Large deviations of the top eigenvalue of random matrices and applications in statistical physics

Grégory SchehrLPTMS, CNRS-Université Paris-Sud XI

Collaborators:

Satya N. Majumdar (LPTMS, Orsay)Peter J. Forrester (Math. Dept., Univ. of Melbourne)Alain Comtet (LPTMS, Orsay)

Journées de Physique StatistiqueParis, January 29-30, 2015

S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014), arXiv:1311.0580

Large spectrum of applications of random matrix theory

Physics: nuclear physics, quantum chaos, disordered systems, mesoscopic transport, quantum entanglement, neural networks, gauge theory, string theory, cosmology, statistical physics (growth models, interface, directed polymers),...

Mathematics: number theory, combinatorics, knot theory, determinantal point processes, integrable systems, free probability,...

Statistics: multivariate statistics, principal component analysis (PCA), image processing, data compression, Bayesian model selection,...

Information theory: signal processing, wireless communications,...

Biology: sequence matching, RNA folding, gene expression networks,...

Economy and finance: time series and big data analysis,...

«The Oxford handbook of random matrix theory», Ed. by G. Akemann, J. Baik and P. Di Francesco (2011)

Spectral statistics in random matrix theory (RMT)

Basic model: real, symmetric, Gaussian random matrix

Invariant under rotationGaussian orthogonal ensemble (GOE)

The matrix has real eigenvalues which are strongly correlated

Spectral statistics in RMT: statistics of

P (M) / exp

2

4�N

2

X

i,j

M2i,j

3

5

/ exp

�N

2

Tr(M2)

�

Largest (top) eigenvalue of random matrices

!

Wigner sea

Density of eigenvalues for N � 1

+p2�

p2

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on

largest eigenvalue

! (", #)

"0

TRACY!WIDOM

WIGNER SEMI!CIRCLE

N

LARGE DEVIATION

LEFT

RIGHT

!2/3

! 2 2

LARGE DEVIATION

Typical fluctuations (small):

Tracy-Widom distribution

Tracy-Widom distribution

100

10-300

10-250

10-200

10-150

10-100

10-50

100

-20 0 20 40 60

⇠ exp

✓�2

3

x

3/2

◆

⇠ exp

✓� 1

24

|x|3◆

logF

0 1(x)

x

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on

largest eigenvalue

! (", #)

"0

TRACY!WIDOM

WIGNER SEMI!CIRCLE

N

LARGE DEVIATION

LEFT

RIGHT

!2/3

! 2 2

LARGE DEVIATION

Typical fluctuations (small):

Tracy-Widom distributionubiquitous

largest eigenvalue of correlation matrices (Wishart-Laguerre) longest increasing subsequence of random permutations directed polymers and growth models in the KPZ universality class continuum KPZ equation sequence alignment problems mesoscopic fluctuations in quantum dots high-energy physics (Yang-Mills theory)...

Experimental observation of TW distributions for GOE and GUE in liquid crystals experiments

Takeuchi & Sano ’10

Takeuchi, Sano, Sasamoto & Spohn ’11

from Takeuchi, Sano, Sasamoto & Spohn, Sci. Rep. (Nature) 1, 34 (2011)

Ubiquity of Tracy-Widom distributions

Q: universality of the Tracy-Widom distributions ?

(Carr-Helfrich instability)

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on

largest eigenvalue

! (", #)

"0

TRACY!WIDOM

WIGNER SEMI!CIRCLE

N

LARGE DEVIATION

LEFT

RIGHT

!2/3

! 2 2

LARGE DEVIATION

Typical fluctuations (small):

Tracy-Widom distributionubiquitous

In this talk: atypical and large fluctuations of

Large deviation functions Third order phase transition

Q: universality of the Tracy-Widom distributions ?

Stability of a large complex system

Stable non-interacting population of species with equilibrium densites

Slightly perturbed densities evolve via

(assuming identical damping times)

Switch on interactions between the species

random interaction matrixcoupling strength

Q: what is the proba. that the system remains stable once the interactions are switched on ?

Linear stability criterion

Linear dynamical system

Eigenvalues of :

The system is stable iff

i.e. iff

Proba. that the system is stable

Stable/Unstable transition for large systems

Assuming that the interaction matrix is real, symmetric and Gaussian

May observed a sharp transition in the limit : stable, weakly interacting phase: unstable, strongly interacting phase

Pstable(!, N) = Prob.["max ! w = 1/!]

wc ="2 w = 1/!STRONG COUPLING

UNSTABLE

STABLE

WEAK COUPLING

0

1

May ’72

Pstable(!, N) = Prob.["max ! w = 1/!]

wc ="2 w = 1/!

LEFT TAIL

0

1

UNSTABLE

STRONG COUPLING

RIGHT TAIL

WEAK COUPLING

STABLE

O(N#2/3)

Stable/Unstable transition for large systems

What happens for finite but large systems, of size ?

Is there any thermodynamic sense to this transition ?

What is the analogue of the free energy ?

What is the order of this transition ?

Coulomb Gas approach

Gaussian random matrix models random matrix :

Standard Dyson ’s ensembles: Orthogonal, Unitary, Symplectic (GOE) (GUE) (GSE)

Gaussian probability measure

Joint PDF of the real eigenvalues

with (GOE), (GUE) and (GSE)

Wigner ’51

Partition function

Coulomb Gas picture and Wigner semi-circle Rewrite the partition function as

2-d Coulomb gas confined to a line, with the inverse temperature

Typical scale of the eigenvalues:

Mean density of eigenvalues

!

Wigner sea

�p2 +

p2

Coulomb Gas with a wall

Cumulative distribution function of

wall

eigenvalues

What happens when the wall is moved ?

Pushed vs. pulled Coulomb gas

Saddle point analysis Dean & Majumdar ’06, ’08

PULLED

!!w(") !!w(")

" "

!!w(")

""2

"2

"2#

"2#

"2#L(w)

w <"2 w =

"2

w ww

w >"2

PUSHED CRITICAL

Mean density of eigenvalues in presence of the wall

Left large deviation function

i.e.

Physically, is the energy to push the Coulomb gas

Left deviation function

whenDean & Majumdar ’06, ’08

limN!1

� 1

�N2F (w,N) = ��(w)

F (w,N) = Pr .[�max

w] =ZN (w)

ZN (w ! 1)

⇠ exp[��N2

��(w)] , w <p2

N2��(w)

PULLED

!!w(") !!w(")

" "

!!w(")

""2

"2

"2#

"2#

"2#L(w)

w <"2 w =

"2

w ww

w >"2

PUSHED CRITICAL

Right large deviation function

The saddle point equation yields a trivial result for

Non trivial corrections requires a different approach

Right tail: pulled Coulomb gas

! max= w2 2!

WIGNER SEMI!CIRCLE

!

: energy to pull a single charge out of the Wigner sea

Majumdar & Vergassola ’09

when

Third order phase transitionunstable

stable This implies the behavior of the free energy

The third derivative of the free energy is discontinuous

Third order phase transition

The crossover, for finite , between the two phases is described by the Tracy-Widom distribution

S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014)

Third order phase transition

1N

0 ! = 1 w12

weakly interacting( )

STABLE

strongly interacting( )

UNSTABLE

crossover

S. N. Majumdar, G. S., J. Stat. Mech. P01012 (2014)

Similar third order phase transition in gauge theory

Similar third order phase transition in gauge theory

1N

0 ! = 1 w12

weakly interacting( )

STABLE

strongly interacting( )

UNSTABLE

crossover

1N

0

crossover

U(N) 2!d lattice gauge theory in

coupling strength g

WEAK STRONG

GROSS!WITTEN!WADIA transition (1980)

gc

Similar transition in lattice gauge theory

Unstable phase = strong coupling phase of Yang-Mills gauge theoryStable phase = weak coupling phase of Yang-Mills gauge theory

Tracy-Widom ditribution describes the crossover between the two regimes (at finite but large ): double scaling regime

Conclusion Largest eigenvalue of a Gaussian random matrix

Application to the stability of large complex system

Proba. distrib. func. (PDF) of : Coulomb gas (CG) with a wall

Tracy-Widom distribution

Physics of large deviation tails Left tail: pushed CGRight tail: unpushed CG

Third order phase transition pushed/unpushedunstable/stable

Similar third order transition in Yang-Mills gauge theories and other systems (conductance fluctuations, complexity in spin glasses, non-intersecting Brownian motions...)