ISSN:1083-589X in PROBABILITY
A note on the tensor product
of two random unitary matrices
Tomasz Tkocz
∗Abstract
In this note we consider the point process of eigenvalues of the tensor product of two independent random unitary matrices of size m×m andn×n. Whenn becomes large, the process behaves like the superposition ofmindependent sine processes. Whenmandngo to infinity, we obtain the Poisson point process in the limit. Keywords: Random matrices; Circular Unitary Ensemble; tensor product; sine point process; Poisson point process.
AMS MSC 2010:Primary 60B20, Secondary 15B52.
Submitted to ECP on December 8, 2012, final version accepted on February 28, 2013.
1
Introduction
In quantum mechanics the time evolution of two noninteracting subsystems can be described by an operatoreitH⊗eitH0, whereH andH0 are Hamiltonians of the
subsys-tems (see e.g. chapters 2.2 and 3.1 in [2]). In applications, the unitary operatoreitH,
which isa priori complicated, is replaced by a random unitary matrix, to make a model tractable. This powerful idea goes back to E. Wigner. Here by an×nrandom unitary matrix we mean a matrix drawn according to the Haar measure on the unitary group
U(n). From this point of view it seems natural to study asymptotic local properties of spectra of the tensor productAm⊗Bn of two independent m×m andn×n random
unitary matrices, to which this short note is devoted. The note, in a sense, continues the investigations commenced in [5].
Some preliminaries are presented in the rest of this section, and the main result is stated. The proofs are provided in the next section. The last section is devoted to some concluding remarks concerning the tensor product of more than two matrices.
1.1 Background and notation
For a simple point processτ onRwe denote itsk-th correlation function, when it exists, byρ(τk) (for the definitions see e.g. [4]). Let us introduce three point processes
Π,Σ, andΞn. ByΠwe shall denotethe Poisson point process onRfor whichρ
(k) Π ≡1
for allk. ByΣwe shall denotethe sine point process onRwhich has the correlation functions
ρ(Σk)(x1, . . . , xk) = det [Q(xi, xj)] k
i,j=1, (1.1)
wherethe sine kernelQ(x, y) =q(x−y)andqreads as follows
q(u) =sin(πu)
πu . (1.2)
Given a n×nrandom unitary matrix with eigenvalueseiξ1, . . . , eiξn, where ξ
i ∈[0,2π)
are eigenphases, we define the point processΞn = {ξ1, . . . , ξn}. It is well known that
this process is determinantal with the kernelSn(x, y) =sn(x−y), where
sn(u) =
1 2π
sin nu2
sin u2, (1.3)
i.e.,
ρ(Ξk)
n(x1, . . . , xk) = det [Sn(xi, xj)] k
i,j=1. (1.4)
Since 2nπsn 2nπu −−−−→
n→∞ q(u), when n becomes large, the process n
2π(Ξn −π) of the
rescaled eigenphases of then×n random unitary matrix locally behaves as the sine processΣ.
Bysuperpositionof two simple point processesΨ ={ψ1, . . . , ψM},Φ ={φ1, . . . , φN},
M, N ≤ ∞we mean the unionΨ∪Φ ={ψ1, . . . , ψM, φ1, . . . , φN}.
1.2 Results
Given two independent m×m and n×n random unitary matrices A and A0 we get two independent point processes of their eigenphasesΞm ={ξ1, . . . , ξm}andΞ0n =
{ξ10, . . . , ξn0}respectively. We define the point processΞm⊗Ξ0nof the eigenphases of the
matrixA⊗A0 as
Ξm⊗Ξ0n={ξi+ξ0jmod2π, i= 1, . . . , m, j= 1, . . . , n}.
It has been recently shown [5, Theorem 1] that the process n2π2(Ξn⊗Ξ0n)behaves locally
as the Poisson point process onR+. We refine this result and investigate what happens
whennbecomes large withmbeing fixed, or when bothmandnbecomes large but not necessarilym=n.
Theorem 1.1. Let Ξmand Ξ0n be point processes of eigenphases of two independent
m×mandn×nrandom unitary matrices. LetΣ1, . . . ,Σmbe independent sine processes and letΠbe a Poisson process onR. Then for eachk≤nthek-th correlation function of the processΞm⊗Ξ0nexists and
(a) ρ(mnk)
2π(Ξm⊗Ξ0n−π)−−−−→n→∞ ρ
(k)
mΣ1∪...∪mΣm,
(b) ρ(mnk)
2π(Ξm⊗Ξ0n−π)−−−−−→m,n→∞ ρ
(k) Π ,
uniformly on all compact sets inRk.
Remark 1.2(Weak convergence). According to [4], by a point process onRwe mean a random variable with values in the metric spaceM(R)ofσ-finite Borel measures onR (counting measures correspond to locally finite subsets ofR) endowed with the topology generated by the functionsµ7→R
fdµfor continuous, compactly supportedf. We say that a sequence of point processes(τn)convergesin distributionto a point processτ if the lawνnofτn converges weakly to that ofτ, sayν, in the spaceM1(M(R))of proba-bility measures onM(R), i.e. R
fdνn →Rfdν for any bounded continuous function on
Remark 1.3(Heuristic behind (a)). In view of the mentioned theorem from [5] result (b) should not be surprising. Neither is (a) as in the simplest casem= 2we have
Ξ2⊗Ξ0n={ξ1+ξ10 mod2π, . . . , ξ1+ξn0 mod2π}
∪ {ξ2+ξ01mod2π, . . . , ξ2+ξn0 mod2π}.
After shifting and rescaling we end up with two families of the rescaled eigenphases of an×nrandom unitary matrix which differ roughly by a large shift 2nπ(ξ1−ξ2)which is independent on the matrix. That makes the families independent and in the limit, according toρ(kn)
2π(Ξn−π)−−−−→n→∞ ρ
(k)
Σ , they look like sine processes.
Remark 1.4(Superposition of many sine processes becomes a Poisson point process). Notice that for any independent copiesΦ1, . . . ,Φmof a point processΦwe have
ρ(Φk1∪) ...∪Φ
m(x1, . . . , xk) = m∧k
X
p=1 X
π∈S(k,p) m! (m−p)!
p
Y
j=1 ρ(]πj)
Φ ((xi)i∈πj),
whereS(k, p)is the collection of all partitions intopnonempty pairwise disjoint subsets of the set{1, . . . , k}. By this we mean that ifπis such a partition thenπ={π1, . . . , πp}, whereπq ={π(q,1), . . . , π(q, ]πq)}is theq-th block of the partitionπ.
Along with the fact that if we rescale, ρ(λkΦ)(x) becomes λ1kρ
(k) Φ
1
λx
, the previous observation yields
ρ(mk)Σ1∪...∪mΣ
m(x) = m∧k
X
p=1 X
π∈S(k,p) 1 mk
m! (m−p)!
p
Y
j=1 ρ(]πj)
Σ 1
m(xi)i∈πj
. (1.5)
Whenmgoes to infinity we thus get
lim
m→∞ρ
(k)
mΣ1∪...∪mΣm(x) = limm→∞ p
Y
j=1 ρ(1)Σ
1
m(xi)i∈πj
= 1 =ρ(Πk).
It retrieves the special case of a quite expected phenomenon put forward in [3]. Namely, the authors say “[...] a Poisson process can be viewed as an infinite superposition of determinantal or permanental point processes” (see Theorem 4 therein and the two preceding paragraphs). Regarding Theorem (a) that implies
lim
m→∞nlim→∞ρ
(k)
mn
2π(Ξm⊗Ξ0n−π)= 1.
Note that in the second part of the theorem we establish a stronger statement, that letting the dimensions of two independent random unitary matrices to infinity reduces all the correlations in their tensor product.
2
Proofs
For the sake of convenience, let us recall a few basic facts which shall be frequently used.
Note the following easy estimate (for the definition see (1.3))
sup
x∈R
2π nsn(x)
= 1. (2.1)
Combined with Hadamard’s inequality (see e.g. (3.4.6) in [1]), it allows us to bound the correlation functions,
sup
x∈Rk
ρ(Ξk)
n(x)≤k k/2ks
nkk∞=
kk/2 (2π)kn
2.1 Proof of Theorem (a)
LetΘm,n= mn2π(Ξm⊗Ξ0n−π). Fix a natural numberk. Since we will letngo to infinity,
we may assume that k ≤ n. First we show that there exists functions ρ(Θk)
m,n:R k −→
[0,∞)so that for any bounded and measurable functionf:Rk−→Rwe have
EXf(θ1, . . . , θk) =
Z
Rk
f(x)ρ(Θk)
m,n(x)dx,
where the summation is over all orderedk-tuples(θ1, . . . , θk)of distinct points ofΘm,n.
This will prove thatρ(Θk)
m,n are the correlation functions ofΘm,n. Then we will deal with
the limit whenn→ ∞.
Fix f. Since for each s = 1, . . . , k, θs = mn2π(ξis +ξ
0
js mod2π−π) for some is ∈
{1, . . . , m},js∈ {1, . . . , n}we can write
EXf(θ1, . . . , θk) =E
X
i∈{1,...,m}k j∈{1,...,n}k
f
mn
2π(ξis+ξ
0
js mod2π−π)
k
s=1
,
where the second sum is subject tok-tuples i, j such that the pairs (i1, j1), . . . ,(ik, jk)
are pairwise distinct. For sure it happens when all the js’s are distinct. Call these
choices ofiandj good and the restbad. So
EX i,j
f =E X goodi,j
f+E X badi,j
f.
First we handle the good sum. Some is’s may overlap and we will control it using
partitions of the set{1, . . . , k} intop ≤k∧mnonempty pairwise disjoint subsets (see Remark 1.4 for the notation) so thatis=itwheneversandtbelong to the same block
of a partition. We have
E X
goodi,j
f =
k∧m
X
p=1 X
π∈S(k,p)
E X
distinct
iπ(1,1),...,iπ(p,1)
X
distinct
j1,...,jk
f.
The sums overi’s andj’s have been separated. Therefore taking advantage of indepen-dence as well as recalling definitions of thep-th andk-th correlation functions of Ξm
andΞ0n we find
E X
goodi,j
f =X
p,π
Z
[0,2π]p
Z
[0,2π]k
f
mn
2π(xπ(s)+ysmod2π−π) k
s=1
ρ(Ξp)
m(x1, . . . , xp)ρ
(k) Ξ0
n(y1, . . . , yk)dx1. . .dxpdy1. . .dyk,
where we note π(s) = q ⇐⇒ s ∈ πq. Finally, we need to address the technicality
concerning the addition mod2π. Keeping in mind that we integrate over [0,2π]p and
[0,2π]kwe consider forη∈ {0,1}k the set
Uη =
x∈[0,2π]p, y∈[0,2π]k; ∀s≤k xπ(s)+ys<2πifηs= 0, and
xπ(s)+ys≥2πifηs= 1
.
Then onUη we havexπ(s)+ysmod2π=xπ(s)+ys−2πηs, thus changing the variables
onUη so thatzs= mn2π(xπ(s)+ys−2πηs−π)we get
E X
goodi,j
f = Z
Rk
f(z) X
p,π,η 1Wη(z)
Z
[0,2π]p
1Vη(x)ρ
(p) Ξm(x)
2π
mn k
ρ(Ξk0)
n(y(z, x))dx
!
whereys(z, x) =mn2πzs−xπ(s)+ 2πηs+π,
Vη=
x∈Rp; ∀s≤k 2π
mnzs+ 2πηs−π≤xπ(s)≤ 2π
mnzs+ 2πηs+π
,
and
Wη=z∈Rk; ∀s≤k zs≤mn/2ifηs= 0, andzs≥ −mn/2ifηs= 1 .
Summarizing, we have just seen that the correlation functionρ(Θk)
m,n(z)takes on the form
ρ(Θk)
m,n(z) =
X
p,π,η 1Wη(z)
Z
[0,2π]p
1Vη(x)ρ(Ξp)
m(x)
2π
mn k
ρ(Ξk0)
n(y(z, x))dx+Bm,n(z), (2.3)
where the termBm,ncorresponds to the sum over bad indicesEPbadi,jf. By the same
kind of reasoning we show that roughly
Bm,n(z) = k
X
p=1
k−1 X
q=1 X
π∈S(k,p)
τ∈S(k,q) X
η
1W˜η(z)
2π mn
kZ
[0,2π]p+q−k
1V˜η(x)ρ
(p)
Ξm(˜x(z, x))
ρ(Ξq0)
n(˜y(z, x))dx,
where the sums are over appropriate partitions and W˜η, V˜η are suitable sets which
appear after changing the variables. Now, by (2.2),
kρ(Ξp)
m·ρ
(q) Ξ0
nk∞≤
pp/2qq/2 (2π)p+qm
pnq, (2.4)
so
Bm,n(z)≤Ck
1 n,
where the constantCk depends only onk(roughly, it equals the number of summands
timeskk). Hence, when takingn→ ∞we will not have to take care aboutB m,n.
Let us have a look at (2.3) and compute now the limit of the first term whenn→ ∞. We observe that1Wη →1pointwise onRk. Moreover,P
η1Vη →1[0,2π)p, and1Vη →0
forη such thatηs6=ηtbutπ(s) =π(t)for somes 6=t. Thus we consider onlyη’s such
thatηs=ηtwheneverπ(s) =π(t)and then the following simple observation
2π mnsn
2π
mnu+v
−−−−→
n→∞
(
0, v6= 0 1
mq u m
, v= 0 (2.5)
yields for all theseη’s,
2π
mn k
ρ(Ξk0)
n(y) = det
2π
mnsn 2π
mn(zs−zt) + 2π(ηs−ηt) +xπ(t)−xπ(s) k s,t=1 −−−−→ n→∞ p Y j=1 det 1 mq z
s−zt
m
s,t∈πj
= 1 mk
p
Y
j=1 ρ(]πj)
Σ 1
m(zi)i∈πj
.
By estimate (2.2), mn2πkρ(Ξk0)
n(y)is bounded byk
k/2/mk, so the integrand in (2.3) can be
simply bounded. Thus by Lebesgue’s dominated convergence theorem
ρ(Θk)
m,n(z)−−−−→n→∞
X p,π 1 mk p Y j=1 ρ(]πj)
Σ 1
m(zi)i∈πj
·
Z
[0,2π]p
ρ(Ξp)
For anyp≤mthe integralR [0,2π)pρ
(p)
Ξm(x)dxjust equalsm!/(m−p)!. Consequently, we
finally obtain
ρ(Θk)
m,n(z1, . . . , zk)−−−−→n→∞
X
p,π
1 mk
m! (m−p)!
p
Y
j=1 ρ(]πj)
Σ 1
m(zi)i∈πj
.
In view of (1.5) this completes the proof.
2.2 Proof of Theorem (b)
Fix a pointz = (z1, . . . , zk)∈Rk. We letmandntend to infinity and want to prove
thatρ(Θk)
mn(z)tends to 1. Recall (2.3) and notice that due to estimate (2.4) all the terms
withp≤k−1are bounded above byCk/m, so we can write
ρ(Θk)
m,n(z) =O
1
m+ 1 n
+X
η
1Wη(z) Z
[0,2π]k 1Vη(x)
2π
mn k
ρ(Ξk)
m(x)ρ
(k) Ξ0
n(y(z, x)) dx.
Using the formulas for the correlation functions and the permutational definition of the determinant, we can put the integrand in the following form
1Vη(x) (2π)k ·det
2π
msm(xs−xt) k
s,t=1
·det
2π
n sn(ys−yt) k
s,t=1
=1Vη(x)
(2π)k 1 +
X
σ6=id orτ6=id
sgnσsgnτ
k
Y
i=1 2π
msm(xi−xσ(i))· 2π
n sn(yi−yτ(i)) !
,
where the second summation runs through permutationsσandτof k indices. The point is that each term in this sum tends to zero withmand ngoing to infinity as we have
2π
msm(xi−xσ(i)) a.e.
−−−−→
m→∞ 0forisuch thati6=σ(i), and
2π
nsn(yi−yτ(i)) a.e.
−−−−→
n→∞ 0ifi6=τ(i)
(see (2.5) and mind the fact that actually y depends on m and n). Recall also that
1Wη →1andP
η1Vη →1[0,2π)k. Moreover, (2.1) yields that the whole sum is bounded
by(k!)2/(2π)k. Therefore by Lebsegue’s dominated convergence theorem we conclude
that
ρ(Θk)
m,n(z)−−−−−→m,n→∞
Z
1[0,2π)k(x)
1
(2π)kdx= 1,
which finishes the proof.
3
Concluding remarks
At the very end we shall discuss the tensor product of more than two matrices. We only briefly sketch what can be easily inferred looking at the proof of the main result.
Let Ξl, Ξ0m, Ξn00 be the point processes of eigenphases of independent l×l, m×m,
andn×nrandom unitary matrices respectively. Proceeding along the same lines as in the proof of Theorem (a), we conclude that the point process lmn2π (Ξl⊗Ξ0m⊗Ξ00n−π)
locally behaves as the Poisson point process onRwhenl is fixed butm andntend to infinity. Indeed, the asymptotics of thek-th correlation functionρ(k)(z)of that process
is governed by the integrals
Z
[0,2π]p+k∩Vη
2π lmn
k ρ(Ξp)
l(x)ρ
(k) Ξ0
m(y)ρ
(k) Ξ00
n(w(x, y, z))dxdy,
which we then sum suitably. Expanding the determinantal correlation functions of
P
p,π
1
lk l!
(l−p)! = 1, where the last identity is due to the well-known combinatorial fact
thatPk
p=1]S(k, p)x(x−1)·. . .(x−p+ 1) =x
k. The same line of reasoning applies also
when in additionl→ ∞. Then the asymptotics depends only on the integral
Z
[0,2π]2k
2π
lmn k
ρ(Ξk)
l(x)ρ
(k) Ξ0
m(y)ρ
(k) Ξ00
n(w(x, y, z))dxdy.
Again, we carry on as in the proof of Theorem (b).
Let A(nii), i = 1, ,2, . . ., be independent ni ×ni random unitary matrices. The other
cases of tensor productsNM i=1A
(i)
ni, when for instance all but one ofni’s are fixed, seem
to be more delicate and we do not wish to go into detail here. Moreover, it looks challenging to consider the tensor products when the number of terms M tends to infinity and(ni)∞i=1is fixed. The simplest case ofni= 2,i≥1has been addressed in [5].
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