Random matrix models and applications S. Hikami Mathematical and - PowerPoint PPT Presentation

Random matrix models and applications S. Hikami Mathematical and Theoretical Physics Unit OIST OIST-iTHES-CTSR July 9, 2016

Talk is based on With E. Brezin • (1) Random Matrix Theory With An External Source, a preparation for book (2016) (2) Random matrix, singularities and open/close intersection numbers, J. Phys. A 48, (2015) 475201.

Applications of random matrix theory -Universal statistics of spacing of energy levels, Wigner surmise - Quantum chaos (Bohigus conjecture) - Distribution of zeros of Riemann zeta function (random) - Central limit theorem

• Topological aspect of random matrix theory - 2D quantum gravity - Riemann surface ---- (closed string)----gravity - boundary (D brane) ---(open string) --- gauge - AdS/CFT, gravity/gauge correspondence ----simple case of random matrix theory - description of Black hole - N=2 supersymmetric CFT, c = 2 – 6/p (c is central charge) * Riemann surface with boundary ---- open/close intersection numbers

Leonardo da Vinci (triangulation of surface)

GAUSSIAN INTEGRALS INTERESTING MODELS ARE NON-GAUSSIAN • Matrix models of 2D gravity (Triangulation of random surface by matrix model ) • Kontsevich ”Airy” matrix model dX exp ( − 1 2Tr Λ X 2 + i Z 6 X 3 ) Z = who proved that F = log Z satisfies Witten’s conjectures. Namely define t n ( Λ ) = − (2 n − 1)!!Tr Λ − (2n − 1) 7

then 1 t k n h τ k 0 0 · · · τ k n n X Y F ( t 0 , · · · , t n , · · · ) = n · · · i k n ! 0 in which the coe ffi cients h τ d 1 · · · τ d n i are the ”intersection numbers of the moduli space of curves” on a Riemann sur- face M g,n , space of inequivalent complex structures on a Riemann surface of genus g with n marked points with 3 g = n � 3 + P n 1 k i . It follows that F satisfies the KdV hierarchy : U = ∂ 2 F/ ∂ t 2 0 ∂ 3 U + 1 ∂ U = U ∂ U ∂ t 3 ∂ t 1 ∂ t 0 12 0

• Penner model Z Z = dX exp − Nt Tr(log(1 − X ) + X ) F = log Z is the generating function of the Euler character of M g,n χ g,n N 2 − 2 g t 2 − 2 g − n X F = from which he obtained χ g,n = ( − 1) n ( n + 2 g − 3)!(2 g − 1) B 2 g n !(2 g )! B 2 g is a Bernoulli number.

Gaussian random matrix model • These interesting non-Gaussian matrix model are obtained from Gaussian matrix model with an external source

Consider Gaussian matrix models with an external matrix source P A ( X ) = 1 e − N 2 tr X 2 − N tr XA Z A A = A † a given matrix with eigenvalues a 1 , · · · a N . N 2 tr A 2 ) but one can tune A to Z A is of course trivial ( Z A = e various situations.

t t t t t y y Density of states (the dots are the eigenvalues of the source) 8

• Euler characteristic χ = V – E + F = 2 (g=0) χ = 2 – 2 g Topology of Riemann surface

Marked points on closed/open Riemann surface

Kontsevich(Ribbon graph) for M 1,1 ( M g,n )

1. Correlation functions in an external source: P A ( X ) = 1 e � N 2 tr X 2 � N tr XA Z A The partition function is trivial, the correlation functions are not, but they are known explicitly (S.Hikami , EB) 1 N K h tre Ns 1 X · · · tre Ns K X i U A ( s 1 , · · · , s K ) = is a function of the eigenvalues a α of the Hermitian source matrix A .

2. A powerful duality Consider the average of the product of K characteristic polynomials det( λ − M ). The K-point function is defined as K 1 Y F K ( λ 1 , ..., λ K ) = det( λ α − X ) > A,X < Z N α =1 K 1 det( λ α · I − X) e − 1 2 Tr(X − A) 2 Z Y = dX Z N α =1

1 det( λ − X ) λ − X = ∂ Tr det( µ − X ) | µ = λ ∂λ Alternatively 1 1 ∂ ∂λ [det( λ − X)] n . tr λ − X = lim n → 0 n

Define the K × K diagonal matrix 0 0 0 λ 1 1 · · · Λ = 0 0 λ 2 B C · · · @ A · · · The duality reads K 1 det( λ α · I − X) e − 1 Z 2 Tr(X − iA) 2 Y [ dX ] N Z N α =1 = N ( − i ) Nk 1 det( a j · I − Y ) e − 1 Z 2 tr(Y+i Λ ) 2 Y [ dY ] K Z k j =1 where Y is a K × K Hermitian matrix. The K-point function with N × N Gaussian random ma- trices (in a source) is equal to an N-point function with

K × K Gaussian matrices in a di ff erent source (like ”color” exchanged with ”flavor”). Trivial example : one-point function 1 [ dX ]det( λ − X) e − 1 2 Tr(X − iA) 2 Z Z N N = ( − i ) N 1 ( a j − y ) e − 1 2 ( y + i λ ) 2 Z Y dy Z j =1

The fact that Kontsevich’s model is dual of a Gaussian model makes it easy to compute the intersection numbers. Duality and replicas If Λ is a multiple of the identity Λ = λ × 1, the coe ffi cients of 1 / λ in the expansion of log Z are proportional to the inter- section numbers of the moduli of curves with one marked point on a Riemann surface of genus g . To recover these numbers one can use replicas, i.e. the simple relation 1 1 ∂ ∂λ [det( λ − X )] n = tr lim λ − X. n n → 0 and < [det( λ − X )] n > A,X = < [det(1 − iY )] N > Λ ,Y 11

where Y is an n ⇥ n random Hermitian matrix. Similarly with two marked points U ( s 1 , s 2 ) = h Tre s 1 X Tre s 2 X i ∂ 2 Z d λ 1 d λ 2 e s 1 λ 1 + s 2 λ 2 h [det(1 � iY )] N i Λ = lim ∂λ 1 ∂λ 2 n 1 ,n 2 ! 0 with Λ = ( λ 1 , · · · , λ 1 , λ 2 , · · · , λ 2 ) degenerate n 1 and n 2 times.

Generalizations : ”spin curves” It is clear that can obtain results for curves with several marked points by the same technique (although it is di ffi cult to go beyong low genera). One can also ”tune” the external source to obtain a gener- alized Kontsevich model. For instance if the source matrix A has N/ 2 eigenvalues equal to +1, and N/ 2 equal to − 1 in the large N appropriate scaling limit N N det( a i − iY ) > = < [det(1 + Y 2 )] Y 2 > < i =1 dY e − N 4 tr Y 4 − iN tr Y Λ Z = This case is nothing but the critical gap closing situation.

J.Kuan (Thesis, Harvard 2015), Pearcey process, non- intersecting Brownian motion

Again an appropriate tuning of the source matrix can yield the p-th generalization of Kontsevich’s Airy matrix-model, a model introduced by Marshakov et al. defined by Z = 1 1 Z p + 1tr(Y p+1 � Λ p+1 ) � tr(Y � Λ ) Λ p ] dY exp[ Z 0 It can be used to recover the ( p, q ) models coupled to grav- ity. The ’free energy’ is the generating function of the general- ized intersection numbers h Q τ m,j i for moduli of curves with ’spin’ j t d m,j τ d m,j m,j X Y Y F = < m,j > d m,j ! d m,j m,j m,j where

m − 1 j − p − m ( p +2) 1 Y 2( p +1) t m,j = ( − p ) ( lp + j + 1)tr Λ mp + j +1 l =0 For instance for the quartic generalized model (p=3) Γ ( g +1 3 ) 1 ,j > g = < τ 8 g − 5 − j Γ ( 2 − j (12) g g ! 3 ) 3 where j = 0 for g = 3 m + 1 and j = 1 for g = 3 m . (For g = 3 m + 2, the intersection numbers are zero). Witten conjecture : the free energy which generates inter- section numbers for spin curves satisfies a Gelfand-Dikii hi- erarchy and indeed this is true for the correlators U ( s 1 , · · · , s K ) providing thereby an alternative definition.

Generalization : higher Kontsevich-Penner models dXe Tr [ X p +1 + k log X + Λ X ] Z Z = • p=2 Airy • p=-1 Penner • p=-2 unitary matrix model

Generating function for the intersection numbers of p-spin curves, U(s 1 ,…,s n )

Up to genus 9 (2016)

[22] p → − 1 limit < τ > g =1 = p − 1 → − 1 24 12 < τ > g =2 → − 1 < τ > g =3 → − 1 120 , 252 χ ( M g, 1 ) = ζ (1 − 2 g ) = − B 2 g 2 g χ ( M g, 1 ) : Euler characteristics, B 2 g : Bernoulli number, B 2 = 1 6 etc.

Intersection numbers for non-orientable surfaces There is no equvalent of the Itzykson-Zuber for real sym- metric matrices which generate non-orientable surfaces. How- ever the HarishChandra formula applies to Lie algebras such as o ( N ) (or sp ( N )) which also produce non-orientable sur- faces. Again one can follow the same path : explicit corre- lation functions and duality. 1. Lie algebra X ∈ o (2 N ) Consider the Lie algebra of o (2 N ), namely real antisymmet- ric matrices. For given real antisymmetric matrices X and Λ the HarishChandra localization formula, for the integral over g ∈ SO (2 N ), reads N (det w )exp[2 w ( x j ) λ j ] P P j =1 Z SO (2 N ) dge tr( gXg − 1 Λ ) = C w ∈ W ( x 2 j − x 2 k )( λ 2 j − λ 2 k ) Q 1 ≤ j<k ≤ N

where C = (2 N � 1)! Q 2 N � 1 (2 j � 1)!, w are elements of the j =1 Weyl group, (permutations followed by reflections ( x i ! ± x i ; i = 1 , · · · , N ) with an even number of sign changes). From this, one derives again the correlation functions in closed form h O ( X ) i A = 1 dX O ( X ) exp (1 Z 2trX 2 + trAX) Z A ! 0 1 A = a 1 v � · · · � a N v, v = i σ 2 = . � 1 0 2 ) 2 � a 2 2 π i (( u + σ e σ u + σ 2 1 1 du u I 2 N < tr e σ X > A = γ ) 4 u 2 � a 2 u + σ N σ 4 γ and similarly for the K-point functions.

2.Duality k N Y Y det( λ α · I − X ) > A = < det( a n · I − Y ) > Λ < α =1 n =1 where X is a 2 N × 2 N real antisymmetric matrix, Y is a 2 k × 2 k real antisymmetric matrix. The matrix source A is also a 2 N × 2 N antisymmetric matrix. The matrix Λ is a 2 k × 2 k antisymmetric matrix, coupled to Y . We assume, without loss of generality, that A and Λ take the canonical form Λ = λ 1 v ⊕ · · · ⊕ λ k v.