Zuber Talk Randommatrices

Free Probability and Large N Limit 1


Introduction to Random Matricesfrom a physicists perspective

Jean-Bernard Zuber

Berkeley March 2007

Random matrices: a major theme in theoretical physics since Wigner (1951);a major theme in mathematical physics and mathematics for the last 20 years.


Plan of talk A short history of the subject Random systems, from nuclear Hamiltonians to mesoscopic systems to

financial markets. . . Large N limit of U(N) gauge theory Topological expansion and the

counting of maps and other planar objects Stat mech models onrandom lattices

Double scaling limit and 2D quantum gravity QCD, Dijkgraaf-Vafa, etc Feynman diagrams and large N limit Counting of maps or triangulations (cf Edouard Maurel-Segalas talk) Computational methods: saddle point; [loop equations (cf EMS)];[orthogonal polynomials (cf Paul Zinn-Justin)]


A short history of the subject1. Random systems [Wigner 1951]Study of spectrum of large size Hamiltonians of big nuclei, regarded asGaussian random matrices, subject to some symmetry or reality property(GOE, GUE, GSE, . . .)

Eugene P. Wigner Freeman Dyson Madan Lal Mehta

1902 - 1995 1923 - 1932 - 2006


Statistics of energy levels of random Hamiltonians

Level spacing histogramfor a large set of nuclearlevels(Nuclear Data Ensemble,

[O. Bohigas et al]).

Other random systems and random matrices : from transport properties (forex. universal fluctuations of electric conductivity) in disordered mesoscopicsystems to financial markets. . .


2. Large N limit of U(N) gauge theories

Gauge theories : quantum field theories ba-sed on a principal fibre bundle with a compactgroup G. Gauge field A (connection) lives inLie algebra of G. For G = U(N), A is a NN(anti-Hermitian) matrix.In the search of a non-trivial approximation, itis natural to look at large N limit, expansionparameter 1/N2, [t Hooft 1974], see below.

Indeed, major simplification in the large N limit of Feynman expansions ofmatrix field theories. . .

Consider toy field theories : integrals over a finite number of (large size)matrices.


Basics of Feynman diagramsConsider a Gaussian integral over n real variables xi, A = AT > 0 def. matrix

Zdnxe

12 xiAi jx j =

(2pi)n/2

det 12 AZ

dnxe 12 xiAi jx j +bixi = (2pi)n/2

det 12 Ae

12 biA1i j b j

Differentiate w.r.t. bixk1 xk2 xk :=

Rdnx xk1 xk2 xke

12 x.A.x

Rdnxe 12 x.A.x

=

bk1 bk

e12 b.A

1.bb=0

= all distinct

pairings P of the k

A1kP1 kP2 A1kP1 kP

propagatorsXX

XXXy

HHHjWick theorem

k

...

Px xk

x k1 2 l P P Pll2P1k k k k

...


Wick theorem also applies to monomials (n = 1 variable for simplicity):p vertices propagator A1

(x4)p= graphs

Non Gaussian integrals (g < 0): power series perturbative expansions

Z =Z

dxe 12 Ax2 +g4! x

4=(2pi

A) 1

2

p=0

gp

p!

Zdx(x4

4!)p

e 12 Ax2

=(2pi

A) 1

2

p=0

graphs G with 2p linesand p 4valent vertices

gp

|AutG |A2p

logZ = connected Feynman diagrams

=g

8A2 +g2

A4( 1

2.4!+

124)

+

= +


Matrix Integrals: Feynman Rules

NN Hermitean matrices M, dM = i dMii i< j deMi j dmMi j

Z =: eF =R

dM eN[ 12 trM2 +g4 trM

4]

Feynman rules: propagator ijlk=

1N i jk [t Hooft]

4-valent vertex :ji kl

mnp

q= gN jkmnpqi

For each connected diagram contributingto logZ: fill each closed index loop with adisk discretized closed 2-surface


Power of N in a connected diagram each vertex N; each double line N1; each loop N.Thus N#vert.#lines+#loops= NEuler()

[t Hooft (1974)]. For example, compare

gN 0gN2

A topological expansion : F = logZ = conn. surf.

N22genus()g#vert.()

symm. factor

=

h=0

N22hF(h)(g).


Thus large N limit of matrix integralR

DMeNtr(M2+g4 M

4)= generating

function of planar 4-valent graphs. . .(cf census of planar maps by Tutte)[Brezin, Itzykson, Parisi, Z. 1978]

limN 1N2 logZ = planar diagramswith n 4verticesgn

symm.factor

or in a dual way, of quadrangulations of 2D surfaces of genus 0

[Kazakov; David; Kazakov-Kostov-Migdal; Ambjrn-Durhuus-Frohlich 85]


Thus large N limit of matrix integralR

DMeNtr(M2+g3 M

3)= generating

function of planar 3-valent graphs. . .[Brezin, Itzykson, Parisi, Z. 1978]

or in a dual way, of triangulations of 2D surfaces of genus 0

[Kazakov; David; Kazakov-Kostov-Migdal; Ambjrn-Durhuus-Frohlich 85]

Triangulated surfaces and discrete 2D gravity

Thus : Large N limit of matrix integrals Counting of planar objects :maps, triangulations, alternating knots and links [P Z-J & J-B Z], etc, orof objects of higher topology . . .


Large N limit of matrix integralR

DADBeNtr (A2+cAB+B2+gA4+gB4)=generating function of bicolored planar 4-valent graphs. . .


Large N limit of matrix integralR

DADBeNtr (A2+cAB+B2+gA4+gB4)=generating function of bicolored planar 4-valent graphs. . .

i.e. describes Ising model on a random quadrangulated sphere[Boulatov-Kazakov 1985]

+ +++++

++

+ + ++

++

+

++

etc etc, many variations on that themeStatistical mechanics models on a random lattice


Two remarks useful in connection with free probabilities. . .

Factorization property 1N trP1 1N trP2= 1N

trP1 1N trP2 disconnected diagrams

+O( 1N2 )

Compare (in Gaussian theory etr(M21+M22)) 1N trM21M22 and 1N trM1M2M1M2)

M1 M2gN 0gN

2

Similar behavior in non Gaussian theory etr(V1(M1)+V2(M2)) providedM1= 0, M2= 0.


If M1 6= 0, M2 6= 0,M1 M2 M1 M2

tr

Compare (a1a2a1a2) = (a21)2(a2)+ 2(a1)(a22) 2(a1)2(a2) for twofree variables


3. Double scaling limit

[Brezin-Kazakov; Douglas-Shenker; Gross-Migdal 1989, . . .]

F(0)(g) and more generally F(h)(g) have a singularity at g = gc,

F(h)(g) (gcg)(2str)(1h)

with str, the string susceptibility, typically equal to 1/2 (for thesimplest models M3 or M4 above), see below.As g gc, # triangles= logFg diverges. Expect to make contact withcontinuum 2D gravity. Keep all genera h in h=0 N22hF(h)(g) by lettinggcg 0 as N in such a way that (gcg)(2str)/2N = fixed.Very interesting limit : appearance of integrable equations (KdV . . .),solutions to Painleve equations . . .

Thus, double scaling limit models of 2D quantum gravity


4. Other physical applications

Cell decomposition of moduli space of Riemann surfaces, intersectionnumbers . . .[Witten, Kontsevich, 1991 . . .]

QCD, the Dirac operator /D = /+/A in the presence of a gauge field andRMT [Verbaarschot et al.]

Dijkgraaf-Vafa 2002 : computing the effective action of supersymmetricgauge theories in terms of matrix integrals

etc etc


Computational techniquesConsider integral over NN Hermitian matrices

Z =Z

dMeNtr V (M) ,

V (M) a polynomial of degree d +1. For ex. V3(M) = ( 12 M2 + g3 M

3) andV4(M) = ( 12 M

2 + g4 M4). Note that multi-traces are excluded, for example

(trM2)2.

Integrand and measure are invariant under U(N) transformationsM UMU. Express both in terms of eigenvalues 1, ,N of M :

Z =Z N

i=1

di i< j

(i j)2eN Ni=1 V (i) ,

Several ways to treat this integral: saddle point approximation ; orthogonalpolynomials; loop equation. . .


1. Saddle point approximation

Rewrite

Z =Z N

i=1

di exp(

2 i< j

log |i j|NN

i=1

V (i))

In the large N limit, if O(1), both terms in exponential are of order N2.Look for the stationary point, i.e. the solution of

2N j 6=i

1i j = V

(i) . ()

To solve this problem, introduce the resolvent

G(x) = 1N

tr

1xM

= 1

N

N

i=1

1xi

.


Computing its square leads after some algebra to

G2(x) = 1N2

i, j=1,,N

1(xi)(x j)

= = 1

NG(x)+V (x)G(x)P(x)

with P(x) := 1N

Ni=1 V(x)V (i)

xi

a polynomial in x of degree d1, i.e.

G2(x)V (x)G(x)+ 1N

G(x)+P(x) = 0 .

(Beware ! Not exact for N finite!) For N large, neglect the 1/N term quadratic equation for G(x), with yet unknown polynomial P, hence

G(x) = 12

(V (x)

V (x)24P(x)

)(minus sign in front of dictated by the requirement that for large |x|,G(x) 1/x.)In that large N limit, the s form a continuous distribution with density


() on a support S , RS d() = 1, and G(x) =R 2a2a

d()x .

For a purely Gaussian potential V () = 12 2, Wigners semi-circle law:() = 12pi

42 on the segment [2,2].

For more general potentials, assume first S to be still a finite segment[2a,2a], in such a way that (*) becomes

2P.P.Z 2a2a

d() = V

() if [2a,2a] .

(P.P.= principal part), expressing that, along its cut,

G(x i) = 12

V () ipi(x) x [2a,2a] . Thus

G(x) = 12

V (x)Q(x)

(x+2a)(x2a)where the coefficients of the polynomial Q(x) and a, a are determined bythe condition that G(x) 1/x for large |x|. Q is of degree d1. Thesolution is unique (under the one-cut assumption).


Example For the quartic potential V () = 12 2 +g4 4, by symmetry

a = a =: a,

G(x) = 12(x+gx3) (1

2+

g2

x2 +ga2)

x24a2

with a2 the solution of3ga4 +a21 = 0 (EQa2)

which goes to 1 as g 0 (a limit where we recover Wigners semi-circlelaw). From this we extract

() = 1pi

(12

+g2

2 +ga2)

4a22

and we may compute all invariant quantities like the free energy or themoments

G2p := 1

NtrM2p

=

Zd 2p () .

For example G2 = (4a2)a2/3, G4 = (3a2)a4, etc. All these functions


of a2 are singular as functions of g at the point gc = 112 where the tworoots of (EQa2) coalesce. For example the genus 0 free energy

F(0)(g) : = limN

(1/N2) log(

Z(g)Z(0)

)=

12

loga2 124

(a21)(9a2)

= p=1

(3gt2

)p(2p1)!p!(p+2)!

[Tutte 62,BIPZ 78]

...

has a power-law singularity

F(0)(g) ggc

|ggc|5/2

which reflects on its series expansion

F(0)(g) =

n=0

fngn , fn n const|gc|

nn7/2 .


Commentsi) Nature of the 1/N2 and of the g expansions, algebraic singularity at

finite gcii) Universal singular behavior at gciii) Extension to several cuts, the role of the algebraic curve (cf Eynard).iv) Connected correlation functions and free (or non crossing) cumulantsv) Factorization property, localization of the matrix integral and the

master field [. . .]


2. Orthogonal polynomials

ZdPm()Pn()eNV() = hnmn

[Mehta, Bessis, . . .]

cf Paul Z-J . . .

3. Loop (or Schwinger-Dyson) equations

ZdM Mi j

{ eNtrV (M)}= 0

and make use of factorization property . . .

cf Edouard M-S . . .


Comments on Free or non-crossing cumulants fn[BIPZ 78, Cvitanovic 81, Voiculescu 85, Speicher 94, Biane, PZ-J]Generating function of moments mn = 1N trM

n

Z( j) = 1N

tr1

1 jM =

n=0

jnmn

or G(u) = u1Z(u1) = n=0 un1mn. The generating function of freecumulants

W (z) = 1+

n=1

zn fn = 1+W (z)

or P(z) = z1W (z), is defined by the relations

W (z) = Z( j(z)) with j(z) = z/W (z)or equivalently

Z( j) = W (z( j)) with z( j) = jZ( j)


Graphically [Cvitanovic]

Z( j) =...

1 + ...

~

Z(j)Z Z Z

W(j)

= W ( jZ( j))

These relations amount to saying that P et G are functional inverses of oneanother PG(u) = u. IndeedP(G(u)) = G1(u)W (G(u)) = uZ1(u1)W (u1Z(u1)) = u, sinceZ(u1) = W (u1Z(u1)).

Using Lagrange formula, one computes

mk = k k!(k+1q)!f 111!

f 222!

or conversely fk =k (k2+q)!k! (m1)1

1!(m2)2

2!


End of Act I

Zuber Talk Randommatrices

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