random matrices and their limits · random matrices and operators deterministic limits of random...

Random Matrices and Their Limits

Roland Speicher

Saarland UniversitySaarbrücken, Germany

supported by ERC Advanced Grant“Non-Commutative Distributions in Free Probability”

Roland Speicher (Saarland University) Random Matrices and Their Limits 1 / 24

Random Matrices and Operators

Section 1




Deterministic limits of random matrices

Fundamental observationMany random matrices show, via concentration, for N →∞ almost surelya deterministic behaviour.

Interesting observationMany random matrices show, via concentration, for N →∞ almost surelya deterministic interesting behaviour.

Interesting observation for the operator algebraic inclinedMany random matrices show, via concentration, for N →∞ almost surelya deterministic behaviour, which can be described by interesting operatorson Hilbert spaces (or their generated C∗-algebras or von Neumannalgebras)



Random matrices and operators

Fundamental observation of Voiculescu (1991)

Limit of random matrices can oftenbe described by “nice” and“interesting” operators on Hilbertspaces(which, in the case of severalmatrices, describe interesting vonNeumann algebras)



Convergence of random matrices

(X(N)1 , . . . , X(N)

m ) −→ (x1, . . . , xm)

random matrices almost surely operators(MN (C), tr) (A, τ),A ⊂ B(H)

Convergence in distributionWe have for all polynomials p in m non-commuting variables

limN→∞

tr[p(X(N)1 , . . . , X(N)

m )] = τ [p(x1, . . . , xm)]

Strong convergenceconvergence in distributionand for all polynomials p

limN→∞

‖p(X(N)1 , . . . , X(N)

m )‖ = ‖p(x1, . . . , xm)‖



Our most beloved example:independent GUE → free semicirculars

One-matrix case: classical commuting caseNote: XN → x means that all moments of XN converge to correspondingmoments of x, hence the distributions (in classical sense of probabilitymeasures on R) converge... so we can just look on pictures of distributions ...

Multi-matrix case: non-commutative case(XN , YN )→ (x, y) means convergence of all moments, but this does notcorrespond to convergence of probability measures on R2

... if we still want to see classical objects and pictures we can look onp(XN , YN )→ p(x, y) for (sufficiently many) functions p in XN , YN ...




One-matrix case: classical commuting caseNote: XN → x means that all moments of XN converge to correspondingmoments of x, hence the distributions (in classical sense of probabilitymeasures on R) converge

... so we can just look on pictures of distributions ...






One-matrix case: classical commuting caseNote: XN → x means that all moments of XN converge to correspondingmoments of x, hence the distributions (in classical sense of probabilitymeasures on R) converge... so we can just look on pictures of distributions ...





One-matrix case: classical random matrix caseXN GUE random matrix

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

x = l + l∗,

l one-sided shift on⊕n≥0Cen

len = en+1

l∗en+1 = en, l∗e0 = 0

τ(a) = 〈e0, ae0〉

XN → x in distribution (Wigner 1955)‖XN‖ → ‖x‖ = 2 (Füredi, Komlós 1981)




−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

x = l + l∗,


len = en+1

l∗en+1 = en, l∗e0 = 0

τ(a) = 〈e0, ae0〉

XN → x in distribution (Wigner 1955)

‖XN‖ → ‖x‖ = 2 (Füredi, Komlós 1981)




−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x

x = l + l∗,


len = en+1

l∗en+1 = en, l∗e0 = 0

τ(a) = 〈e0, ae0〉

XN → x in distribution (Wigner 1955)‖XN‖ → ‖x‖ = 2 (Füredi, Komlós 1981)



Multi-matrix case: non-commutative caseXN , YN independent GUE

p(x, y) = xy + yx+ x2

−6 −4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x = l1 + l∗1, y = l2 + l∗2

two copies of one-sided shift indifferent directions (creation andannihilation operators on fullFock space; Cuntz algebra)

τ(a) = 〈Ω, aΩ〉

XN , YN → x, y in distribution (Voiculescu 1991)p(XN , YN ) → p(x, y) in distribution‖p(XN , YN )‖ → ‖p(x, y)‖ (Haagerup and Thorbjørnsen 2005)



Multi-matrix case: non-commutative caseXN , YN independent GUE

p(x, y) = xy + yx+ x2

−6 −4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉

XN , YN → x, y in distribution (Voiculescu 1991)

p(XN , YN ) → p(x, y) in distribution‖p(XN , YN )‖ → ‖p(x, y)‖ (Haagerup and Thorbjørnsen 2005)



Multi-matrix case: non-commutative caseXN , YN independent GUE p(x, y) = xy + yx+ x2

−6 −4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉

XN , YN → x, y in distribution (Voiculescu 1991)p(XN , YN ) → p(x, y) in distribution

‖p(XN , YN )‖ → ‖p(x, y)‖ (Haagerup and Thorbjørnsen 2005)



Multi-matrix case: non-commutative caseXN , YN independent GUE p(x, y) = xy + yx+ x2

−6 −4 −2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉

XN , YN → x, y in distribution (Voiculescu 1991)p(XN , YN ) → p(x, y) in distribution‖p(XN , YN )‖ → ‖p(x, y)‖ (Haagerup and Thorbjørnsen 2005)



Goal: Go over from Polynomials to More GeneralClasses of Functions

For m = 1, one hasfor all continuous f :

limN→∞ tr[f(X(N))] = τ [f(x)]

limN→∞ ‖f(X(N))‖ = ‖f(x)‖

Which classes of functions in non-commuting variables

continuous ???analytic free analysisrational !!!polynomials !!!






limN→∞ ‖f(X(N))‖ = ‖f(x)‖

Which classes of functions in non-commuting variables

continuous ???analytic free analysisrational !!!

polynomials !!!






limN→∞ ‖f(X(N))‖ = ‖f(x)‖

Which classes of functions in non-commuting variablescontinuous ???

analytic free analysisrational !!!

polynomials !!!






limN→∞ ‖f(X(N))‖ = ‖f(x)‖

Which classes of functions in non-commuting variablescontinuous ???analytic free analysis

rational !!!

polynomials !!!






limN→∞ ‖f(X(N))‖ = ‖f(x)‖

Which classes of functions in non-commuting variablescontinuous ???analytic free analysisrational !!!polynomials !!!


Non-Commutative Rational Functions

Section 2




Non-commutative rational functionsNon-commutative rational functions (Amitsur 1966, Cohn 1971)

A rational function r(y1, . . . , ym) in non-commuting variablesy1, . . . , ym is anything we can get by algebraic operations, includinginverses, from y1, . . . , ym,

like

(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1

moduloI identifying “algebraically equivalent” expressionsI not inverting 0

this is not obvious to decide, e.g., one has rational identities like

y−12 + y−12 (y−13 y−11 − y−12 )−1y−12 − (y2 − y3y1)−1 = 0

after all, it works and gives a skew field

C (<y1, . . . , ym )> “free field”




A rational function r(y1, . . . , ym) in non-commuting variablesy1, . . . , ym is anything we can get by algebraic operations, includinginverses, from y1, . . . , ym, like

(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1



y−12 + y−12 (y−13 y−11 − y−12 )−1y−12 − (y2 − y3y1)−1 = 0


C (<y1, . . . , ym )> “free field”




A rational function r(y1, . . . , ym) in non-commuting variablesy1, . . . , ym is anything we can get by algebraic operations, includinginverses, from y1, . . . , ym, like

(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1



y−12 + y−12 (y−13 y−11 − y−12 )−1y−12 − (y2 − y3y1)−1 = 0


C (<y1, . . . , ym )> “free field”Roland Speicher (Saarland University) Random Matrices and Their Limits 11 / 24


A rational function r(y1, . . . , ym) in non-commuting variables y1, . . . , ymcan be realized more systematically by going over to matrices

r(y1, . . . , ym) = uQ−1v

where, for some N ,I u is 1×NI Q is N ×N and invertible in MN (C (<y1, . . . , ym )>)I v is N × 1I and all entries are polynomials (can actually be chosen with degree ≤ 1)

(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1

=(12 0

)·(−1 + 1

4y114y2

14y2 −1 + 1

4y1

)−1·(

120

)this is essentially (in the case of polynomials) the “linearization trick”which we use in free probability, for example, to calculate distributionsof polynomials in free variables




r(y1, . . . , ym) = uQ−1v


(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1

=(12 0

)·(−1 + 1

4y114y2

14y2 −1 + 1

4y1

)−1·(

120

)

this is essentially (in the case of polynomials) the “linearization trick”which we use in free probability, for example, to calculate distributionsof polynomials in free variables




r(y1, . . . , ym) = uQ−1v


(4− y1)−1 + (4− y1)−1y2((4− y1)− y2(4− y1)−1y2

)−1y2(4− y1)−1

=(12 0

)·(−1 + 1

4y114y2

14y2 −1 + 1

4y1

)−1·(

120

)this is essentially (in the case of polynomials) the “linearization trick”which we use in free probability, for example, to calculate distributionsof polynomials in free variables



Historical remarkNote that this linearization trick is a well-known idea in many othermathematical communities, known under various names like

Higman’s trick (Higman “The units of group rings”, 1940)recognizable power series (automata theory, Kleene 1956,Schützenberger 1961)linearization by enlargement (ring theory, Cohn 1985; Cohn andReutenauer 1994, Malcolmson 1978 )descriptor realization (control theory, Kalman 1963; Helton,McCullough, Vinnikov 2006)linearization trick (Haagerup, Thorbjørnsen 2005 (+Schultz 2006);Anderson 2012)



Applying nc rational functions to operators

... there are a couple of issues arising ...Not every relation in C (<y1, . . . , ym )> must necessarily hold in any algebra(even if it makes sense there)

In C (<y1, y2 )> we have

y1(y2y1)−1y2 = 1

Let v be an isometry which is not unitary, i.e. vv∗ = 1, v∗v 6= 1 Thenwe have

v∗(vv∗)−1v = v∗v 6= 1

Solution: By Cohn, we know that the above is the only issue, hencewe are okay if we avoid non-unitary isometries (also in matrices overthe algebra);

I hence only work in stably finite algebrasI this is the case in our situations, where we have a tracial state






y1(y2y1)−1y2 = 1


v∗(vv∗)−1v = v∗v 6= 1


I hence only work in stably finite algebras

I this is the case in our situations, where we have a tracial state






y1(y2y1)−1y2 = 1


v∗(vv∗)−1v = v∗v 6= 1


I hence only work in stably finite algebrasI this is the case in our situations, where we have a tracial state




... there are a couple of issues arising ...In C (<y1, . . . , ym )> every r(y1, . . . , ym) 6= 0 is invertible. This is ofcourse not true when we apply r to operators:

I r(x1, . . . , xm) = 0 could happen for 0 6= r ∈ C (<y1, . . . , ym )>I even if r(x1, . . . , xm) 6= 0, it does not need to be invertible in general

Possible solutions:I consider only r for which r(x1, . . . , xm) is invertible as bounded

operatorI or allow also unbounded operators





I r(x1, . . . , xm) = 0 could happen for 0 6= r ∈ C (<y1, . . . , ym )>

I even if r(x1, . . . , xm) 6= 0, it does not need to be invertible in generalPossible solutions:

I consider only r for which r(x1, . . . , xm) is invertible as boundedoperator

I or allow also unbounded operators







operator

I or allow also unbounded operators



Rational functions of random matrices and theirlimitProposition (Sheng Yin 2017)

Consider random matrices (X(N)1 , . . . , X

(N)m ) which converge to operators

(x1, . . . , xm) in the strong sense: for any polyomial p ∈ C〈y1, . . . , ym〉 wehave

limN→∞ tr[p(X(N)1 , . . . , X

(N)m )] = τ [p(x1, . . . , xm)]

limN→∞ ‖p(X(N)1 , . . . , X

(N)m )‖ = ‖p(x1, . . . , xm)‖

Then this strong convergence remains also true for rational functions: Letr ∈ C (<y1, . . . , ym )>, such that r(x1, . . . , xm) is defined as boundedoperator.Then we have almost surely that

r(X(N)1 , . . . , X

(N)m ) is defined for sufficiently large N

limN→∞ tr[r(X(N)1 , . . . , X

(N)m )] = τ [r(x1, . . . , xm)]

limN→∞ ‖r(X(N)1 , . . . , X

(N)m )‖ = ‖r(x1, . . . , xm)‖







limN→∞ tr[p(X(N)1 , . . . , X

(N)m )] = τ [p(x1, . . . , xm)]

limN→∞ ‖p(X(N)1 , . . . , X

(N)m )‖ = ‖p(x1, . . . , xm)‖

Then this strong convergence remains also true for rational functions: Letr ∈ C (<y1, . . . , ym )>, such that r(x1, . . . , xm) is defined as boundedoperator.

Then we have almost surely that

r(X(N)1 , . . . , X


limN→∞ tr[r(X(N)1 , . . . , X

(N)m )] = τ [r(x1, . . . , xm)]

limN→∞ ‖r(X(N)1 , . . . , X

(N)m )‖ = ‖r(x1, . . . , xm)‖







limN→∞ tr[p(X(N)1 , . . . , X

(N)m )] = τ [p(x1, . . . , xm)]

limN→∞ ‖p(X(N)1 , . . . , X

(N)m )‖ = ‖p(x1, . . . , xm)‖

Then this strong convergence remains also true for rational functions: Letr ∈ C (<y1, . . . , ym )>, such that r(x1, . . . , xm) is defined as boundedoperator.Then we have almost surely that

r(X(N)1 , . . . , X


limN→∞ tr[r(X(N)1 , . . . , X

(N)m )] = τ [r(x1, . . . , xm)]

limN→∞ ‖r(X(N)1 , . . . , X

(N)m )‖ = ‖r(x1, . . . , xm)‖



Proofby recursion on complexity of formulas with respect to inversionsmain step: controlling taking inverse, by approximations bypolynomials, uniformly in approximating matrices and limit operators

convergence for polynomials convergence for rational functions

strong =⇒ strongin distribution 6=⇒ in distribution





strong =⇒ strong

in distribution 6=⇒ in distribution





strong =⇒ strongin distribution

6=⇒ in distribution





strong =⇒ strongin distribution 6=⇒ in distribution



Distribution of random matrices and their limit forr(x, y) = (4− x)−1 + (4− x)−1y

((4− x)− y(4− x)−1y

)−1y(4− x)−1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5

6

7

XN , YN independent GUE

x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉

r(XN , YN ) → r(x, y) in distribution‖r(XN , YN )‖ → ‖r(x, y)‖




((4− x)− y(4− x)−1y

)−1y(4− x)−1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5

6

7 XN , YN independent GUE

x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉





((4− x)− y(4− x)−1y

)−1y(4− x)−1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5

6


x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉

r(XN , YN ) → r(x, y) in distribution

‖r(XN , YN )‖ → ‖r(x, y)‖




((4− x)− y(4− x)−1y

)−1y(4− x)−1

0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.650

1

2

3

4

5

6


x = l1 + l∗1, y = l2 + l∗2


τ(a) = 〈Ω, aΩ〉



Unbounded Operators

Section 3

Unbounded Operators


Unbounded Operators

In the limit (x1, . . . , xm) ⊂ (A, τ) we are in a II1-situationunbounded operators U(A) affiliated to vN algebra A form a ∗-algebraand a ∈ U(A) is invertible if and only if a has no zero divisors

(a has zero divisor: ∃b ∈ U(A) with b 6= 0 and ab = 0)

we expectr(x1, . . . , xm) for r ∈ C (<y1, . . . , ym )> is well-defined and has, for r 6= 0,no zero divisors for

x1, . . . , xm free semicircularsmore general, limit operators of “nice” random matrix models(which should be operators having maximal free entropy dimension)

we know (Charlesworth-Shlyakhtenko and Mai-Speicher-Weber)p(x1, . . . , xm) for 0 6= p ∈ C〈y1, . . . , ym〉 has no zero divisors for

x1, . . . , xm free semicircularsx1, . . . , xm with maximal free entropy dimension = m


Unbounded Operators

In the limit (x1, . . . , xm) ⊂ (A, τ) we are in a II1-situationunbounded operators U(A) affiliated to vN algebra A form a ∗-algebraand a ∈ U(A) is invertible if and only if a has no zero divisors(a has zero divisor: ∃b ∈ U(A) with b 6= 0 and ab = 0)






Unbounded Operators



x1, . . . , xm free semicircularsmore general, limit operators of “nice” random matrix models

(which should be operators having maximal free entropy dimension)




Unbounded Operators







Unbounded Operators

What do we expect of nice operators

working definition of “nice”operators (x1, . . . , xm) are nice if δ(x1, . . . , xm) = m.

(X(N)1 , . . . , X

(N)m ) → (x1, . . . , xm)

nice random matrices → nice operators

nice operators should be without algebraicrelations in a very general sense

we knowI no polynomial relationsI no “local” polynomial relations

we don’t know (but would like to)I no rational relationsI no “local rational” relations


Unbounded Operators

What do we expect of nice operators

working definition of “nice”operators (x1, . . . , xm) are nice if δ(x1, . . . , xm) = m.

(X(N)1 , . . . , X

(N)m ) → (x1, . . . , xm)

nice random matrices → nice operatorsnice operators should be without algebraicrelations in a very general sense

we knowI no polynomial relationsI no “local” polynomial relations

we don’t know (but would like to)I no rational relationsI no “local rational” relations


Unbounded Operators

no polynomial relations 6=⇒ no rational relations

There exist many embeddings

C〈y1, . . . , ym〉 ⊂ skew field K

The free field C (<y1, . . . , ym )> is the universal field of fractions, every otherK as above is a localization of C (<y1, . . . , ym )>

Why no rational relations for nice operators?0 = r ∈ C (<y1, . . . ym )>⇐⇒ r(A1, . . . , Am) = 0 for all matrices

A1, . . . , Am of all sizesnice operators have many matrix tuples approximating them, thusshould behave like a “generic” matrix tuple of arbitrary size


Unbounded Operators






A1, . . . , Am of all sizes

nice operators have many matrix tuples approximating them, thusshould behave like a “generic” matrix tuple of arbitrary size


Unbounded Operators






A1, . . . , Am of all sizesnice operators have many matrix tuples approximating them, thusshould behave like a “generic” matrix tuple of arbitrary size


Unbounded Operators

QuestionHow much “regularity” of the distribution of (x1, . . . , xm) ⊂ (A, τ) isnecessary to have

C (<x1, . . . , xm )> ≡ free field

where C (<x1, . . . , xm )> is the division closure of C〈x1, . . . , xm〉 inunbounded operators, i.e., the smallest algebra

containing C〈x1, . . . , xm〉being closed under taking inverse, when possible (i.e, when no zerodivisor)


Unbounded Operators

What do we know about the division closure in concrete casesif u1, . . . , um are free unitary elements, then (Linnell 1993)

C (<u1, . . . , um )> ≡ free field

if x1, . . . , xm are free and the distribution of each xi has no atoms,then we have (Shlyakhtenko, Skoufranis; Dykema, Pascoe; Yin)

C (<x1, . . . , xm )> ≡ skew field (hence no zero divisors)?≡ free field

there are examples where there are non-trivial rational relations, butno zero-divisors, like for free semicirculars x1, x2 (Dykema, Pascoe)

a := x1, b := x1x2x1, c := x1x22x1

a, b, c are algebraically free, but satisfy rational relation ba−1b = c

what in the general nice case of maximal entropy dimension???


Unbounded Operators

What do we know about the division closure in concrete casesif u1, . . . , um are free unitary elements, then (Linnell 1993)

C (<u1, . . . , um )> ≡ free field

if x1, . . . , xm are free and the distribution of each xi has no atoms,then we have (Shlyakhtenko, Skoufranis; Dykema, Pascoe; Yin)

C (<x1, . . . , xm )> ≡ skew field (hence no zero divisors)?≡ free field

there are examples where there are non-trivial rational relations, butno zero-divisors, like for free semicirculars x1, x2 (Dykema, Pascoe)

a := x1, b := x1x2x1, c := x1x22x1

a, b, c are algebraically free, but satisfy rational relation ba−1b = c

what in the general nice case of maximal entropy dimension???Roland Speicher (Saarland University) Random Matrices and Their Limits 24 / 24

Unbounded Operators

1

ISBN 978-1-4939-6941-8

The Fields Institute for Research in Mathematical Sciences

Fields Institute MonographsFields Institute Monographs 35

James A. MingoRoland Speicher

Free Probability and Random Matrices

Free Prob

ability an

d

Ran

dom

Matrices

Min

go · Speich

er

35

James A. Mingo · Roland Speicher Free Probability and Random Matrices

Th is volume opens the world of free probability to a wide variety of readers. From

its roots in the theory of operator algebras, free probability has intertwined with

non-crossing partitions, random matrices, applications in wireless communications,

representation theory of large groups, quantum groups, the invariant subspace problem,

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relation of free probability to random matrices, but also touches upon the operator

algebraic, combinatorial, and analytic aspects of the theory.

Th e book serves as a combination textbook/research monograph, with self-contained

chapters, exercises scattered throughout the text, and coverage of important ongoing

progress of the theory. It will appeal to graduate students and all mathematicians

interested in random matrices and free probability from the point of view of operator

algebras, combinatorics, analytic functions, or applications in engineering and statistical

physics.

9 781493 969418


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