Random Matrix Theory and applications in cosmology Sbastien - PowerPoint PPT Presentation

Random Matrix Theory and applications in cosmology Sébastien Renaux-Petel CNRS - Institut d’Astrophysique de Paris IESC Cargèse, 5th and 6th of September 2018 School Analytics, Inference, and Computation in Cosmology: Advanced methods

Outline 1. Introduction 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5. High-energy landscape Renaux-Petel, IAP

Outline Introduction 1. 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5. High-energy landscape Renaux-Petel, IAP

Random Matrix Theory and applications … Economy Physics & finance RMT Mathematics Biology Statistics Information theory « The Oxford handbook of random matrix theory » Renaux-Petel, IAP

Wigner’s surmise Hard to compute from first principles! Heavy nucleus 1956: possible shape of distribution Q: of the spacing of energy levels? p ( s ) = s 4 s 2 2 e − 1 A: Wigner: Renaux-Petel, IAP

Wigner and Dyson’s idea Energy levels = eigenvalues of Hamiltonian Hamiltonian: large complicated Hermitian matrix Let us model it as a random matrix! Renaux-Petel, IAP

Wigner’s semicircle law From math-ph:1712.07903 « Introduction to Random Matrices - Theory and Practice » Renaux-Petel, IAP

Universality Universal behaviors emerge independently of details of distributions of entries Take real symmetric matrix - independent entries - - entries decay sufficiently fast at infinity - (normalization) Renaux-Petel, IAP

Classification Renaux-Petel, IAP

Outline 1. Introduction 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5. High-energy landscape Renaux-Petel, IAP

Gaussian ensemble: construction Simplest starting point: N  1 �� Y − M 2 � ρ [ M ] ≡ ρ ( M 11 , . . . , M NN ) = 2 π exp ij / 2 √ i,j =1 H = 1 2( M + M T ) Symmetrization: " # Y " # N � − ( H ) 2 � � � − ( H ) 2 exp exp ii / 2 Independent ij Y ρ ( H ) = √ √ π entries 2 π i =1 i<j Rotational 2 Tr( H 2 ) ρ ( H ) ∝ exp − 1 invariance Renaux-Petel, IAP

Gaussian ensemble: pdf of eigenvalues ρ ( x 1 , . . . , x N ) = 1 P N i =1 x 2 Z e − 1 i Y | x j − x k | β Result: 2 j<k β = (1 , 2 , 4) Dyson index (GOE, GUE, GSE) Sketch of proof: with { X = diag( x 1 , . . . , x N ) H = OXO T OO T = 1 Change of variables H → { x , O } ρ ( x , O ) = ρ ( H ( x , O )) | J ( H → { x , O } ) | ˆ Renaux-Petel, IAP

Gaussian ensemble: pdf of eigenvalues Sketch of proof (ctd): Vandermonde Y J ( H → { x , O } ) = ( x j − x i ) determinant i<j For rotationally invariant ensembles ρ ( H ) = ϕ (Tr H, . . . Tr( H N )) Integrating over O is trivial ⇣X ⌘ Y X x N ρ ( x 1 , . . . , x N ) ∝ ϕ | x j − x k | x i , . . . , i j<k Renaux-Petel, IAP

Gaussian ensemble: pdf of eigenvalues ρ ( x 1 , . . . , x N ) = 1 P N i =1 x 2 Z e − 1 i Y | x j − x k | β 2 j<k Kills configurations Kills configurations with ‘large’ xi where any two eigenvalues are ‘close’ Interplay between confinement and eigenvalue repulsion Renaux-Petel, IAP

Spectral density N n ( x ) ≡ 1 Counting function X δ ( x − x i ) of eigenvalues N i =1 For random matrix, n becomes a random measure Z h n ( x ) i = d x 2 . . . d x N ρ ( x, x 2 , . . . , x N ) ⌘ ρ ( x ) Spectral density = marginal density of jpdf of eigenvalues N →∞ ρ SC ( x ) = 1 p p p 2 − x 2 β N ρ N ( β Nx ) − Result: − − − → π How to prove this? Coulomb gas and resolvent Renaux-Petel, IAP

Outline 1. Introduction 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5. High-energy landscape Renaux-Petel, IAP

The Coulomb gas p Rescaling β N x i → x i N Z dx j e − β N 2 V [ x ] Y Z N, β ∝ R N j =1 with 1 1 X X x 2 V [ x ] = ln | x i − x j | i − 2 N 2 N 2 i i 6 = j Z = Canonical partition function of fluid of particles of positions xi on a line - β N 2 in equilibrium at inverse temperature - with quadratic potential - and repulsive logarithmic potential - Renaux-Petel, IAP

The Coulomb gas 2D static fluid of charged particles confined on a line, with quadratic potential Large N 0 temperature limit To find equilibrium position at 0 temperature 1 Minimize intensive free energy F = − β N 2 ln Z N, β Z D n ( x ) e − β N 2 F [ n ( x )] Z N, β = 1) continuum description Aim: Functional integral n ? ( x ) = ρ ( x ) over counting functions 2) saddle-point Renaux-Petel, IAP

Continuum description 1). Coarse-graining - sum over micro states compatible with given Idea: macrostate (counting function) - then sum over all possible counting functions " # N Z n ( x ) − 1 X 1 = D [ n ( x )] δ δ ( x − x i ) N i =1 " # N N n ( x ) − 1 Z Z dx j e − β N 2 V [ x ] δ Y X D [ n ( x )] δ ( x − x i ) Z ∼ N R N j =1 i =1 Renaux-Petel, IAP

Continuum description 2). From sums to integrals N n ( x ) f ( x ) dx = 1 Z X f ( x i ) N R i =1 N 1 ZZ X R 2 dxdx 0 n ( x ) n ( x 0 ) g ( x, x 0 ) = g ( x i , x j ) N 2 i,j =1 " # N N Z Z n ( x ) − 1 Y X D [ n ( x )] e − β N 2 V [ n ( x )] δ ( x − x i ) Z ∼ dx j δ N R N j =1 i =1 | {z } with I N [ n ( x )] V [ n ( x )] = 1 dx x 2 n ( x ) − 1 Z ZZ R 2 dxdx 0 n ( x ) n ( x 0 ) ln | x − x 0 | 2 2 R + regularization Renaux-Petel, IAP

Continuum description 3). Compute I N [ n ( x )]  � Z I N [ n ( x )] ∼ exp [entropy] ∼ exp dx n ( x ) ln n ( x ) − N Summary: Z Z dk e − β N 2 S [ n ( x ) ,k ]+ o ( N 2 ) D [ n ( x )] Z ∼ R ✓Z ◆ S [ n ( x ) , κ ] = V [ n ( x )] − k dx n ( x ) − 1 Renaux-Petel, IAP

Saddle-point Z ∼ exp( − β N 2 S [ n ? ( x ) , k ? ]) 8 � R dx 0 n ? ( x 0 ) ln | x − x 0 | − k ? , k = k ? = x 2 � R 0 = � n ( x ) S [ n ( x ) , k ] � 2 − � n = n ? > < � @ R dx n ? ( x ) = 1 , R 0 = @ k S [ n ( x ) , k ] � k = k ? ⇒ � n = n ? > : n ? ( x ) ( a, b ) Look for and support ! n ? ( x ; a, b ) 1) find 2) find optimal (a,b) by minimizing free energy F = S [ n ? ( x ; a, b )] Renaux-Petel, IAP

Saddle-point dx 0 n ? ( x 0 ) Z PV x − x 0 = x Tricomi's theorem: √ R b ( t � a )( b � t ) Z b dt C − PV g ( t ) dx 0 f ( x 0 ) a x � t π PV x − x 0 = g ( x ) ⇒ f ( x ) = p ( x − a )( b − x ) π a  � 1 1 − x 2 + 1 2( a + b ) x + 1 8( b − a ) 2 1) n ? ( x ) = p ( x − a )( b − x ) π n ? ( x ) ≡ ρ SC ( x ) = 1 √ p 2 − x 2 2) − a = b = 2 π Renaux-Petel, IAP

Outline 1. Introduction 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5. High-energy landscape Renaux-Petel, IAP

Resolvent: generalities N G N ( z ) ≡ 1 1 random complex function X with poles at eigenvalues’s location N z − x i i =1 Resolvent dx 0 ρ ( x 0 ) Z z � x 0 ⌘ G ( av ) Stieltjes transform h G N ( z ) i ! 1 ( z ) Green’s function ⇢ ( x ) = 1 ✏ → 0 + Im G ( av ) ⇡ lim ∞ ( x − i ✏ ) Easy to deduce the spectral density from the resolvent Renaux-Petel, IAP

Resolvent for Gaussian Ensemble 1 = 1 1 1 x i ∂ V [ x ] = 0 ⇒ x i = 1 1 X X X X N ( z − x i ) ∂ x i N z − x i N x i − x j N x i − x j j 6 = i i i j 6 = i − 1 + zG N ( z ) = 1 N ( z ) + 1 2 G 2 2 N G 0 N ( z ) negligible in large N limit G ( av )2 ( z ) − 2 zG ( av ) ∞ ( z ) + 2 = 0 ∞ p G ( av ) z 2 − 2 ∞ ( z ) = z ± √ for x 2 < 2 1 1 ⇢ 2 − x 2 ✏ → 0 + ⇡ Im G ( av ) ∞ ( x − i ✏ ) − → ⇡ 0 otherwise Renaux-Petel, IAP

Another classical ensemble: Wishart-Laguerre W = HH † ( M ≥ N ) with matrix H : N × M positive eigenvalues Entries: Eigenvalues: Renaux-Petel, IAP

Wishart-Laguerre through the resolvent kept fixed Large N,M with With rescaling where Renaux-Petel, IAP

Wishart-Laguerre through the resolvent where Renaux-Petel, IAP

Marcenko-Pastur density 0.5 0.4 c = 1 / 8 0.3 c = 1 / 2 ρ MP ( x ) 0.2 0.1 0.0 0 5 10 15 x Renaux-Petel, IAP

Marcenko-Pastur density 1.0 0.8 c = 1 0.6 c = 1 / 8 ρ MP ( x ) 0.4 c = 1 / 2 0.2 0.0 0 5 10 15 x Particular case c=1. Accumulation of eigenvalues near 0 Renaux-Petel, IAP

Outline 1. Introduction 2. The Gaussian Ensemble(s) 3. The Coulomb gas approach 4. The resolvent approach 5.High-energy landscape Renaux-Petel, IAP

High-energy landscape High-energy physics: large number of fields, complex interactions Statistical predictions? Universal behavior in large N limit? With landscape modeled as multi-dimensional scalar potential (say Gaussian random field) Vacua = local minima of potential = critical points+ all Hessian eigenvalues positive Dependence of dynamics of inflation on smallest eigenvalues of the Hessian Key Q: Hessian eigenvalue distribution of Gaussian random fields Renaux-Petel, IAP

Gaussian Random Fields (GRF) Gaussian random field, static, statistically homogeneous h V ( ϕ ) i = ¯ and isotropic V ( φ ) with V All information in 2 pt correlation function: 1 Z V 2 = h V ( ϕ 1 ) V ( ϕ 2 ) i � ¯ d N k P ( k ) e i k · ( ϕ 1 − ϕ 2 ) (2 π ) N 1 Z d N k k 2 n P ( k ) σ 2 Define n ≡ (2 π ) N and correlation length For amplitude V 0 Λ ✓ N ◆ n σ 2 n ∼ V 2 0 Λ 2 Renaux-Petel, IAP