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

Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI Analytical Results in Statistical Physics, in memory of Bernard Jancovici

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

Grégory Schehr LPTMS, CNRS-Université Paris-Sud XI

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

Analytical Results in Statistical Physics, in memory of Bernard Jancovici

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

Large spectrum of applications of random matrix theory

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),...

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,...

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,...

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,...

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,...

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,...

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

P(M) ∝ exp  −N 2

• i,j

M 2

i,j

  ∝ exp

• −N

2 Tr(M 2)

Spectral statistics in random matrix theory (RMT)

Basic model: real, symmetric, Gaussian random matrix Invariant under rotation Gaussian orthogonal ensemble (GOE)

P(M) ∝ exp  −N 2

• i,j

M 2

i,j

  ∝ exp

• −N

2 Tr(M 2)

Spectral statistics in random matrix theory (RMT)

Basic model: real, symmetric, Gaussian random matrix Invariant under rotation Gaussian orthogonal ensemble (GOE) The matrix has real eigenvalues which are strongly correlated

P(M) ∝ exp  −N 2

• i,j

M 2

i,j

  ∝ exp

• −N

2 Tr(M 2)

Spectral statistics in random matrix theory (RMT)

Basic model: real, symmetric, Gaussian random matrix Invariant under rotation Gaussian orthogonal ensemble (GOE) The matrix has real eigenvalues which are strongly correlated Spectral statistics in RMT: statistics of

P(M) ∝ exp  −N 2

• i,j

M 2

i,j

  ∝ exp

• −N

2 Tr(M 2)

Largest (top) eigenvalue of random matrices

λ Wigner sea

Density of eigenvalues for N ≫ 1

+ √ 2 − √ 2

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on largest eigenvalue

ρ (λ, Ν)

λ

TRACY−WIDOM

WIGNER SEMI−CIRCLE

N LARGE DEVIATION LEFT RIGHT −2/3

− 2

2 LARGE DEVIATION

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on largest eigenvalue

ρ (λ, Ν)

λ

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

20 40 60

∼ exp

• −2

3x3/2

• ∼ exp
• − 1

24|x|3

• log F ′

1(x)

x

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on largest eigenvalue

ρ (λ, Ν)

λ

TRACY−WIDOM

WIGNER SEMI−CIRCLE

N LARGE DEVIATION LEFT RIGHT −2/3

− 2

2 LARGE DEVIATION

Typical fluctuations (small): Tracy-Widom distribution ubiquitous

Largest (top) eigenvalue of random matrices

Recent excitements in statistical physics and mathematics on largest eigenvalue

ρ (λ, Ν)

λ

TRACY−WIDOM

WIGNER SEMI−CIRCLE

N LARGE DEVIATION LEFT RIGHT −2/3

− 2

2 LARGE DEVIATION

Typical fluctuations (small): Tracy-Widom distribution ubiquitous

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

ρ (λ, Ν)

λ

TRACY−WIDOM

WIGNER SEMI−CIRCLE

N LARGE DEVIATION LEFT RIGHT −2/3

− 2

2 LARGE DEVIATION

Typical fluctuations (small): Tracy-Widom distribution ubiquitous 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

Stability of a large complex system

Stable non-interacting population of species with equilibrium densites

Stability of a large complex system

Stable non-interacting population of species with equilibrium densites Slightly perturbed densities evolve via

(assuming identical damping times)

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 matrix coupling strength

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

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

Linear stability criterion

Linear dynamical system

Linear stability criterion

Linear dynamical system Eigenvalues of :

Linear stability criterion

Linear dynamical system Eigenvalues of : The system is stable iff i.e. iff

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

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(α, N) = Prob.[λmax ≤ w = 1/α] wc = √ 2 w = 1/α STRONG COUPLING UNSTABLE STABLE WEAK COUPLING 1

May ’72

Pstable(α, N) = Prob.[λmax ≤ w = 1/α] wc = √ 2 w = 1/α LEFT TAIL 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 ?

Pstable(α, N) = Prob.[λmax ≤ w = 1/α] wc = √ 2 w = 1/α LEFT TAIL 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 :

Gaussian random matrix models

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

Gaussian random matrix models

random matrix : Standard Dyson ’ s ensembles: Orthogonal, Unitary, Symplectic (GOE) (GUE) (GSE) Gaussian probability measure

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

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

Coulomb Gas picture and Wigner semi-circle

Rewrite the partition function as 2-d Coulomb gas confined to a line, with the inverse temperature

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:

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

λ Wigner sea

− √ 2 + √ 2

Coulomb Gas with a wall

Cumulative distribution function of

Cumulative distribution function of

wall eigenvalues

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 ρ∗

w(λ)

ρ∗

w(λ)

λ λ ρ∗

w(λ)

λ √ 2 √ 2 √ 2 − √ 2 − √ 2 −L(w) w < √ 2 w = √ 2 w w w w > √ 2 PUSHED CRITICAL

Mean density of eigenvalues in presence of the wall

Left large deviation function

i.e.

lim

N→∞ −

1 βN 2 F(w, N) = Φ−(w) F(w, N) = Pr .[λmax ≤ w] = ZN(w) ZN(w → ∞) ∼ exp[−βN 2Φ−(w)] , w < √ 2

Left large deviation function

i.e. Physically, is the energy to push the Coulomb gas

lim

N→∞ −

1 βN 2 F(w, N) = Φ−(w) F(w, N) = Pr .[λmax ≤ w] = ZN(w) ZN(w → ∞) ∼ exp[−βN 2Φ−(w)] , w < √ 2 N 2Φ−(w)

Left large deviation function

i.e. Physically, is the energy to push the Coulomb gas Left deviation function when

Dean & Majumdar ’06, ’08

lim

N→∞ −

1 βN 2 F(w, N) = Φ−(w) F(w, N) = Pr .[λmax ≤ w] = ZN(w) ZN(w → ∞) ∼ exp[−βN 2Φ−(w)] , w < √ 2 N 2Φ−(w)

PULLED ρ∗

w(λ)

ρ∗

w(λ)

λ λ ρ∗

w(λ)

λ √ 2 √ 2 √ 2 − √ 2 − √ 2 −L(w) w < √ 2 w = √ 2 w w w w > √ 2 PUSHED CRITICAL

Right large deviation function

The saddle point equation yields a trivial result for

PULLED ρ∗

w(λ)

ρ∗

w(λ)

λ λ ρ∗

w(λ)

λ √ 2 √ 2 √ 2 − √ 2 − √ 2 −L(w) w < √ 2 w = √ 2 w w w 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

λmax = w

2 2

−

WIGNER SEMI−CIRCLE

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

Right tail: pulled Coulomb gas

λmax = w

2 2

−

WIGNER SEMI−CIRCLE

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

Majumdar & Vergassola ’09

when

Third order phase transition

unstable stable

Third order phase transition

unstable stable This implies the behavior of the free energy

Third order phase transition

unstable stable This implies the behavior of the free energy The third derivative of the free energy is discontinuous Third order phase transition

Third order phase transition

unstable 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

1 N

α = 1 w 1 2

weakly interacting

( ) STABLE

strongly interacting

( ) UNSTABLE

crossover

Third order phase transition

1 N

α = 1 w 1 2

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

1 N

α = 1 w 1 2

weakly interacting

( ) STABLE

strongly interacting

( ) UNSTABLE

crossover

1 N

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 theory Stable phase = weak coupling phase of Yang-Mills gauge theory

Similar third order phase transition in gauge theory

1 N

α = 1 w 1 2

weakly interacting

( ) STABLE

strongly interacting

( ) UNSTABLE

crossover

1 N

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 theory Stable 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

Conclusion

Largest eigenvalue of a Gaussian random matrix

Conclusion

Largest eigenvalue of a Gaussian random matrix Application to the stability of large complex system

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

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

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 CG Right tail: unpushed CG

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 CG Right tail: unpushed CG Third order phase transition pushed/unpushed unstable/stable

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 CG Right tail: unpushed CG Third order phase transition pushed/unpushed unstable/stable Similar third order transition in Yang-Mills gauge theories and

• ther systems (conductance fluctuations, complexity in spin

glasses, non-intersecting Brownian motions...)