near extreme eigenvalues of large random gaussian matrices and …... · 2016. 5. 14. · near...

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


At the edge


, 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


Density of near extreme eigenvalues for GUE


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




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




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


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

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




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)

Short Courses

Jean-Philippe Bouchaud (CFM, Ecole Polytechnique) Alain Comtet (Université P. et M. Curie, Paris VI) Bertrand Eynard (Commissariat à l’énergie atomique, Saclay) Jonathan Keating (School of Mathematics, University of Bristol) Gernot Akemann (Universität Bielefeld) Aris Moustakas (University of Athens) Henning Schomerus (Lancaster University) Vincent Vargas (CNRS, Ecole Normale Supérieure, Paris) Hans Weidenmüller (Universität Heidelberg) Anton Zabrodin (ITEP, Moscow)

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

Scientific Program

The connections between stochastic processes and random matrix theory (RMT) have been a rapidly evolving subject during the last ten years where the continuous development of new tools has led to an avalanche of new results. The most emblematic example of such successful connections is provided by the theory of growth phenomena in the Kardar-Parisi-Zhang (KPZ) universality class. Other important instances include non-intersecting Brownian motions or processes governed by "1/f" noise, which are at the heart of disordered systems exhibiting a freezing transition. These breakthroughs have been made possible thanks, to a large extent, to the recent development of various new techniques in RMT. The goal of the school thus is to present the state of the art concepts of the field, with a special emphasis on the large spectrum of techniques and applications of RMT. Finally an important aspect of this school is that the topic of stochastic processes and RMT is at the interface of statistical physics and mathematics. Therefore this school aims at bringing together students (and researchers) from both communities.

Registration

Applications must reach the School before March 1, 2015 in order to be considered by the selection committee. The full cost per participant, including housing, meals and the book of lecture notes, is 1500€ (we should be able to provide financial aid to a limited number of students). All information and application form can be found directly on Les Houches website: http://houches.ujf-grenoble.fr/.

One can also contact the School at

ECOLE D’ETE DE PHYSIQUE THEORIQUE La Côte des Chavants 74310 LES HOUCHES, France

Director: Leticia Cugliandolo Phone: +33 4 50 54 40 59 – Fax: +33 4 50 55 53 25 Email: [email protected]

Les Houches is a village located in Chamonix valley, in the French Alps. Established in 1951, the Physics School is situated at 1150m above sea level in natural surroundings, with breathtaking view on the Mont-Blanc range. A quiet place, ideal for intellectual activity.

Les Houches Physics School is affiliated with Université Joseph Fourier and Institut National Polytechnique de Grenoble, and is supported by the Ministère de l’Education Nationale et de la Recherche, the Centre National de la Recherche Scientifique (CNRS), the Direction des Sciences de la Matière du Commissariat à l’Energie Atomique (CEA/DSM).

http://lptms.u-psud.fr/workshop/randmat/

near extreme eigenvalues of large random gaussian matrices and …... · 2016. 5. 14. · near...

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