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Near extreme eigenvalues of large random Gaussian matrices and applications
Grégory SchehrLPTMS, CNRS-Université Paris-Sud XI
A. Perret, G. S., J. Stat. Phys. 156, pp. 843-876 (2014)
Groupe de Travail «Matrices et graphes aléatoires» Paris (IHP), January 16, 2015
Collaborators:
Yan V. Fyodorov (Queen Mary, London)Anthony Perret (LPTMS, Orsay)
A. Perret, Y. V. Fyodorov, G. S., in preparation
Extreme value statistics
Statement of the problem
random variables, X1, X2, · · · , XN : N Pjoint
(X1
, X2
, · · · , XN )
Xmax
= max
1iNXi, FN (M) = P(Xmax M)
Xi
Three different universality classes depending on the pdf of : Gumbel, Fréchet, Weibull
Fully understood for i.i.d. random variables
Q: N ! 1 ?
Extreme value statistics
Random walks
Random matrices
Xmax
= max
1iNXi, FN (M) = P(Xmax M)
X1, X2, · · · , XN : N random variables, Pjoint
(X1
, X2
, · · · , XN )
Q: N ! 1 ?
Statement of the problem
Very few exact results for strongly correlated variables
NXmax
is interesting BUT concerns a single variable among
Statistics of Near-Extremes
Statistical PhysicsEnergy levels
E0 Ground state{T > 0
T = 0
Natural sciences (e.g. seismology)
Branching Brownian motion
1/f noise
see also:
Brownian motion
...
Crowding near the extremes
Statistics of Near-Extremes
How to quantify the crowding close to extreme values ?
Look at (higher) order statistics: k maximum th
Xmax
= M1,N > M2,N > · · · > MN,N = Xmin
and in particular the spacings (gaps) dk,N = Mk,N �Mk+1,N
Consider the density of near-extremes Sabhapandit, Majumdar ’07
Near-extreme eigenvalues of random matrices
Let be a real symmetric (or complex Hermitian) random matrix
The matrix has real eigenvalues which are strongly correlated
Density of near extreme eigenvalues
see also Witte, Bornemann, Forrester ’13
A related quantity is the gap between the two largest eigenvalues
Largest eigenvalue
N ⇥N
Application: minimizing a quadratic form on the sphere
Quadratic form on the -dimensional sphere
Minimisation of this quadratic form on the sphere
are the eigenvalues of with
introduce a Lagrange multiplier
with
Eigenvalues of the Hessian matrix at the minimum
spectrum of is
is the mean eigenvalue density of
with
Reminding that
Application: minimizing a quadratic form on the sphere
Dynamics of the spherical fully connected spin-glass model (spherical Sherrington-Kirkpatrick model)
Application to spherical fully connected spin-glasses
Cugliandolo, Dean ’95
where
and belongs to the GOE ensemble of RMT with variance
Random initial condition at time :
temp.
infinitesimal mag. field
normalized eigenvectors of :
Relaxational dynamics characterized by two-time quantities
Ben Arous, Dembo, Guionnet ’06
Relaxational dynamics characterized by two-time quantities
Application to spherical fully connected spin-glasses
Correlation function Response function
In the quasi-stationary regime, for fixed
Perret, Fyodorov, G. S. ’14
see also Kurchan, Laloux ’96
Near-extreme eigenvalues of random matrices
rdr
λΛ1,N = λmax
Λ2,N
Wigner sea
Density of near extreme eigenvalues
1 gapst
⇢DOS(r,N) =1
N � 1
NX
i=1i 6=i
max
�(�max � �i � r)
Pjoint
(�1
,�2
, ...,�N ) =1
ZN
Y
i
Density of near extreme eigenvalues in GUE
Fluctuations of the largest eigenvalue �max
= max
1iN�i
Tracy-Widom
Depending on one expects two different regimes(r,N)
bulk regime
edge regime
A detour by the density of eigenvalues of random matrices
Two regimes: bulk and edge regime
-1.0 -0.5 0.5 1.0
0.1
0.2
0.3
0.4
-10 -8 -6 -4 -2 2
0.2
0.4
0.6
0.8
1.0
Bowick & Brézin ’91 , Forrester ’93
Two regimes: bulk and edge regime
A detour by the density of eigenvalues of GUE random matrices
Matching between the bulk and edge regimesmatching with Wigner semi-circle
coincides with the right tail of TW
Density of near extreme eigenvalues: results
In the bulk :
is insensitive to the fluctuations of
shifted Wigner semi-circle
A. Perret, G. S. ’14
Density of near extreme eigenvalues: results
At the edge
A. Perret, G. S. ’14
, a non trivial function
Consequences on the dynamics of the spherical fully connected spin-glass model
In the quasi-stationary regime, for
two regimes:
Two temporal regimes for the response function
Cugliandolo, Dean ’95
Perret, Fyodorov, G. S. ’15
Ben Arous, Dembo, Guionnet ’06
Consequences on the dynamics of the spherical fully connected spin-glass model
For
Asymptotic behaviors
Can one compute the full universal function
universal !
?an exact calculation for GUE Perret, G. S. ’14
,
Density of near extreme eigenvalues for GUE
is related to a solution of the Lax pair associated to Painlevé XXXIV
Perret, G. S. ’14
is related to a solution of the Lax pair associated to Painlevé XXXIV
Density of near extreme eigenvalues for GUE
Density of near extreme eigenvalues: results
Asymptotic behaviors
ρ̃edge(r̃)
r̃00
10
1
Outline
Exact formulas for for finite
Asymptotic analysis for large
Comparison with existing results
Conclusion and related open problems
Outline
Asymptotic analysis for large
Comparison with existing results
Conclusion and related open problems
Exact formulas for for finite
An exact formula for
wall
two-point correlations for conditioned eigenvalues
eigenvalues
After some manipulations one obtains
wih the kernel
⇢DOS(r,N) =1
N � 1
NX
i=1i 6=i
max
�(�max � �i � r)
An exact formula for
wall
eigenvalues
Finally a useful formula is 2-point correlations
⇢DOS(r,N) =1
N � 1
NX
i=1i 6=i
max
�(�max � �i � r)
An exact formula for the PDF of the first gap
wall
eigenvalues
Reminding that
one gets
Outline
Orthogonal polynomials (OPs) on the semi-infinite real line
Exact formulas for for finite
Comparison with existing results
Conclusion and related open problems
Orthogonal polynomials
Pjoint
(�1
,�2
, ...,�N ) =1
ZN
Y
i
Physical picture associated to the OP sytem
Coulomb gas with a wall located in
Moving the wall through the edge: the density
ρy(λ)
λ λ λ
PUSHED CRITICAL PULLEDy >
√2N
√2N−
√2N
yyy
−L(y)√2N
y =√2N
−√2N
√2N
y <√2N
ρy(λ) ρy(λ)
described by a double scaling limitsee also T. Claeys, A. Kuijlaars ’08, «...when the soft edge meets the hard edge»
for a review see S. N. Majumdar, G. S. ’13
⇢DOS(r,N) =1
N � 1
NX
i=1i 6=i
max
h�(�max � �i � r)i
Large analysis of
Edge regime :
Analysis of the kernel in the double scaling limit
where
Find a solution of the recurrence in the double scaling limit
⇢DOS(r,N) =1
N � 1
NX
i=1i 6=i
max
h�(�max � �i � r)i
Large analysis of
In the double scaling limit the recurrence relation is solved by
A. Perret, G. S. ’14
with
In the double scaling limit, one has C. Nadal, S. N. Majumdar ’13
Large analysis of
After some more computations...
see T. Claeys, A. Kuijlaars ’07 for a (rigorous) derivation using RH
where solve the Lax pair for Painlevé XXXIV
is related to a solution of the Lax pair associated to Painlevé XXXIV
Density of near extreme eigenvalues: results
Typical fluctuations of the gap: resultsA. Perret, G. S. ’13
using
we obtain
from which we obtain the asymptotic behaviors
withand complicated integral involving
Outline
Orthogonal polynomials (OPs) on the semi-infinite real line
Exact formulas for for finite
Comparison with existing results
Conclusion and related open problems
Relations with existing results
Density of near extreme eigenvalues was not studied in RMT (to my knowledge)
Forrester ’93
Witte, Bornemann, Forrester ’13
Previous studies of the PDF of the first gap
an expression in terms of a Fredholm determinant
an expression in terms of Painlevé transcendents
and a numerical computation of the formula in terms of a Fredholm determinant
PDF of the first gap: the formula of Witte, Bornemann, Forrester
with
and
Showing that this formula coincides with ours is still challenging...
with
PDF of the first gap: the numerical evaluation of Witte, Bornemann, Forrester
logp̃ty
p(r̃)
log r̃
0.001
0.01
0.1
0.1
1
Conclusion and related open questions
Exact results for the statistics of near extreme eigenvalues of GUE
A new formula for the PDF of the first gap in terms of Painlevé transcendents (precise asymptotics)
What about GOE and GSE (skew orthogonal polynomials) ?
Applications to the relaxational dynamics of mean-field spin glass
What about Laguerre-Wishart matrices ?
Stochastic processes and random matrices July 6-31, 2015
Session CIV!
Long Courses
Organizing committee
Integrability and the Kardar-Parisi-Zhang universality class An introduction to free probability Replicas, renormalization and integrability in random systems Recent applications of random matrices in statistical physics The Kardar-Parisi-Zhang equation – a statistical physics perspective β-ensembles and random Schrödinger operators
Alexander Altland Yan Fyodorov Neil O’Connell Grégory Schehr
Institut für Theoretische Physik, Universität Köln, Germany School of Mathematical Sciences, Queen Mary University of London, Great-Britain Mathematics Institute, University of Warwick, Great-Britain Laboratoire de Physique Théorique et Modèles Statisitques, CNRS, Université Paris-Sud, France
Alexei Borodin (Massachusetts Institute of Technology) Alice Guionnet (Massachusetts Institute of Technology) Pierre Le Doussal (CNRS, Ecole Normale Supérieure, Paris) Satya Majumdar (CNRS, Université Paris-Sud, Orsay) Herbert Spohn (Universität München) Bálint Virág (University of Toronto)
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Random matrix theory and (big) data analysis Schrödinger operators in random potentials Topological recursion in random matrices and combinatorics of maps Random matrix theory and number theory Matrix models and quantum chromo-dynamics Random matrix theory methods for telecommunication systems Random matrix theory approaches to open quantum systems Gaussian multiplicative chaos and Liouville quantum gravity Historical overview: random matrix theory and its applications Some aspects of integrability and quantum-classical correspondence
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