large deviations of the top eigenvalue of random matrices...
TRANSCRIPT
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
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 ?
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
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
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...)