Guilherme Silva - Unicamp · Guilherme Silva University of Michigan - USA Guilherme Silva Random matrices. Random matrices I Random matrix: matrix whose entries are random variables

Random matrices

Guilherme SilvaUniversity of Michigan - USA

Guilherme Silva Random matrices

Random matrices

I Random matrix: matrix whose entries are randomvariables

I First systematic studies in the 1950’s by Dyson in thecontext of high energy physics

I Deep connections with number theory, statisticalmechanics, PDE’s, telecommunications, dynamicalsystems, approximation theory, among many others


Random matrices





Random matrices





Random matrices





Basic (asymptotic) questions

• We are typically interested in large random matrices

• Questions of interest when n→∞

I Distribution of eigenvalues and eigenvectors

I Behavior of extremal eigenvalues

I Typical location of eigenvectors

I Correlations between nearby eigenvalues

I Typical spacing between consecutive eigenvalues
























































First part:

Random Matrix Theory, why?


Examples and applications - CUE

Circular Unitary Ensemble: space U(N) of N ×N unitarymatrices equipped with uniform distribution (normalized Haarmeasure)







I Eigenvalues eiθ1 , . . . , eiθN (0 ≤ θ1 ≤ . . . ≤ θN < 2π) ofU ∈ U(N) distribute according to

1

ZN

∏j<k

|eiθk − eiθk |2dθ1 . . . dθN ,

and consequently

E(θk+1 − θk) =1

2πN




I Eigenvalues eiθ1 , . . . , eiθN (0 ≤ θ1 ≤ . . . ≤ θN < 2π) ofU ∈ U(N) distribute according to

1

ZN

∏j<k

|eiθk − eiθk |2dθ1 . . . dθN ,

and consequently

E(θk+1 − θk) =1

2πN




I Rescale φk = θk2πN so that

E (φk+1 − φk) = 1

I For 0 ≤ α < β [Dyson, 1962]

limN→∞

1

NE (#(φn, φm) | α ≤ φm − φn ≤ β)

=

∫ β

α

(1−

(sinπx

πx

)2)dx




I Rescale φk = θk2πN so that

E (φk+1 − φk) = 1

I For 0 ≤ α < β [Dyson, 1962]

limN→∞

1

NE (#(φn, φm) | α ≤ φm − φn ≤ β)

=

∫ β

α

(1−

(sinπx

πx

)2)dx


Examples and applications - CUE and RiemannHyphothesis

The Riemann zeta function

ζ(s) =

∞∑n=1

1

ns=

∏p prime

(1− 1

ps

)−1

, Re s > 1,

has a meromorphic extension to C with simple pole at s = 1and no other poles.




ζ(s) =

∞∑n=1

1

ns=

∏p prime

(1− 1

ps

)−1

, Re s > 1,


I It has simple zeros at s = −2,−4,−6 . . . (the so calledtrivial zeros)

I It has no other zeros outside the strips ∈ C | 0 < Re s < 1




ζ(s) =

∞∑n=1

1

ns=

∏p prime

(1− 1

ps

)−1

, Re s > 1,


I It has simple zeros at s = −2,−4,−6 . . . (the so calledtrivial zeros)

I It has no other zeros outside the strips ∈ C | 0 < Re s < 1




ζ(s) =

∞∑n=1

1

ns=

∏p prime

(1− 1

ps

)−1

, Re s > 1,


I It has an infinite number of zeros on the vertical lineL := s ∈ C | Re s = 1/2

I The Riemann Hypothesis conjectures that every nontrivialzero of ζ belongs to L




ζ(s) =

∞∑n=1

1

ns=

∏p prime

(1− 1

ps

)−1

, Re s > 1,


I It has an infinite number of zeros on the vertical lineL := s ∈ C | Re s = 1/2

I The Riemann Hypothesis conjectures that every nontrivialzero of ζ belongs to L



Denote by σn = 12 + itn the zeros of ζ on L with tn+1 > tn > 0.

I The rescaled quantities wn = 12π tn log tn

2π have the propertythat

limW→∞

1

W#wn ∈ [0,W ] = 1,

so that E(wn+1 − wn) ≈ 1.

I Conjecture [Montgomery, 1973]: For 0 < α < β,

limW→∞

1

W#(wn, wm) ∈ [0,W ]2 | α ≤ wm − wn ≤ β

=

∫ β

α

(1−

(sinπx

πx

)2)dx






limW→∞

1

W#wn ∈ [0,W ] = 1,



limW→∞

1

W#(wn, wm) ∈ [0,W ]2 | α ≤ wm − wn ≤ β

=

∫ β

α

(1−

(sinπx

πx

)2)dx






limW→∞

1

W#wn ∈ [0,W ] = 1,



limW→∞

1

W#(wn, wm) ∈ [0,W ]2 | α ≤ wm − wn ≤ β

=

∫ β

α

(1−

(sinπx

πx

)2)dx


Examples and applications - Random Schrodingeroperators and sparse matrices

I The Anderson model (or random Schrodinger operator) isthe operator

H = −∆ + λV

acting on l2(Zd), where V is a random potential, λ > 0 is a(disorder) parameter and

∆φ(u) =∑|u−v|=1

(φ(u)− φ(v))

is the discrete Laplacian

I Labelling the vertices of Zd, the operator H can berepresented as a two-sided infinite random band matrix,with random entries along the diagonal and 1’s in theremaining non-zero entries



I The Anderson model (or random Schrodinger operator) isthe operator

H = −∆ + λV

acting on l2(Zd), where V is a random potential, λ > 0 is a(disorder) parameter and

∆φ(u) =∑|u−v|=1

(φ(u)− φ(v))

is the discrete Laplacian

I Labelling the vertices of Zd, the operator H can berepresented as a two-sided infinite random band matrix,with random entries along the diagonal and 1’s in theremaining non-zero entries



I For example, if we assume d = 1, the (truncated) matrixrepresentation H of H is

H :=

V1 11 V2 1

1 V3 1. . . . . .

1 Vn−1 11 Vn

where the Vk’s are random



For d ≥ 3, one of the forms of the universality conjectureassures that

I If λ is sufficiently large (but still finite) every eigenfunction φis localized: |φ(u)| is exponentially decreasing with u

I If λ is sufficiently small (but still positive) everyeigenfunction is delocalized












Examples and applications - Non-intersectingBrownian motions

I The Brownian motion is a stochastic process W = Wt

(t ≥ 0) on R satisfying the following properties

• W0 = 0 with probability 1• With probability 1, t 7→Wt is continuous• For any t1 < · · · < tn, the random variablesWt1 ,Wt2−t1 , . . . ,Wtn−tn−1

are independent• If s < t, the random variables Wt−s and Wt −Ws are

(equally distributed) centered normal variables withvariance t− s

I Describes the motion of a particle suspended in a fluid

I It is the natural limit for random walks

I Appears in stochastic calculus, financial market models,statistical mechanics, biology...




(t ≥ 0) on R satisfying the following properties

• W0 = 0 with probability 1• With probability 1, t 7→Wt is continuous• For any t1 < · · · < tn, the random variablesWt1 ,Wt2−t1 , . . . ,Wtn−tn−1









(t ≥ 0) on R satisfying the following properties• W0 = 0 with probability 1

• With probability 1, t 7→Wt is continuous• For any t1 < · · · < tn, the random variablesWt1 ,Wt2−t1 , . . . ,Wtn−tn−1









(t ≥ 0) on R satisfying the following properties• W0 = 0 with probability 1• With probability 1, t 7→Wt is continuous

• For any t1 < · · · < tn, the random variablesWt1 ,Wt2−t1 , . . . ,Wtn−tn−1









(t ≥ 0) on R satisfying the following properties• W0 = 0 with probability 1• With probability 1, t 7→Wt is continuous• For any t1 < · · · < tn, the random variablesWt1 ,Wt2−t1 , . . . ,Wtn−tn−1

are independent

• If s < t, the random variables Wt−s and Wt −Ws are(equally distributed) centered normal variables withvariance t− s












































Consider n Brownian motions B1, . . . , Bn on [0, 1] with thefollowing conditions

I Bk(0) = Bk(1) = 1, k = 1, . . . , n

I B1(t) < B2(t) < · · · < Bn(t), ∀t ∈ (0, 1)




I Bk(0) = Bk(1) = 1, k = 1, . . . , n

I B1(t) < B2(t) < · · · < Bn(t), ∀t ∈ (0, 1)




I Bk(0) = Bk(1) = 1, k = 1, . . . , n

I B1(t) < B2(t) < · · · < Bn(t), ∀t ∈ (0, 1)




Examples and applications - Non-intersectingBrownian motions and the GUE


I Bk(0) = Bk(1) = 1, k = 1, . . . , n

I B1(t) < B2(t) < · · · < Bn(t), ∀t ∈ (0, 1)

The distribution of (B1, . . . , Bn) at time t is the same as thedistribution of eigenvalues of the Gaussian unitary ensemble(GUE): the spaceMn of n× n hermitian matrices M withdistribution

1

Zne−Tr M2√

2t(1−t)dM


Examples and applications - random permutations

I For a permutation σ ∈ Sn, denote by ln(σ) the length of itslongest increasing subsequence

σ = (1 3 5 4 2 6)

Thus ln(σ) = 4

I Equip the set of permutations of length n, say Sn, with theuniform distribution P(σ ∈ Sn) = 1/(n!)

I Law of large numbers for ln(σ) [Kerov & Vershik, Logan &Shepp, 1977]:

limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2




σ = (1 3 5 4 2 6)

Thus ln(σ) = 4



limn→∞

E(ln(σ))√n

= 2


Examples and applications - random permutationsand the GUE

I Roughly, the law of large numbers for ln(σ) tells us that

ln(σ) = 2√

(n) + o(n1/2)

I It is natural to ask about computing the next order error(fluctuations) ln(σ)− 2

√n

I Denote by λmax the largest eigenvalue of the GUE

1

Zne−

M2

2 dM

I [Baik, Deift & Johansson, 1998]

limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)




ln(σ) = 2√

(n) + o(n1/2)


√n


1

Zne−

M2

2 dM


limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)




ln(σ) = 2√

(n) + o(n1/2)


√n


1

Zne−

M2

2 dM


limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)




ln(σ) = 2√

(n) + o(n1/2)


√n


1

Zne−

M2

2 dM


limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)




limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)

I More precisely, these limits are equal to the celebratedTracy-Widom distribution F2(t),

F2(t) := 1 +

∞∑k=1

(−1)k

k!

∫[0,t]k

det1≤j,l≤k

(A(xk, xl))dx1 . . . dxk

where

A(x, y) :=Ai(x) Ai′(y)−Ai(y) Ai′(x)

x− yand Ai is the Airy function




limn→∞

P(ln(σ)− 2

√n

n1/6≤ t)

= limn→∞

P(λmax − 2

√n

n−1/6≤ t)

I More precisely, these limits are equal to the celebratedTracy-Widom distribution F2(t),

F2(t) := 1 +

∞∑k=1

(−1)k

k!

∫[0,t]k

det1≤j,l≤k

(A(xk, xl))dx1 . . . dxk

where

A(x, y) :=Ai(x) Ai′(y)−Ai(y) Ai′(x)

x− yand Ai is the Airy function


TIME FOR A BREAK?


Second part

Random matrix theory: basic techniques foruniversality


Universality - a known example

Given a sequence of i.i.d. random variables X1, X2, . . . withE(Xj) = µ, Var(Xj) = σ2 > 0, the average

X1 + . . .+Xn − nµσ√n

converges in distribution to N (0, 1)

I After scaling, the limit above does not depend on preciseinformation on the Xj ’s!

I This phenomenon is one instance of universality




X1 + . . .+Xn − nµσ√n







X1 + . . .+Xn − nµσ√n





Universality - a not so known example

The Benford law assures that about 30% (more preciselylog10 2) of numerical data statistics’ start with the digit 1

I 30% of the cities have a population size that starts withdigit 1

I 30% of the house numbers in Campinas start with digit 1

I 30% of cash withdraws amounts start with digit 1

So 30% of the numbers drawn for “mega-sena” start with digit1??






























Unitary ensembles

I Unitary ensembles: space Hn of n× n hermitian matricesequipped with probability distribution

1

Zne−nTrV (M)dM, (1)

where V is a real polynomial of even degree and dM is theLebesgue measure on Hn ' Rn2

.

• Unitary because (1) is invariant under unitary conjugationM 7→ UMU∗

• When V (x) = x2/2, entries are independent Gaussianrandom variables (GUE)

• The factor n makes sure that eigenvalues remain bounded• The results that will be discussed also hold true for more

general V ’s


Unitary ensembles


1



.




general V ’s


Unitary ensembles


1



.




general V ’s


Unitary ensembles


1



.




general V ’s


Unitary ensembles


1



.



• The factor n makes sure that eigenvalues remain bounded

• The results that will be discussed also hold true for moregeneral V ’s


Unitary ensembles


1



.




general V ’s


Unitary ensembles - techniques

I We can see the diagonalization

M = U

λ1 0 · · · 00 λ2 · · · 0...

.... . .

...0 0 · · · λn

U∗

as a change of variables

Hn 3M 7→ (λ,U) ∈ Rn × U(n)/Tn

where λ = (λ1, . . . , λn) and T = eiθ | θ ∈ R



I Computing the Jacobian of the change of variables we getthat

1

Zne−nTrV (M)dM =

1

Zn

∏j<k

(λj − λk)2∏j

e−nV (λj)dλ1 . . . dλn dU

where dU is the Haar measure on Un

I Consequences:

• Eigenvalues and eigenvectors are statistically independent• Eigenvectors are completely delocalized (uniformly

distributed on U(n))• Eigenvalues exhibit local repulsion




1

Zne−nTrV (M)dM =

1

Zn

∏j<k

(λj − λk)2∏j


where dU is the Haar measure on UnI Consequences:






1

Zne−nTrV (M)dM =

1

Zn

∏j<k

(λj − λk)2∏j



• Eigenvalues and eigenvectors are statistically independent

• Eigenvectors are completely delocalized (uniformlydistributed on U(n))

• Eigenvalues exhibit local repulsion




1

Zne−nTrV (M)dM =

1

Zn

∏j<k

(λj − λk)2∏j




distributed on U(n))

• Eigenvalues exhibit local repulsion




1

Zne−nTrV (M)dM =

1

Zn

∏j<k

(λj − λk)2∏j






Unitary ensembles and orthogonal polynomials

After some massage, we get that1

Zn

∏j<k

(λj − λk)2∏j

e−nV (λj) = det(Kn(λk, λj))1≤k,j≤n

where Kn is the correlation kernel

Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y),

pk = pn,k’s are the orthonormal polynomials for e−nV (x)dx,∫pj(x)pk(x)e−nV (x)dx = δjk.

Furthermore, for some hn > 0,

pn(x) =1

hnE [det(Ix−M)]

Main message: all information is encoded in the OP’s!




Zn

∏j<k

(λj − λk)2∏j



Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y),



pn(x) =1

hnE [det(Ix−M)]





Zn

∏j<k

(λj − λk)2∏j



Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y),



pn(x) =1

hnE [det(Ix−M)]





Zn

∏j<k

(λj − λk)2∏j



Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y),



pn(x) =1

hnE [det(Ix−M)]

Main message: all information is encoded in the OP’s!Guilherme Silva Random matrices

Unitary ensembles - global behavior of eigenvalues

We can rewrite

1

Zn

∏j<k

(λj − λk)2∏j

e−nV (λj) =1

Zne−n

2H(λ1,...,λn)

where

H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Thus the most likely eigenvalues should be the onesminimizing H(λ1, . . . , λn)



We can rewrite

1

Zn

∏j<k

(λj − λk)2∏j

e−nV (λj) =1

Zne−n

2H(λ1,...,λn)

where

H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Thus the most likely eigenvalues should be the onesminimizing H(λ1, . . . , λn)



H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Denoting

νn,λ =1

n

n∑j=1

δλj , λ = (λ1, . . . , λn)

we can alternatively write

H(λ) =

∫∫x6=y

log1

|x− y|dνn,λ(x)dνn,λ(y) +

∫V (x)dνn,λ(x)

I So when n→∞ the eigenvalues should follow thedistribution µV that minimizes∫∫

log1

|x− y|dν(x)dν(y) +

∫V (x)dν(x)

over all probability measures ν on R



H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Denoting

νn,λ =1

n

n∑j=1

δλj , λ = (λ1, . . . , λn)


H(λ) =

∫∫x6=y

log1


∫V (x)dνn,λ(x)


log1


∫V (x)dν(x)




H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Denoting

νn,λ =1

n

n∑j=1

δλj , λ = (λ1, . . . , λn)


H(λ) =

∫∫x6=y

log1


∫V (x)dνn,λ(x)


log1


∫V (x)dν(x)




H(λ1, . . . , λn) =1

n2

∑j 6=k

log1

|λj − λk|+

1

n

∑j

Q(λj)

I Denoting

νn,λ =1

n

n∑j=1

δλj , λ = (λ1, . . . , λn)


H(λ) =

∫∫x6=y

log1


∫V (x)dνn,λ(x)


log1


∫V (x)dν(x)

over all probability measures ν on RGuilherme Silva Random matrices


I For a real polynomial V of even degree, there exists aunique measure µV , called the equilibrium measure, thatminimizes the energy∫∫

log1


∫V (x)dν(x)

over all probability measures on R

I The measure µV is absolutely continuous with continuousdensity ρV .

I For instance, if V (x) = x2/2, then

ρV (x) = ρsc(x) =1

2π

√4− x2χ[−2,2](x)




log1


∫V (x)dν(x)

over all probability measures on RI The measure µV is absolutely continuous with continuous

density ρV .

I For instance, if V (x) = x2/2, then

ρV (x) = ρsc(x) =1

2π

√4− x2χ[−2,2](x)




log1


∫V (x)dν(x)

over all probability measures on RI The measure µV is absolutely continuous with continuous

density ρV .I For instance, if V (x) = x2/2, then

ρV (x) = ρsc(x) =1

2π

√4− x2χ[−2,2](x)



I Limiting distribution of eigenvalues: Let λ1, . . . , λn be theeigenvalues of a matrix M , distributed according to the UEwith potential V . Then the limit

1

n

n∑j=1

δλj∗→ µV

holds true with probability one.

I In particular, for an interval J ⊂ R

limn→∞

E (# eigenvalues in J)

n=

∫JdµV



I Limiting distribution of eigenvalues: Let λ1, . . . , λn be theeigenvalues of a matrix M , distributed according to the UEwith potential V . Then the limit

1

n

n∑j=1

δλj∗→ µV

holds true with probability one.I In particular, for an interval J ⊂ R

limn→∞

E (# eigenvalues in J)

n=

∫JdµV



I Limiting zero distribution: Let pn be the n-th orthonormalpolynomial for e−nV (x)dx. Then

1

n

∑pn(w)=0

δw∗→ µV

I Thus, at the macroscopic scale, zeros and eigenvalueshave the same large n behavior







equilibrium measure µV


Unitary ensembles - Bulk universality

Recall that Kn is the Christoffel-Darboux kernel,

Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y)

= e−n2

(V (x)+V (y)) pn(x)pn−1(y)− pn(y)pn−1(x)

x− y

I Sine universality: Fix x∗ for which ρV (x∗) > 0. Usingasymptotics of OP’s,

limn→∞

1

nρV (x∗)Kn

(x∗ +

u

nρV (x∗), x∗ +

v

nρV (x∗)

)=

sinπ(u− v)

π(u− v)


Unitary ensembles - Bulk universality

Recall that Kn is the Christoffel-Darboux kernel,

Kn(x, y) = e−n2

(V (x)+V (y))n−1∑k=0

pk(x)pk(y)

= e−n2

(V (x)+V (y)) pn(x)pn−1(y)− pn(y)pn−1(x)

x− y

I Sine universality: Fix x∗ for which ρV (x∗) > 0. Usingasymptotics of OP’s,

limn→∞

1

nρV (x∗)Kn

(x∗ +

u

nρV (x∗), x∗ +

v

nρV (x∗)

)=

sinπ(u− v)

π(u− v)


Unitary ensembles - Fluctuations at the edge

I suppµV is a finite union of compact intervals, and

ρV (x) ∼ c |x− p|4k+1

2 , k = k(p) ≥ 0

at each endpoint p of the connected components ofsuppµV .

I Tracy-Widom Universality Let x∗ be the right-most endpointof suppµV and suppose that k(x∗) = 0. Then

P(λn ≤ x∗ +

t

n2/3

)= F2(t)

where F2 is the Tracy-Widom distribution



I suppµV is a finite union of compact intervals, and

ρV (x) ∼ c |x− p|4k+1

2 , k = k(p) ≥ 0

at each endpoint p of the connected components ofsuppµV .

I Tracy-Widom Universality Let x∗ be the right-most endpointof suppµV and suppose that k(x∗) = 0. Then

P(λn ≤ x∗ +

t

n2/3

)= F2(t)

where F2 is the Tracy-Widom distribution



I The condition k(x∗) = 0, that is,

ρV (x) ∼ c (x∗ − x)1/2, x x∗

is “generically” satisfied. For instance, it holds true if V isconvex.

I The Tracy-Widom distribution admits the representation

F2(t) = exp

(−∫ ∞t

(y − t)q20(y)dy

),

where q0 is a particular (a.k.a. Hastings-McLeod) solutionto the Painleve II equation

q′′(y) = yq(y) + 2q(y)3

I For k(x∗) 6= 0, the limiting distribution is expressed in termsof solutions to higher-order ODE’s (members of the PIIhierarchy)




ρV (x) ∼ c (x∗ − x)1/2, x x∗



F2(t) = exp

(−∫ ∞t

(y − t)q20(y)dy

),


q′′(y) = yq(y) + 2q(y)3





ρV (x) ∼ c (x∗ − x)1/2, x x∗



F2(t) = exp

(−∫ ∞t

(y − t)q20(y)dy

),


q′′(y) = yq(y) + 2q(y)3



Main technique: asymptotics for OP’s via theRiemann-Hilbert approach

Let qn be the monic OP of degree n for e−nV (x)dx

qn = Y11, where Y is the solution to the followingRiemann-Hilbert problem

• Y : C \ Σ→ C2×2 is analytic;

• Y+(x) = Y−(x)

(1 e−nV (x)

0 1

), x ∈ R;

• Y (z) = (I +O(z−1))

(zn 00 z−n

), z →∞.






• Y+(x) = Y−(x)

(1 e−nV (x)

0 1

), x ∈ R;

• Y (z) = (I +O(z−1))

(zn 00 z−n

), z →∞.






• Y+(x) = Y−(x)

(1 e−nV (x)

0 1

), x ∈ R;

• Y (z) = (I +O(z−1))

(zn 00 z−n

), z →∞.






• Y+(x) = Y−(x)

(1 e−nV (x)

0 1

), x ∈ R;

• Y (z) = (I +O(z−1))

(zn 00 z−n

), z →∞.






• Y+(x) = Y−(x)

(1 e−nV (x)

0 1

), x ∈ R;

• Y (z) = (I +O(z−1))

(zn 00 z−n

), z →∞.


Deift-Zhou Steepest Descent method

The idea is to perform a number of (invertible) transformations

Y 7→ X 7→ · · · 7→ S

so that at the end JS ≈ I

and consequently

S ≈ I, n→∞.

Reverting the transformations, one gets asymptotics for Y , andconsequently for qn.

This has been first developed by Deift, Kriecherbauer,McLaughlin, Venakides and Zhou (1997), and since then hasbeen explored and generalized to countless different contextsand problems




Y 7→ X 7→ · · · 7→ S

so that at the end JS ≈ I and consequently

S ≈ I, n→∞.






Y 7→ X 7→ · · · 7→ S


S ≈ I, n→∞.






Y 7→ X 7→ · · · 7→ S


S ≈ I, n→∞.




Time to wrap things up!

I We motivated the study of RM’s through their appearancein other problems of math and physics

I We gave an overview on unitary ensembles

I We briefly commented on how orthogonal polynomials canbe used to study universality

I We showed recent developments on the normal matrixmodel

I There are many other different random matrix modelswhere OP’s and their generalizations appear. For instance,the normal matrix model, two-matrix models, the externalsource model, products of random matrices...






























The normal matrix model

I Normal matrix model = space of n× n normal randommatrices (MM∗ = M∗M ) with probability distribution of theform

1

Znexp

(− nt0

TrV(M)

)dM

for some polynomial V on M and M∗ with V(M) = V(M)∗.I Distribution of eigenvalues (λ1, . . . , λn) ∈ Cn is

Pn(λ1, . . . , λn)dλ1 . . . dλn

=1

Zn

∏j<k

|λk − λj |2∏j

e− nt0V(λj)dλ1 · · · dλn




1

Znexp

(− nt0

TrV(M)

)dM

for some polynomial V on M and M∗ with V(M) = V(M)∗.

I Distribution of eigenvalues (λ1, . . . , λn) ∈ Cn is

Pn(λ1, . . . , λn)dλ1 . . . dλn

=1

Zn

∏j<k

|λk − λj |2∏j





1

Znexp

(− nt0

TrV(M)

)dM

for some polynomial V on M and M∗ with V(M) = V(M)∗.I Distribution of eigenvalues (λ1, . . . , λn) ∈ Cn is

Pn(λ1, . . . , λn)dλ1 . . . dλn

=1

Zn

∏j<k

|λk − λj |2∏j



The limiting shape of eigenvalues

I In the large n limit, the eigenvalues accumulate on a setΩt0 = Ωt0,V

V(z) = |z|2 + 2 log1

|z − 1|

t0 = 1 t0 = 4


Laplacian growth

Laplacian growth is a model of evolution for planar domains.


Laplacian growth



Laplacian growth


I For a bounded simply connected domain Ω ⊂ CC, itsassociated Green function (with pole at∞) gΩ is thesolution to the Dirichlet problem

4gΩ(z) = 0, z ∈ C \ Ω

gΩ(z) = 0, z ∈ ∂Ω

gΩ(z) = log |z|+O(1), z →∞


Laplacian growth


I

4gΩ(z) = 0, z ∈ C \ Ω

gΩ(z) = 0, z ∈ ∂Ω

gΩ(z) = log |z|+O(1), z →∞


Laplacian growth


I

4gΩ(z) = 0, z ∈ C \ Ω

gΩ(z) = 0, z ∈ ∂Ω

gΩ(z) = log |z|+O(1), z →∞

I A bounded simply connected domain Ω = Ωt0 evolvesaccording to the Laplacian growth if

Area(Ωt0) = πt0

∂t0ηt0(z) = const∇gΩt0(z)

Ωt0

ηt0


Laplacian growth and normal matrices - afterWiegmann et al.

At the formal level,

I The harmonic moments

tk := − 1

π

∫∫C\Ωt0

dA(z)

zk, k = 1, 2, 3, · · ·

are preserved by the Laplacian growth.I The domain Ωt0 coincides with the limiting domain for the

eigenvalues of the normal matrix model

1

Zne− nt0

Tr(MM∗−V (W )−V (W ∗))dM, V (z) =

∞∑k=1

tkkzk





tk := − 1

π

∫∫C\Ωt0

dA(z)

zk, k = 1, 2, 3, · · ·

are preserved by the Laplacian growth.

I The domain Ωt0 coincides with the limiting domain for theeigenvalues of the normal matrix model

1

Zne− nt0

Tr(MM∗−V (W )−V (W ∗))dM, V (z) =

∞∑k=1

tkkzk





tk := − 1

π

∫∫C\Ωt0

dA(z)

zk, k = 1, 2, 3, · · ·

are preserved by the Laplacian growth.I The domain Ωt0 coincides with the limiting domain for the

eigenvalues of the normal matrix model

1

Zne− nt0

Tr(MM∗−V (W )−V (W ∗))dM, V (z) =

∞∑k=1

tkkzk


The NMM with cubic potential (jointly with PavelBleher - IUPUI)

I For now on, we specify to the NMM

1

Zne− nt0

Tr(MM∗−V (W )−V (W ∗))dM

with

V (z) =z3

3+ t1z, −3

4< t1 <

1

4

I Symmetric case t1 = 0 studied by Bleher & Kuijlaars (2012)




1

Zne− nt0


with

V (z) =z3

3+ t1z, −3

4< t1 <

1

4





1

Zne− nt0


with

V (z) =z3

3+ t1z, −3

4< t1 <

1

4



Limiting domain of eigenvalues

Theorem (Bleher & S., 2016)There exists t0,crit = t0,crit(t1) > 0 for which

I For certain functions r = r(t0, t1) and a0 = a0(t0, t1),

∂Ωt0 = z = h(eiθ) | θ ∈ [0, 2π],

h(w) := rw + a0 +2a0r

w+r2

w2

In particular, ∂Ωt0 is analytic for t0 < t0,crit

I Ωt0 solves the Laplacian growth problem with harmonicmoments

t1 ∈ (−3/4, 1/4), t3 = 1, tk = 0, k = 2, 4, 5, . . .





∂Ωt0 = z = h(eiθ) | θ ∈ [0, 2π],

h(w) := rw + a0 +2a0r

w+r2

w2



t1 ∈ (−3/4, 1/4), t3 = 1, tk = 0, k = 2, 4, 5, . . .





∂Ωt0 = z = h(eiθ) | θ ∈ [0, 2π],

h(w) := rw + a0 +2a0r

w+r2

w2



t1 ∈ (−3/4, 1/4), t3 = 1, tk = 0, k = 2, 4, 5, . . .


Evolution of the boundary for t1 = 116 (left) and t1 = −1

4

(right)


Phase diagram


Average characteristic polynomial

For n even and t1 ≥ 0 we also study the (regularized) averagecharacteristic polynomials

pn(z) := E [det(Iz −M)]


Associated orthogonality

Theorem (Bleher & S., 2016)The polynomial pn satisfies the multiple orthogonality∫

Γpn(s)ske

nτV (s)y(s)ds = 0, k = 0, · · · , n

2− 1∫

Γpn(s)ske

nτV (s)y′(s)ds = 0, k = 0, · · · , n

2− 1

where Γ = Γ0 ∪ Γ1 ∪ Γ2, y∣∣Γj≡ yj and

Γ0

Γ2

Γ1

y0(s) := Ai(s),

y1(s) := ω2 Ai(ω2s),

y2(s) := ωAi(ωs),

ω := e2πi3 ,

s :=(nτ

)2/3(s− t)


Large n behavior of zeros

Theorem (Bleher & S., 2016)For t1 ∈ (−3/4, 1/4), the zeros of pn accumulate on a contourΣ∗ (explicitly determined).


Large n behavior of zeros

Theorem (Bleher & S., 2016)For t1 ∈ (−3/4, 1/4), the zeros of pn accumulate on a contourΣ∗ (explicitly determined).


Basic references

I General aspects of random matrix theory (including mostaspects discussed on applications of RMT):

• Madan L. Mehta. Random matrices - Third edition (2004),Academic Press - Series Pure and Applied Mathematics(book 142).

• Handbook of Random Matrix Theory (Edited by G.Akemann, J. Baik and P. Di Francesco), 960 pages, OxfordUniversity Press.

• Terence Tao. Topics in Random Matrix Theory, AMS, 2012.

• Percy Deift and Dimitri Gioev. Random matrix theory:invariant ensembles and universality. Courant LectureNotes in Mathematics 18, 2009.


Technical references

I Number theory:


• J.P. Keating. Random matrices and the Riemannzeta-function – a review, Applied Mathematics Entering the21st Century: Invited Talks from the ICIAM 2003 Congress.



I Brownian motion:

• Dyson, F. J.. A Brownian-motion model for the eigenvaluesof a random matrix, J. Math. Phys. 3(1962), 1191–1198.


• C. Tracy and H. Widom. Nonintersecting Brownianexcursions, The Annals of Applied Probability 2007, Vol.17, No. 3, 953–979.

• E. Daems, A. Kuijlaars and W. Veys Asymptotics ofnon-intersecting Brownian motions and a 4 x 4Riemann-Hilbert problem, J. Approx. Theory 153 (2008),225–256.



I Random permutations:

• J. Baik, P. Deift and K. Johansson. On the Distribution ofthe Length of the Longest Increasing Subsequence ofRandom Permutations, J. Amer. Math. Soc., 12,no.4:1119-1178, 1999

• D. Romik. The Surprising Mathematics of LongestIncreasing Subsequences, Cambridge University Press2015. Available online for free in author’s webpage



I Unitary ensembles:

• P. Deift. Orthogonal polynomials and random matrices: aRiemann-Hilbert approach. Courant Lecture Notes 1999.

• D. Lubinsky. A New Approach to Universality LimitsInvolving Orthogonal Polynomials, Annals of Mathematics,170(2009), 915-939.

• T. Claeys, A. Its, and I. Krasovsky. Higher order analoguesof the Tracy-Widom distribution and the Painleve IIhierarchy, Comm. Pure Appl. Math. 63 (2010), 362-412



I Normal Matrix Model:

• P. Elbau. Random Normal Matrices and Polynomial Curves,Ph.D. Thesis, ETH Zurich, 2006, 36 pages,arXiv:0707.0425

• P. Bleher and G. Silva. The mother body phase transition inthe normal matrix model, 2016, 127 pages,arXiv:1601.05124


Thank you!


Thank you!


Thank you!


Guilherme Silva - Unicamp · Guilherme Silva University of Michigan - USA Guilherme Silva Random matrices. Random matrices I Random matrix: matrix whose entries are random variables

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