇢ Zd finite :
P sup |V(x)
x2⇤
|<" >0
For 2 supp2 (P) typical realizations of the random process will exhibit arbitrarily large regions where x 7! V(x, !) is "-close to . Those regions accommodate approximate eigenstates of energies in the range + [0, 4d]. For
ergodic processes the general relation is supp2 (P) ⇢ supp1 (P), with equality
in case of iid random variables and strict inequality for certain processes with
strong anti-correlations.
The above comment on Weyl sequences summarizes the proof idea of the
following result of H. Kunz and B. Souillard [163], generalization of which
can be found in [130].
Theorem 3.10 (Kunz/Souillard) For the family H(!) =
operators on `2 (Zd )
[0, 4d] + supp2 (P) ⇢
for P-almost all !.
+V(!) of ergodic
(H(!)) ⇢ [0, 4d] + supp1 (P) ,
(3.16)
3.3 The density of states
35
Proof The proof of the second inclusion is based on (A.14) in which we set
A = H(!) and B = V(!) + 2d such that kA Bk  2d, cf. (A.11). Since
(B) = (V(!)) + 2d, the assertion follows from
(V(!)) = V(x, !) | x 2 Zd = supp1 (P) .
For a proof of the first inclusion, we pick 2 ( ) = [0, 4d] and a corresponding Weyl sequence ('n ) ⇢ `2 (Zd ). In fact, by a suitable smooth truncation
we may even assume that 'n has compact support. For each n 2 N we consider
the event
(
)
1
d
⌦n := There is j 2 Z : sup |V(x + j) µ| < n .
x2supp 'n
If µ 2 supp2 (P) these events have a non-zero probability and are invariant under
the shifts (S x ) x2Zd . By ergodicity we conclude that they are almost certain,
P (⌦n ) = 1. Therefore their intersection
\
⌦0 :=
⌦n
n2N
is almost certain too. By construction, for any ! 2 ⌦0 there is a sequence
( jn ) ⇢ Zd such that
n
is a Weyl sequence for H(!) and
k(H(!)
µ)
nk
 k(
 k(
:= 'n (·
jn )
+ µ 2 (H(!)). This follows from
)
nk
+ k(V(!)
µ)
nk
) 'n k + sup |V(x + jn )
x2supp 'n
which goes to zero as n ! 1.
µ| k'n k ,
⇤
In case of iid random variables, it is an easy exercise to show that supp1 (P) =
supp2 (P). Therefore the almost-sure spectrum may be determined explicitly.
Corollary 3.11 In case (V(x)) x2Zd are iid with common distribution P0 (dv) :=
P (V(x) 2 dv),
(H(!)) = [0, 4d] + supp P0
(3.17)
for P-almost all !.
3.3 The density of states
The density of states (DOS) measure ⌫ is a quantity of great importance in
statistical and condensed matter physics. For homogeneous quantum systems,
Random Operators c M. Aizenman & S. Warzel
DRAFT
36
Ergodic operators
it roughly speaking measures the number of single-particle states per volume.
Its knowledge allows to predict all basic thermal-equilibrium properties of the
corresponding system of non-interacting particles. For instance, in case of a
homogenous gas of non-interacting (spinless) fermions the specific free energy is given by
"
#
Z
⇣
⌘
f ( , n̄) := sup µ n̄ T
log 1 + e(µ E) ⌫(dE)
(3.18)
µ2R
R
as a function of the absolute temperature 1 > 0 (multiplied by Boltzmann’s
constant) and the fermion concentration n̄ > 0. As we will see later, ⌫ also
enters formulae for conductance.
This section is devoted to a precise mathematical definition and some basic
properties of the DOS which in the ergodic setup is a non-random measure.
Most importantly, we will identify it as a deterministic infinite-volume limit of
finite-volume quantities.
3.3.1 Definition for standard ergodic operators
For a finite matrix, or an operator in a Hilbert space of finite dimension N, the
DOS is the point measure
⌫=
1
N
X
En
(3.19)
.
En 2 (H)
There are di↵erent ways in which one could envision extensions of this notion
to random operators in the infinite volume setup. For the following class of
ergodic operators a number of such natural extensions coincide.
Definition 3.12 Ergodic operators H(!) acting in `2 (G) over the vertex set
G of a transitive graph whose graph homomorphisms have the representation
(T x ) x2I on the underlying probability space (⌦, A, P), will be called a standard
ergodic operators if
h 0 , f (H(!))
0i
= h
Tx0,
f H(T x !)
Tx0i
(3.20)
for all x 2 I and all f 2 C0 (R).
Examples are the random Schrödinger operators with or without constant
magnetic field in Examples 3.5 and 3.6. Let us therefore start from the following as the definition of their DOS.
3.3 The density of states
37
Definition 3.13 The density of states measure of a standard ergodic operator
H(!) is the average of the spectral measure associated with 0 2 `2 (G):
⌫ := E h 0 , P(·) (H)
0i
.
(3.21)
By the covariance property (3.20), 0 may be replaced in (3.21) by any x 2
`2 (G). Moreover, the next theorem asserts that for ergodic operators on the
finite-dimensional lattice, the expectation over the disorder E[·] can, almost
surely, be replaced by a volume-average of the spectral measures.
Theorem 3.14 Let H(!) be a standard ergodic operator acting in `2 (Zd ).
Then there exists a set ⌦0 ⇢ ⌦ of full measure, P(⌦0 ) = 1, such that for all
! 2 ⌦0 and all f 2 C0 (R):
Z
⇥
⇤
1
lim
tr 1⇤L f (H(!)) =
f (u) ⌫(du) .
(3.22)
L!1 |⇤L |
R
Proof
To establish (3.22), we write, for f 2 C0 (R):
⇥
⇤
1
1 X
tr 1⇤L f (H(!)) =
h x , f (H(!)) x i
|⇤L |
|⇤L | x2⇤
L
1 X
=
h 0 , f (H(S x !)) 0 i
|⇤L | x2⇤
(3.23)
L
Birkho↵’s ergodic theorem (Proposition 3.2) is applicable to this expression.
It implies that for every f 2 C0 (R) there exists ⌦ f 2 A with P(⌦ f ) = 1 such
that the above average converges to the non-random value
Z
E (h 0 , f (H(S x !)) 0 i) =
f (u) ⌫(du) .
One may conclude that for any countable collection D ⇢ C0 (R) this converT
gence holds simultaneously for a full measure set, ⌦0 = f 2D ⌦ f . By the
Stone-Weierstrass theorem [203, 233], this applies, for example, to the collection of functions f (u) = (u z) 1 , with Re z and Im z , 0 rational. Since this is
a uniformly dense subset whose linear span is dense in the uniform topology,
we conclude that with probability one, the convergence holds simultaneously
for all f 2 C0 (R). This establishes (3.22).
⇤
3.3.2 Finite-volume density of states and its limit
Definition 3.13 involves the infinite-volume operator H(!). The next statement
asserts that ⌫ also coincides also with limits of the densities of states of finitevolume versions of H(!).
Random Operators c M. Aizenman & S. Warzel
DRAFT
38
Ergodic operators
Theorem 3.15 Let H(!) be a standard ergodic operator in `2 (Zd ), and HL (!)
a sequence of operators such that for P-almost all ! 2 ⌦:
1. 1⇤L HL (!) = HL (!) 1⇤L , where 1⇤L are projections to `2 (⇤L ),
2. the di↵erence 1⇤L HL (!) 1⇤L H(!) is trace class, satisfying the uniform
bound:
tr | 1⇤L HL (!)
1⇤L H(!)|  ✏ (L) |⇤L | ,
(3.24)
with some ✏ (L) which vanishes as L ! 1.
Then there exists a set ⌦0 ⇢ ⌦ of full measure, P(⌦0 ) = 1, such that for all
! 2 ⌦0 and all f 2 C0 (R):
Z
1
lim
tr 1⇤L f (HL (!)) =
f (u) d⌫(u) .
(3.25)
L!1 |⇤L |
R
Since HL (!) commutes with 1⇤L , the trace in the left side only depends on
the restriction of HL (!) to the subspace `2 (⇤L ). Equation (3.25) can be restated as saying that the finite-volume density of states measure given by
tr P(·) (HL (!)| `2 (⇤L ) ) almost surely and after normalization converges weakly
to ⌫.
Theorem 3.15 can be rephrased in terms of the integrated density of states
n(E) := ⌫ ( 1, E) ,
(3.26)
and its finite-volume counterparts, N(HL ; E) := tr P( 1,E) (HL | `2 (⇤L ) ). The latter counts (including multiplicity) the number of eigenvalues of HL (!) when
restricted to `2 (⇤L ) below E.
Corollary 3.16 For any ergodic operator with the structure described in Theorem 3.15 there exists a full measure set ⌦0 2 A (i.e., with P(⌦0 ) = 1) such
that
1
lim
N(HL (!); E) = n(E)
(3.27)
L!1 |⇤L |
for all ! 2 ⌦0 and all E 2 R except the (at most countably many) points of
discontinuity of n.
Proof Since weak convergence implies the convergence of the distribution
function at all points of continuity of the limit [29], this is an immediate consequence of Theorem 3.15.
⇤
We now turn to the proof of Theorem 3.15.
3.3 The density of states
39
Proof of Theorem 3.15: By Lemma A.7 it suffices to establish the convergence for resolvents, i.e., function of the form f (u) = (u z) 1 , with z 2 C\R.
For such a function we have, using the resolvent identity:
1
1
tr 1⇤L f (HL (!))
tr 1⇤L f (H(!))
|⇤L |
|⇤L |
"
#
1
1
1
=
tr 1⇤L
|⇤L |
HL (!) z H(!) z
1
1
1
=
tr 1⇤L
[HL (!) 1⇤L H(!)]
|⇤L |
HL (!) z
H(!)
1
1

tr | 1⇤L HL (!) 1⇤L H(!)|
2
| Im z| |⇤L |
✏ (L)

! 0
(for L ! 1).
| Im z|2
z
(3.28)
We already know, by Theorem 3.14, that for P-a.e. !, |⇤1L | tr 1⇤L f (H(!)) tends
R
to (x z) 1 ⌫(du). The uniform bound on the di↵erence (3.28) permits to conclude that so does |⇤1L | tr 1⇤L f (HL (!)).
⇤
Theorem 3.15 can be applied in several ways:
1. HL (!) can be just the minimal modification of H(!) obtained by decoupling
⇤L from its exterior, i.e.
HL (!) = 1⇤L H(!) 1⇤L + 1⇤cL H(!) 1⇤cL .
(3.29)
2. HL (!) restricted to `2 (⇤L ) can also be taken to be any of the many finitevolume versions of the operator H(!) with rather arbitrary choices of boundary conditions: free, periodic, or “wired” in some way. In order to relate
such approximations to the infinite operator, it is convenient to set HL (!)
to zero in the complement of ⇤L since the trace in the left side of (3.25) is
anyway independent of this choice.
It should be appreciated that for the latter situation, the theorem states in particular, that the boundary conditions in HL (!) do not a↵ect the limiting DOS,
as long as the trace condition (3.24) is met.
The most prominent examples of self-adjoint operators arising from putting
boundary conditions on some domain ⇤L are the Dirichlet and Neumann operators, which will be discussed next.
Random Operators c M. Aizenman & S. Warzel
DRAFT
40
Ergodic operators
3.3.3 Dirichlet and Neumann Laplacians
The properties of Dirichlet and Neumann Laplacians are best explained in the
context of arbitrary graphs with vertex set G and edge set EG . The corresponding graph Laplacian is defined by
X
( G ) (x) :=
( (y)
(x)) .
(3.30)
[x,y][x,y]2EG
In case the degree of the vertices, degG (x) := |{y 2 G | [x, y] 2 EG }|, are
2
unifomly bounded,
G is a bounded non-negative operator on ` (G) with
k G k  sup x2G degG (x).
We will be interested in graphs (G0 , EG0 ) which are realized as a subgraph
of a given graph (G, EG ). In this situation at least two types of Laplacians can
be associated with the subgraph.
1. The Neumann Laplacian on `2 (G0 ) is given by
X
⇣
⌘
N
( (y)
(x)
:=
0
G
(x)) .
[x,y]2EG0
2. The Dirichlet Laplacian on `2 (G0 ) is
D
G0
:=
N
G0
+ 2 (DG0
DG ) ,
where (DG ) (x) := degG (x) (x) stands for the degree operator.
The Neumann Laplacian coincides with the natural graph Laplacian on the
subgraph. The Dirichlet Laplacian on the other hand assigns to vertices at the
boundary of the subgraph an extra weight in terms of the di↵erence of degrees
N
D
D
such that 0 
G0 . The terminology Dirichlet Laplacian for G0 is
G0 
borrowed from [215] and is not consistently used throughout the literature. The
virtue of this choice, which seems to go back to [215], becomes clear in the
next proposition.
Proposition 3.17 (Dirichlet-Neumann bracketing) For disjoint subgraphs
(G1 , EG1 ), (G2 , EG2 ) of a graph (G, EG ) with uniformly bounded degree one
has
N
G1
N
G2
on `2 (G1 [ G2 ) = `2 (G1 )

N
G1 [G2

D
G1 [G2

D
G1
D
G2
`2 (G2 ).
Proof The Neumann Laplacian on (G1 [G2 , EG1 [G2 ) coincides with the quadratic
form
X
h , GN1 [G2 i =
(y)|2 ,
| (x)
[x,y]2EG1 [G2
3.3 The density of states
41
where we note that edges are not oriented, which explains the lack of the factor
1
2 in comparison to (A.7). Omitting all terms in the sum which neither belong
to EG1 nor EG2 yields the first inequality. Since the second one just restates the
domination of the negative Dirichlet over the Neumann operator, it remains to
proof the last inequality. The latter is based on the observation that
D
G1
h ,
=
X
D
G2
i
h ,
| (x) + (y)|2
D
G1 [G2
0,
i
[x,y]2EG1 [G2
[x,y]<EG1 [EG2
which completes the proof.
⇤
Readers familiar with the Dirichlet and Neumann Laplacian on L2 (⇤) over
a domain ⇤ ⇢ Rd with piecewise smooth boundary, will recognize Proposition 3.17 as the exact analogue of the Dirichlet-Neumann bracketing available
there [201].
Let us now turn to applications and consider Schrödinger operators restricted
to ⇤ ⇢ Zd arising from the Neumann (X = N) or Dirichlet (X = D) Laplacian,
X
i.e., H⇤X :=
+ V on `2 (⇤). The following is an immediate consequence of
Proposition 3.17.
Corollary 3.18
Then
Let ⇤ ⇢ Zd be the disjoint union of ⇤ j , with j = 1, . . . , N.
N
M
j=1
H⇤N j  H⇤N  H⇤D 
N
M
j=1
H⇤Dj .
In the standard ergodic setup, it is an easy exercise to proof that
HLX (!) =
X
⇤L
+ V(!) ,
when naturally embedded in the full Hilbert space, satisfy the requirements
of Theorem 3.15 for both X = N, D. Thus |⇤L | 1 N(H⇤XL , E) almost surely
converges to n(E). In view of Corollary 3.18, this convergence happens to be
monotone at least on average.
Lemma 3.19 For an ergodic operator H(!) =
+ V(!) in `2 (Zd ) one has
for all ` > 0 and all points of continuity E 2 R of the integrated DOS:
h
i
h
i
1
1
E N(H`D ; E)  n(E) 
E N(H`N ; E) .
|⇤` |
|⇤` |
Random Operators c M. Aizenman & S. Warzel
(3.31)
DRAFT
42
Ergodic operators
Proof Consider a large region ⇤L , which appears in the representation (3.27)
of the integrated DOS and assume that ⇤L is a disjoint union of smaller cubes
⇤`, j of fixed side length ` > 0. Applying Corollary 3.18 together with the
min-max principle, Proposition 3.20 and (3.35) we obtain:
X
N(H`,Dj (!); E)  N(HLD (!); E)
(3.32)
j
X
 N(HLN (!); E) 
N(H`,N j (!); E) .
(3.33)
j
We now divide by |⇤L | and take the expectation value. Since there are |⇤L |/|⇤` |
terms in each of the two sums which are equally distributed, the left and right
side in the above inequality yield the respective left and right side in (3.31).
In the limit L ! 1 the two terms in the middle converge to n(E) by Corollary 3.16.
⇤
The previous proof was based on the min-max principle by E. Fischer [90]
and R. Courant [60] for self-adjoint operators A : D(A) ! H which are
bounded from below, i.e., h , A i a k k2 for some a > 1 and all 2 D(A).
For its formulation we denote by E0 (A)  E1 (A)  · · ·  En (A)  . . .
the eigenvalues of A taking into account multiplicity by repeating the eigenvalue) below the essential spectrum ess (A) with the convention that E N (A) =
inf ess (A) in case the number of these eigenvalues is N 2 N0 .
Proposition 3.20 (Min-max principle) Let A : D(A) ! H be a self-adjoint
operator which is bounded from below. Then
E0 (A) =
En (A) =
inf
h', A'i
'2D(A), k'k=1
sup
1 ,..., n 2H
inf
h', A'i ,
'2D(A), k'k=1
'?span{ 1 ,..., n }
n2N.
(3.34)
Proof Without loss of generality we assume that there are n eigenvalues
E0 , . . . , En 1 below En and we denote by '0 , . . . , 'n 1 the corresponding orthonormal eigenfunctions. Then for any ' ? span{'0 , . . . , 'n 1 } with ' 2
D(A), one has ' = P[En ,1) (A)' such that by the spectral representation
Z 1
h' , A'i = h' , A P[En ,1) (A)'i =
µ' (d ) En k'k2 .
En
Picking '0 , . . . , 'n 1 for 1 , . . . , n in the supremum in (3.34) shows that the
left side is bounded from above by the right side. To establish the converse
inequality, we pick for 'n ? span{'0 , . . . , 'n 1 } either a normalized eigenfunction corresponding to En or, in case the latter is the bottom of the esssential
Exercises
43
spectrum, an approximate normalized eigenfunction, i.e. k(A En )'n k < ",
P
cf. Proposition 3.9. Then any normalized linear combination, ' = nj=0 ↵ j ' j
obeys:
n
X
h' , A'i 
|↵ j |2 E j + "  En + " .
j=0
Since at least one of the ' j ’s in belongs to span{
chosen arbitrarily small, this completes the proof.
1, . . . ,
n}
?
and " can be
⇤
The proof shows that the infima and suprema are attained as long as En (A)
is an eigenvalue. The special case n = 0 is called the Rayleigh-Ritz principle.
Proposition 3.20 implies that ranked eigenvalues form monotone functions of
self-adjoint operators. Thus for any pair of operators A  B, and N(A; E) :=
tr P( 1,E) (A) (which equals 1 for E > inf ess (A)) one has
and
En (A)  En (B)
N(B; E)  N(A; E)
for all n
(3.35)
for all E .
(3.36)
More generally, tr F(A) for monotone functions F form monotone functions of
the operator.
Exercises
3.1
A family of measure preserving transformations (S x ) x2Zd on a probability
space (⌦, A, P) is called mixing if for any pair of events A, B 2 A:
⇣
⌘
P A \ S x 1 B ! P(A) P(B) as |x| ! 1.
1. Show that mixing implies the ergodicity of (S x ) x2Zd .
d
2. Show that the shifts (3.3) defined on the canonical realization (RZ , B, P)
of the probability space corresponding to iid random variables, (!(⇠))⇠2Zd ,
are mixing.
3.2
3.3
Let X : ⌦ ! R [ {1} be a random variable, which is invariant under a
family (T x ) x2I of ergodic transformations.
1. Show that the distribution function ( 1, 1] 3 t 7! P (X  t) takes
values in the set {0, 1} only.
2. Show that P (X = c) = 1 for c := inf {t 2 ( 1, 1] | P (X  t) = 1}.
Derive a relation between the Laplacian with a constant magnetic field (3.7)
and the Harper operator (3.10).
Random Operators c M. Aizenman & S. Warzel
DRAFT
44
Ergodic operators
Notes
Our definition of ergodic operators is slightly more general (and not standard)
in comparison to what is usually discussed in the literature [122, 50]. In fact,
our notion of standard ergodic operators agree with the usual ergodic operators.
In the standard case, much more can be said: the range of the spectral projections are almost surely either zero or infinity. Therefore the discrete spectrum
is empty. For details see [122, 50].
The existence of the DOS measure as a limit of finite-volume quantities can
be established in various ways. One alternative starting point is the sub- respectively superadditivity of the Dirchlet- respectively Neumann finite-volume
integrated DOS as expressed in (3.32). Here the convergence (3.27) follows
for these boundary conditions from the Akcoglu-Krengel subadditive ergodic
theorem [158]. For details, see [128, 122]. Another approach is based on the
representation of the Laplace transform of the DOS in terms of suitable integrals over Brownian motion, cf. [196, 50].
For random Schrödinger operators on L2 (Rd ) the DOS measure ⌫ may be
defined in similar way [122, 50]. The only di↵erence is that ⌫ is no longer
a bounded measure. Therefore the notion of weak convergence should be replaces by vague convergence. For a recent survey about the density of states,
see [132].