Large-N Quantum Field Large-N Quantum Field Theories and Nonlinear Theories and Nonlinear

Random ProcessesRandom Processes

Pavel BuividovichPavel Buividovich

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

ITEP Lattice Seminar, 16.09.2010ITEP Lattice Seminar, 16.09.2010

MotivationMotivationProblemsProblems for modernfor modern Lattice QCDLattice QCD

simulations(based onsimulations(based on standard Monte-Carlostandard Monte-Carlo):):

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

• Finite-volumeFinite-volume effectseffects• Difficult analysis ofDifficult analysis of excited statesexcited states• Critical slowing-downCritical slowing-down• Large-NLarge-N extrapolationextrapolation

Look forLook for alternative numerical algorithmsalternative numerical algorithms

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

• Standard Monte-Carlo:Standard Monte-Carlo: directly evaluate the directly evaluate the path integralpath integral

• Diagrammatic Monte-Carlo:Diagrammatic Monte-Carlo: stochastically sumstochastically sum all the terms in the all the terms in the perturbative expansionperturbative expansion

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

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

Example: Example: Ising modelIsing model

x,yx,y – – head and head and tail of the wormtail of the worm

Applications:Applications: • Discrete symmetry groups Discrete symmetry groups a-la Isinga-la Ising [Prokof’ev, Svistunov][Prokof’ev, 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 symmetrysymmetry groups ... groups ...

Typical problems:Typical problems:• NonconvergenceNonconvergence of perturbative expansion of perturbative expansion (non-compact variables)(non-compact variables)

• Compact variables (Compact variables (SU(N), O(N), CP(N-1)SU(N), O(N), CP(N-1) etc.): etc.): finite convergence radiusfinite convergence radius for for strong strong couplingcoupling

• Algorithm complexity Algorithm complexity grows withgrows with N N

• Weak-coupling expansionWeak-coupling expansion (=lattice (=lattice perturbation theory): perturbation theory): complicated, volume-complicated, volume-dependent...dependent...

Large-N Quantum Field Large-N Quantum Field TheoriesTheoriesSituation might be Situation might be better at large N ...better at large N ...

• Sums over Sums over PLANAR DIAGRAMSPLANAR DIAGRAMS typically typically converge at weak couplingconverge at weak coupling

• Large-N theoriesLarge-N theories are quite nontrivial – are quite nontrivial – confinement, asymptotic freedom,confinement, asymptotic freedom, ... ...

• Interesting for Interesting for AdS/CFTAdS/CFT, , quantum gravityquantum gravity, , AdS/condmatAdS/condmat ... ...

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

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

Noncompact VariablesNoncompact Variables Itzykson-Zuber integralsItzykson-Zuber integrals Weingarten modelWeingarten model = Random surfaces = Random surfaces • Stochastic resummationStochastic resummation of divergent of divergent series:series:random processes with memoryrandom processes with memory

O(N) sigma-modelO(N) sigma-model outlook: outlook: non-Abelian large-N theoriesnon-Abelian large-N theories

Schwinger-Dyson equations at large NSchwinger-Dyson equations at large N

• Example:Example: φφ44 theory, theory, φφ – hermitian NxN matrix – hermitian NxN matrix• Action:Action:

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

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

Schwinger-Dyson equations at large NSchwinger-Dyson equations at large N• Closed equations for w(kClosed equations for w(k11, ..., k, ..., knn):):

• Always Always 22ndnd order equations order equations ! !• Infinitely many unknownsInfinitely 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(k11, ..., k, ..., knn) – probability) – probability

Schwinger-Dyson equations at large NSchwinger-Dyson equations at large N

““Genetic’’ Random ProcessGenetic’’ Random ProcessAlso: Also: Recursive Markov ChainRecursive Markov Chain [[Etessami, Etessami, Yannakakis, 2005Yannakakis, 2005]]

• Let Let XX be some be some discrete setdiscrete set• Consider Consider stackstack of the of the elements of Xelements of X• At each At each process stepprocess step::

Create:Create: with probability with probability PPcc(x) create new x(x) create new x and push it to stackand push it to stack

Evolve:Evolve: with probability with probability PPee(x|y) replace y (x|y) replace y on the top of the stack with xon the top of the stack with x

Merge:Merge: with probability with probability PPmm(x|y(x|y11,y,y22) pop two ) pop two elements yelements y11, y, y22 from the stack and from the stack and push xpush x into into the stackthe stack

Otherwise restart!!!Otherwise restart!!!

““Genetic’’ Random Process: Genetic’’ Random Process: Steady State and Propagation of ChaosSteady State and Propagation of Chaos• ProbabilityProbability to find to find n elements xn elements x11 ... x ... xnn in the in the stack:stack:

W(xW(x11, ..., x, ..., xnn))

• Propagation of chaosPropagation of chaos [McKean, 1966][McKean, 1966] ( = ( = factorization at large-Nfactorization at large-N [tHooft, Witten, [tHooft, Witten, 197x]):197x]):

W(xW(x11, ..., x, ..., xnn) = w) = w00(x(x11) w(x) w(x22) ... w(x) ... w(xnn))

• Steady-state equationSteady-state equation (sum over y, z): (sum over y, z):

w(x) = Pw(x) = Pcc(x) + P(x) + Pee(x|y) w(y) + P(x|y) w(y) + Pmm(x|y,z) w(y) (x|y,z) w(y) w(z) w(z)

““Genetic’’ Random Process and Genetic’’ Random Process and Schwinger-Dyson equationsSchwinger-Dyson equations

• Let Let XX = set of = set of all sequences {kall sequences {k11, ..., k, ..., knn}}, k – momenta, k – momenta

• Steady state equationSteady state equation for for “Genetic” Random Process“Genetic” Random Process = = Schwinger-Dyson equationsSchwinger-Dyson equations, IF:, IF:

• Create:Create: push a pair {k, -k}, P ~ G push a pair {k, -k}, P ~ G00(k)(k)• Merge:Merge: pop two sequences and merge them pop two sequences and merge them• Evolve:Evolve: add a pair {k, -k}, P ~ G0(k)add a pair {k, -k}, P ~ G0(k) sum up three momentasum up three momenta on top of the stack, P ~ on top of the stack, P ~ λλ GG00(k) (k)

Examples: Examples: drawing drawing diagramsdiagrams““Sunset” diagramSunset” diagram “Typical” “Typical” diagramdiagram

Only planar diagrams are drawn in this way!!!Only planar diagrams are drawn in this way!!!

Examples: Examples: tr tr φφ4 4 Matrix Matrix ModelModel

ExactExact answer knownanswer known [Brezin, Itzykson, Zuber][Brezin, Itzykson, Zuber]

Examples: Examples: tr tr φφ44 Matrix Matrix ModelModel

• Autocorrelation timeAutocorrelation time vs. coupling: vs. coupling: No critical slowing–downNo critical slowing–down• Peculiar: Peculiar: only g < ¾ gonly g < ¾ gcc can be reached!!! can be reached!!! NotNot a a dynamicaldynamical, but an , but an algorithmic limitation...algorithmic limitation...

Examples: Examples: tr tr φφ4 Matrix 4 Matrix ModelModel

Sign problemSign problem vs. coupling: vs. coupling: No severe sign No severe sign problem!!!problem!!!

Examples: Examples: Weingarten Weingarten modelmodelWeak-coupling expansionWeak-coupling expansion = sum over bosonic = sum over bosonic random surfacesrandom surfaces [Weingarten, 1980][Weingarten, 1980]ComplexComplex NxN matrices NxN matrices on on lattice linkslattice links::

““Genetic” random Genetic” random process:process:• Stack ofStack of loops! loops!• Basic steps:Basic steps: Join loopsJoin loops Remove Remove plaquetteplaquette

Loop equations:Loop equations:

Examples: Examples: Weingarten Weingarten modelmodel

Randomly evolving Randomly evolving loopsloops sweep outsweep out all possible all possible surfacessurfaces with with spherical topologyspherical topology

The process mostly produces The process mostly produces “spiky” loops = “spiky” loops = random walksrandom walks Noncritical stringNoncritical string theory degenerates into theory degenerates into scalar particlescalar particle [Polyakov 1980][Polyakov 1980]

““Genetic” random Genetic” random process:process:• Stack ofStack of loops! loops!• Basic steps:Basic steps: Join loopsJoin loops Remove Remove plaquetteplaquette

Examples: Examples: Weingarten modelWeingarten modelCritical index vs. dimensionCritical index vs. dimension

Peak around D=26 !!! (Preliminary)Peak around D=26 !!! (Preliminary)

Some historical remarksSome historical remarks““Genetic” algorithmGenetic” algorithm vs. vs. branching random processbranching random process

Probability to find Probability to find some configurationsome configurationof branches obeys nonlinearof branches obeys nonlinearequationequation

Steady state due to creationSteady state due to creationand mergingand merging

Recursive Markov Chains Recursive Markov Chains [[Etessami, Yannakakis, 2005Etessami, Yannakakis, 2005]]

Also some modification of Also some modification of McKean-Vlasov-Kac modelsMcKean-Vlasov-Kac models[McKean, Vlasov, Kac, 196x][McKean, Vlasov, Kac, 196x]

““Extinction probability”Extinction probability” obeys obeys nonlinear equationnonlinear equation[Galton, Watson, 1974][Galton, Watson, 1974]““Extinction of peerage”Extinction of peerage”

Attempts to solve Attempts to solve QCD QCD loop loop equations equations [Migdal, Marchesini, 1981][Migdal, Marchesini, 1981]

““Loop extinction”:Loop extinction”:No importance samplingNo importance sampling

Compact variables? QCD, Compact variables? QCD, CP(N),...CP(N),...• Schwinger-Dyson equations: still quadraticSchwinger-Dyson equations: still quadratic• Problem: alternating signs!!!Problem: alternating signs!!!• Convergence only at strong couplingConvergence only at strong coupling• Weak coupling is most interesting...Weak coupling is most interesting...

Observables:Observables:

Example: O(N) sigma model on the latticeExample: O(N) sigma model on the lattice

O(N) O(N) σσ-model: -model: Schwinger-Schwinger-DysonDysonSchwinger-Dyson equations:Schwinger-Dyson equations:

Rewrite as:Rewrite as:

Now define aNow define a “probability” w(x): “probability” w(x):

Strong-coupling expansion does NOT converge !!!Strong-coupling expansion does NOT converge !!!

O(N) O(N) σσ-model: -model: Random walkRandom walkIntroduce the Introduce the “hopping parameter”:“hopping parameter”:

Schwinger-Dyson equationsSchwinger-Dyson equations = Steady-state equation for = Steady-state equation for Bosonic Random Walk:Bosonic Random Walk:

Random walks with memoryRandom walks with memory““hopping parameter” hopping parameter” depends on thedepends on the return return probability w(0):probability w(0):

Iterative solution:Iterative solution:• Start with some Start with some initial initial hopping parameterhopping parameter• Estimate w(0) from previous history Estimate w(0) from previous history memorymemory

• Algorithm A:Algorithm A: continuously updatecontinuously 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: convergenceconvergence

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

Random walks with memory: Random walks with memory: condensatescondensates and and renormalonsrenormalons

• <n(0)<n(0)∙∙n(+/- e n(+/- e μμ)> – )> – “Condensate”“Condensate”• Non-analyticNon-analytic dependence on dependence on λλ• O(N) O(N) σσ-model = -model = Random Walk in its own Random Walk in its own “condensate”“condensate”

• O(N) O(N) σσ-model at large N: -model at large N: divergent strong divergent strong coupling expansioncoupling expansion• Absorb Absorb divergencedivergence into a into a redefined redefined expansion parameterexpansion parameter• Similar to Similar to renormalonsrenormalons [Parisi, [Parisi, Zakharov, ...]Zakharov, ...]• Nice Nice convergent expansionconvergent expansion

Outlook: Outlook: large-N gauge theorylarge-N gauge theory

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

• Self-consistent condensates = Lagrange Self-consistent condensates = Lagrange multipliers for “Zigzag symmetry” multipliers for “Zigzag symmetry” [Kazakov [Kazakov 93]: “String project in multicolor QCD”, 93]: “String project in multicolor QCD”, ArXiv:ArXiv:hep-th/9308135hep-th/9308135

““QCD String”QCD String” in its own in its own condensate???condensate???

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

• AdS: “Zigzag symmetry” at the boundary AdS: “Zigzag symmetry” at the boundary [Gubser, Klebanov, Polyakov 98],[Gubser, Klebanov, Polyakov 98], ArXiv:ArXiv:hep-hep-th/9802109th/9802109

SummarySummary• Stochastic summation of planar Stochastic summation of planar

diagramsdiagrams at large N is possible at large N is possible

Random process of “Genetic” typeRandom process of “Genetic” type

• Useful also for Useful also for Random SurfacesRandom Surfaces

• Divergent expansions:Divergent expansions: absorb absorb divergences into divergences into redefined self-redefined self-consistent expansion parametersconsistent expansion parameters

• Solving for Solving for self-consistencyself-consistency

Random process with memoryRandom process with memory