1009.4033] ] : 1009.4033 [ArXiv ArXiv: [ Pavel Buividovich - - PowerPoint PPT Presentation

Large Large-

• N Quantum Field Theories

N Quantum Field Theories and Nonlinear Random Processes and Nonlinear Random Processes [ [ArXiv ArXiv: :1009.4033

1009.4033] ]

Pavel Pavel Buividovich Buividovich (ITEP, Moscow and JINR, (ITEP, Moscow and JINR, Dubna Dubna) )

Bogoliubov Bogoliubov readings, readings, Dubna Dubna, 23.09.2010 , 23.09.2010

Motivation Motivation

Problems Problems for for modern modern Lattice Lattice QCD QCD simulations simulations( (based based

• n
• n standard

standard Monte Monte-

• Carlo

Carlo): ):

• Sign problem

Sign problem (finite chemical potential, fermions (finite chemical potential, fermions etc.) etc.)

• Finite

Finite-

• volume

volume effects effects

• Difficult

Difficult analysis analysis of

• f excited

excited states states

• Critical slowing

Critical slowing-

• down

down

• Large

Large-

• N

N extrapolation extrapolation ( (AdS AdS/CFT, /CFT, AdS AdS/QCD) /QCD) Look Look for for alternative alternative numerical numerical algorithms algorithms

Motivation: Motivation: Diagrammatic MC, Diagrammatic MC, Worm Algorithm, ... Worm Algorithm, ...

• Standard Monte

Standard Monte-

• Carlo:

Carlo: directly evaluate the directly evaluate the path path integral integral

• Diagrammatic Monte

Diagrammatic Monte-

• Carlo:

Carlo: stochastically sum stochastically sum all the all the terms in the terms in the perturbative perturbative expansion expansion

Motivation: Motivation: Diagrammatic MC, Diagrammatic MC, Worm Algorithm, ... Worm Algorithm, ...

• Worm Algorithm

Worm Algorithm [ [Prokof Prokof’ ’ev ev, , Svistunov Svistunov]: ]: Directly sample Directly sample Green functions, Dedicated simulations!!! Green functions, Dedicated simulations!!!

Example: Example: Ising Ising model model x,y x,y – – head and tail head and tail

• f the worm
• f the worm

Applications: Applications:

• Discrete symmetry groups

Discrete symmetry groups a a-

• la

la Ising Ising [ [Prokof Prokof’ ’ev ev, , Svistunov Svistunov] ]

• O(N)/CP(N) lattice theories

O(N)/CP(N) lattice theories [Wolff] [Wolff] – – difficult and limited difficult and limited

Extension to QCD? Extension to QCD?

And other quantum field theories with And other quantum field theories with continuous continuous symmetry symmetry groups ... groups ... Typical problems: Typical problems:

• Nonconvergence

Nonconvergence of

• f perturbative

perturbative expansion (non expansion (non-

• compact variables)

compact variables)

• Compact variables (

Compact variables (SU(N), O(N), CP(N SU(N), O(N), CP(N-

• 1)

1) etc.): etc.): finite convergence radius finite convergence radius for for strong coupling strong coupling

• Algorithm complexity

Algorithm complexity grows with grows with N N

• Weak

Weak-

• coupling expansion

coupling expansion (=lattice perturbation (=lattice perturbation theory): theory): complicated, volume complicated, volume-

• dependent...

dependent...

Large Large-

• N Quantum Field Theories

N Quantum Field Theories

Situation might be Situation might be better at large N ... better at large N ...

• Sums over

Sums over PLANAR DIAGRAMS PLANAR DIAGRAMS typically typically converge converge at weak coupling at weak coupling

• Large

Large-

• N theories

N theories are quite nontrivial are quite nontrivial – – confinement, asymptotic freedom, confinement, asymptotic freedom, ... ...

• Interesting for

Interesting for AdS AdS/CFT /CFT, , quantum gravity quantum gravity, , AdS/condmat AdS/condmat ... ...

Main results to be presented: Main results to be presented:

• Stochastic summation over planar graphs:

Stochastic summation over planar graphs: a a general general “ “genetic genetic” ” random processes random processes

• Noncompact

Noncompact Variables Variables

• Itzykson

Itzykson-

• Zuber

Zuber integrals integrals

• Weingarten model

Weingarten model = Random surfaces = Random surfaces

• Stochastic

Stochastic resummation resummation of divergent series:

• f divergent series:

random processes with memory random processes with memory

• O(N) sigma

O(N) sigma-

• model

model

• utlook:
• utlook: non

non-

• Abelian

Abelian large large-

• N theories

N theories

Schwinger Schwinger-

• Dyson equations at large N

Dyson equations at large N

• Example:

Example: φ φ4

4 theory,

theory, φ φ – – hermitian hermitian NxN NxN matrix matrix

• Action:

Action:

• Observables = Green functions (Factorization!!!):

Observables = Green functions (Factorization!!!):

• N, c

N, c – – “ “renormalization constants renormalization constants” ”

Schwinger Schwinger-

• Dyson equations at large N

Dyson equations at large N

• Closed equations for w(k

Closed equations for w(k1

1, ...,

, ..., k kn

n):

):

• Always

Always 2 2nd

nd order equations

• rder equations !

!

• Infinitely many unknowns

Infinitely many unknowns, but simpler than at finite N , but simpler than at finite N

• Efficient

Efficient numerical solution? numerical solution? Stochastic! Stochastic!

• Importance sampling:

Importance sampling: w(k w(k1

1, ...,

, ..., k kn

n)

) – – probability probability

Schwinger Schwinger-

• Dyson equations at large N

Dyson equations at large N

“ “Genetic Genetic’’ ’’ Random Process Random Process

Also: Also: Recursive Markov Chain Recursive Markov Chain [ [Etessami, Etessami, Yannakakis, 2005 Yannakakis, 2005] ]

• Let

Let X X be some be some discrete set discrete set

• Consider

Consider stack stack of the

• f the elements of X

elements of X

• At each

At each process step process step: :

• Create:

Create: with probability with probability P Pc

c(x

(x) create new x ) create new x and and push it to stack push it to stack

• Evolve:

Evolve: with probability with probability P Pe

e(x|y

(x|y) replace y on the ) replace y on the top of the stack with x top of the stack with x

• Merge:

Merge: with probability with probability P Pm

m(x|y

(x|y1

1,y

,y2

2) pop two

) pop two elements y elements y1

1, y

, y2

2 from the stack and

from the stack and push x push x into the into the stack stack

Otherwise restart!!! Otherwise restart!!!

“ “Genetic Genetic’’ ’’ Random Process: Random Process: Steady State and Propagation of Chaos Steady State and Propagation of Chaos

• Probability

Probability to find to find n elements x n elements x1

1 ...

... x xn

n in the stack:

in the stack: W(x W(x1

1, ...,

, ..., x xn

n)

)

• Propagation of chaos

Propagation of chaos [McKean, 1966] [McKean, 1966] ( = ( = factorization at large factorization at large-

• N

N [ [tHooft tHooft, , Witten Witten, 197x]): , 197x]): W(x W(x1

1, ...,

, ..., x xn

n) = w

) = w0

0(x

(x1

1) w(x

) w(x2

2) ...

) ... w(x w(xn

n)

)

• Steady

Steady-

• state equation

state equation (sum over y, z): (sum over y, z): w(x w(x) = ) = P Pc

c(x

(x) + ) + P Pe

e(x|y

(x|y) ) w(y w(y) + ) + P Pm

m(x|y,z

(x|y,z) ) w(y w(y) ) w(z w(z) )

“ “Genetic Genetic’’ ’’ Random Process and Random Process and Schwinger Schwinger-

• Dyson equations

Dyson equations

• Let

Let X X = set of = set of all sequences {k all sequences {k1

1, ...,

, ..., k kn

n}

}, k , k – – momenta momenta

• Steady state equation

Steady state equation for for “ “Genetic Genetic” ” Random Process Random Process = = Schwinger Schwinger-

• Dyson equations

Dyson equations, IF: , IF:

• Create:

Create: push a pair {k, push a pair {k, -

• k}, P ~ G

k}, P ~ G0

0(k)

(k)

• Merge:

Merge: pop two sequences and merge them pop two sequences and merge them

• Evolve:

Evolve:

• add a pair {k,

add a pair {k, -

• k}, P ~ G0(k)

k}, P ~ G0(k)

• sum up three

sum up three momenta momenta

• n top of the stack, P ~
• n top of the stack, P ~ λ

λ G G0

0(k)

(k)

Examples: Examples: drawing diagrams drawing diagrams

“ “Sunset Sunset” ” diagram diagram “ “Typical Typical” ” diagram diagram Only planar diagrams are drawn in this way!!! Only planar diagrams are drawn in this way!!!

Examples: Examples: tr tr φ φ4

4 Matrix Model

Matrix Model

Exact Exact answer known answer known [ [Brezin Brezin, , Itzykson Itzykson, , Zuber Zuber] ]

Examples: Examples: tr tr φ φ4

4 Matrix Model

Matrix Model

• Autocorrelation time

Autocorrelation time vs. coupling:

• vs. coupling:

No critical slowing No critical slowing– –down down

• Peculiar:

Peculiar: only g <

• nly g < ¾

¾ g gc

c can be reached!!!

can be reached!!! Not Not a a dynamical dynamical, but an , but an algorithmic limitation... algorithmic limitation...

Examples: Examples: tr tr φ φ4

4 Matrix Model

Matrix Model

Sign problem Sign problem vs. coupling:

• vs. coupling: No severe sign

No severe sign problem!!! problem!!!

Examples: Examples: Weingarten model Weingarten model

Weak Weak-

• coupling expansion

coupling expansion = sum over = sum over bosonic bosonic random surfaces random surfaces [Weingarten, 1980] [Weingarten, 1980] Complex Complex NxN NxN matrices matrices on

• n lattice links

lattice links: : “ “Genetic Genetic” ” random random process: process:

• Stack of

Stack of loops! loops!

• Basic steps:

Basic steps:

• Join loops

Join loops

• Remove

Remove plaquette plaquette Loop equations: Loop equations:

Examples: Examples: Weingarten model Weingarten model

Randomly evolving Randomly evolving loops loops sweep out sweep out all possible all possible surfaces surfaces with with spherical topology spherical topology The process mostly produces The process mostly produces “ “spiky spiky” ” loops = loops = random walks random walks Noncritical Noncritical string string theory degenerates into theory degenerates into scalar scalar particle particle [ [Polyakov Polyakov 1980] 1980] “ “Genetic Genetic” ” random random process: process:

• Stack of

Stack of loops! loops!

• Basic steps:

Basic steps:

• Join loops

Join loops

• Remove

Remove plaquette plaquette

Examples: Examples: Weingarten model Weingarten model Critical index vs. dimension Critical index vs. dimension

Peak around D=23, close to D Peak around D=23, close to Dc

c=26 !!!

=26 !!!

Some historical remarks Some historical remarks

“ “Genetic Genetic” ” algorithm algorithm vs.

• vs. branching random process

branching random process

Some historical remarks Some historical remarks

“ “Genetic Genetic” ” algorithm algorithm vs.

• vs. branching random process

branching random process

Probability to find Probability to find some configuration some configuration

• f branches obeys nonlinear
• f branches obeys nonlinear

equation equation Steady state due to creation Steady state due to creation and merging and merging Recursive Markov Chains Recursive Markov Chains [ [Etessami, Yannakakis, 2005 Etessami, Yannakakis, 2005] ] Also some modification of Also some modification of McKean McKean-

• Vlasov

Vlasov-

• Kac

Kac models models [McKean, [McKean, Vlasov Vlasov, , Kac Kac, 196x] , 196x] “ “Extinction probability Extinction probability” ” obeys

• beys

nonlinear equation nonlinear equation [ [ Galton Galton, Watson, 1974] , Watson, 1974] “ “Extinction of peerage Extinction of peerage” ” Attempts to solve Attempts to solve QCD loop QCD loop equations equations [ [ Migdal Migdal, , Marchesini Marchesini, 1981] , 1981] “ “Loop extinction Loop extinction” ”: : No importance sampling No importance sampling

Compact variables? QCD, CP(N),... Compact variables? QCD, CP(N),...

• Schwinger

Schwinger-

• Dyson equations: still quadratic

Dyson equations: still quadratic

• Problem: alternating signs!!!

Problem: alternating signs!!!

• Convergence only at strong coupling

Convergence only at strong coupling

• Weak coupling is most interesting...

Weak coupling is most interesting... Observables: Observables: Example: O(N) sigma model on the lattice Example: O(N) sigma model on the lattice

O(N) O(N) σ σ-

• model:

model: Schwinger Schwinger-

• Dyson

Dyson

Schwinger Schwinger-

• Dyson equations:

Dyson equations: Rewrite as: Rewrite as: Now define a Now define a “ “probability probability” ” w(x w(x): ): Strong Strong-

• coupling expansion does NOT converge !!!

coupling expansion does NOT converge !!!

O(N) O(N) σ σ-

• model:

model: Random walk Random walk

Introduce the Introduce the “ “hopping parameter hopping parameter” ”: : Schwinger Schwinger-

• Dyson equations

Dyson equations = Steady = Steady-

• state equation for

state equation for Bosonic Bosonic Random Walk: Random Walk:

Random walks with memory Random walks with memory

“ “hopping parameter hopping parameter” ” depends on the depends on the return return probability w(0): probability w(0): Iterative solution: Iterative solution:

• Start with some

Start with some initial initial hopping parameter hopping parameter

• Estimate w(0) from previous history memory

Estimate w(0) from previous history memory

• Algorithm A:

Algorithm A: continuously update continuously update hopping hopping parameter and w(0) parameter and w(0)

• Algorithm B:

Algorithm B: iterations iterations

Random walks with memory: Random walks with memory: convergence convergence

Random walks with memory: Random walks with memory: asymptotic freedom in 2D asymptotic freedom in 2D

Random walks with memory: Random walks with memory: condensates condensates and and renormalons renormalons

• <n(0)

<n(0)· ·n(+/ n(+/-

• e

e μ

μ)>

)> – – “ “Condensate Condensate” ”

• Non

Non-

• analytic

analytic dependence on dependence on λ λ

• O(N)

O(N) σ σ-

• model =

model = Random Walk in its own Random Walk in its own “ “condensate condensate” ”

• O(N)

O(N) σ σ-

• model at large N:

model at large N: divergent strong divergent strong coupling expansion coupling expansion

• Absorb

Absorb divergence divergence into a into a redefined expansion redefined expansion parameter parameter

• Similar to

Similar to renormalons renormalons [ [Parisi Parisi, , Zakharov Zakharov, ...] , ...]

• Nice

Nice convergent expansion convergent expansion

Outlook: Outlook: large large-

• N gauge theory

N gauge theory

• |

|n(x n(x)|=1 = )|=1 = “ “Zigzag symmetry Zigzag symmetry” ”

• Self

Self-

• consistent condensates = Lagrange multipliers

consistent condensates = Lagrange multipliers for for “ “Zigzag symmetry Zigzag symmetry” ” [ [Kazakov Kazakov 93]: 93]: “ “String project String project in multicolor QCD in multicolor QCD” ”, , ArXiv ArXiv: :hep hep-

• th/9308135

th/9308135 “ “QCD String QCD String” ” in its own in its own condensate??? condensate???

• AdS

AdS/QCD: /QCD: String in its own gravitation field String in its own gravitation field

• AdS

AdS: : “ “Zigzag symmetry Zigzag symmetry” ” at the boundary at the boundary [ [Gubser Gubser, , Klebanov Klebanov, , Polyakov Polyakov 98], 98], ArXiv: ArXiv:hep hep-

• th/9802109

th/9802109

Summary Summary

• Stochastic summation of planar diagrams

Stochastic summation of planar diagrams at at large N is possible large N is possible Random process of Random process of “ “Genetic Genetic” ” type type

• Useful also for

Useful also for Random Surfaces Random Surfaces

• Divergent expansions:

Divergent expansions: absorb divergences absorb divergences into into redefined self redefined self-

• consistent expansion

consistent expansion parameters parameters

• Solving for

Solving for self self-

• consistency

consistency Random process with memory Random process with memory