1. Large-N Quantum Field Theories and Nonlinear Random Processes [ArXiv: 1009.4033, 1011.2664] Pavel Buividovich (ITEP, Moscow and JINR, Dubna) Theory Seminar @ UNI Regensburg

2. Motivation Problems for modern Lattice QCD simulations(based on standard Monte-Carlo): Sign problem (finite chemical potential, fermions etc.) Sign problem (finite chemical potential, fermions etc.) Finite-volume effects Finite-volume effects Difficult analysis of excited states Difficult analysis of excited states Critical slowing-down Critical slowing-down Large-N extrapolation (AdS/CFT, AdS/QCD) Large-N extrapolation (AdS/CFT, AdS/QCD) Difficulties with SUSY Difficulties with SUSY Look for alternative numerical algorithms Look for alternative numerical algorithms

3. Motivation: Diagrammatic MC, Worm Algorithm, FUN methods,... Standard Monte-Carlo: directly evaluate the path integral Standard Monte-Carlo: directly evaluate the path integral Diagrammatic Monte-Carlo: stochastically sum all the terms in the perturbative expansion Diagrammatic Monte-Carlo: stochastically sum all the terms in the perturbative expansion

4. Motivation: Diagrammatic MC, Worm Algorithm, FUN methods,... Worm Algorithm [Prokof’ev, Svistunov]: Worm Algorithm [Prokof’ev, Svistunov]: Directly sample Green functions, Dedicated simulations!!! Example: Ising model x,y – head and tail of the worm Applications: Discrete symmetry groups a-la Ising [Prokof’ev, Svistunov] Discrete symmetry groups a-la Ising [Prokof’ev, Svistunov] O(N)/CP(N) lattice theories [Wolff] – difficult and limited O(N)/CP(N) lattice theories [Wolff] – difficult and limited

5. Motivation: Diagrammatic MC, Worm Algorithm, FUN methods,... FUNctional methods: Numerical solution of Schwinger-Dyson equations Numerical solution of Schwinger-Dyson equations Exact renormalization group Exact renormalization group Infinite set of equations Infinite set of equations Truncation required Truncation required Truncations typically break Truncations typically break gauge invariance... gauge invariance...

6. Systematic Extension to QCD? And other quantum field theories with continuous symmetry groups... Typical problems: Nonconvergence of perturbative expansion (non- compact variables) Nonconvergence of perturbative expansion (non- compact variables) Compact variables (SU(N), O(N), CP(N-1) etc.): finite convergence radius for strong coupling Compact variables (SU(N), O(N), CP(N-1) etc.): finite convergence radius for strong coupling Algorithm complexity grows with N Algorithm complexity grows with N Weak-coupling expansion (=lattice perturbation theory): complicated, volume-dependent...

7. Systematic Extension to QCD? QCD: perturbative series in g CONDENSATES (1/g) terms Power Corrections [SVZ, Parisi,...] Even planar diagrams do not converge E.g. perturbative expansion of TEK action (takes Jacobian into account): How to sum both contributions?

8. Large-N Quantum Field Theories Situation might be better at large N... In simple theories (such as φ 4 ) sums over PLANAR DIAGRAMS typically converge at weak coupling In simple theories (such as φ 4 ) sums over PLANAR DIAGRAMS typically converge at weak coupling Large-N theories are quite nontrivial – confinement, asymptotic freedom,... Large-N theories are quite nontrivial – confinement, asymptotic freedom,... Lattice SUSY is easier at large N(Reduced models) Lattice SUSY is easier at large N(Reduced models) Interesting for AdS/CFT, quantum gravity, AdS/condmat... Interesting for AdS/CFT, quantum gravity, AdS/condmat...

9. Main results to be presented: Two separate ingredients for QCD: Stochastic summation over planar graphs: a general “genetic” random processes (planar φ 4 ) Stochastic summation over planar graphs: a general “genetic” random processes (planar φ 4 ) Stochastic resummation of divergent series: Stochastic resummation of divergent series: random processes with memory (O(N) sigma-model) Join two algorithms into one: to be done!!!

10. Schwinger-Dyson equations at large N Example: φ 4 theory, φ – hermitian NxN matrix Example: φ 4 theory, φ – hermitian NxN matrix Action: Action: Observables = Green functions (Factorization!!!): Observables = Green functions (Factorization!!!): N, c – “renormalization constants” N, c – “renormalization constants”

11. Schwinger-Dyson equations at large N Closed equations for w(k 1,..., k n ): Closed equations for w(k 1,..., k n ): Always 2 nd order equations ! Always 2 nd order equations ! Infinitely many unknowns, but simpler than at finite N Infinitely many unknowns, but simpler than at finite N Efficient numerical solution? Stochastic! Efficient numerical solution? Stochastic! Importance sampling: w(k 1,..., k n ) – probability Importance sampling: w(k 1,..., k n ) – probability

12. Schwinger-Dyson equations at large N

13. “Genetic’’ Random Process Also: Recursive Markov Chain [Etessami, Yannakakis, 2005] Let X be some discrete set Let X be some discrete set Consider stack of the elements of X Consider stack of the elements of X At each process step: At each process step:  Create: with probability P c (x) create new x and push it to stack  Evolve: with probability P e (x|y) replace y on the top of the stack with x  Merge: with probability P m (x|y 1,y 2 ) pop two elements y 1, y 2 from the stack and push x into the stack Otherwise restart!!!

14. “Genetic’’ Random Process: Steady State and Propagation of Chaos Probability to find n elements x 1... x n in the stack: Probability to find n elements x 1... x n in the stack: W(x 1,..., x n ) W(x 1,..., x n ) Propagation of chaos [McKean, 1966] Propagation of chaos [McKean, 1966] ( = factorization at large-N [tHooft, Witten, 197x]): ( = factorization at large-N [tHooft, Witten, 197x]): W(x 1,..., x n ) = w 0 (x 1 ) w(x 2 )... w(x n ) W(x 1,..., x n ) = w 0 (x 1 ) w(x 2 )... w(x n ) Steady-state equation (sum over y, z): Steady-state equation (sum over y, z): w(x) = P c (x) + P e (x|y) w(y) + P m (x|y,z) w(y) w(z) w(x) = P c (x) + P e (x|y) w(y) + P m (x|y,z) w(y) w(z)

15. “Genetic’’ Random Process and Schwinger-Dyson equations Let X = set of all sequences {k 1,..., k n }, k – momenta Let X = set of all sequences {k 1,..., k n }, k – momenta Steady state equation for “Genetic” Random Process = Steady state equation for “Genetic” Random Process = Schwinger-Dyson equations, IF: Schwinger-Dyson equations, IF: Create: push a pair {k, -k}, P ~ G 0 (k) Create: push a pair {k, -k}, P ~ G 0 (k) Merge: pop two sequences and merge them Merge: pop two sequences and merge them Evolve: Evolve:  add a pair {k, -k}, P ~ G0(k)  sum up three momenta on top of the stack, P ~ λ G 0 (k) on top of the stack, P ~ λ G 0 (k)

16. Examples: drawing diagrams “Sunset” diagram“Typical” diagram “Sunset” diagram “Typical” diagram Only planar diagrams are drawn in this way!!!

17. Examples: tr φ 4 Matrix Model Exact answer known [Brezin, Itzykson, Zuber]

18. Examples: tr φ 4 Matrix Model Autocorrelation time vs. coupling: No critical slowing–down No critical slowing–down Convergence limit (a-la GW) cannot be reached Convergence limit (a-la GW) cannot be reached but not very interesting for large-N LGT but not very interesting for large-N LGT

19. Examples: tr φ 4 Matrix Model Sign problem vs. coupling: No severe sign problem!!!

20. Comment on asymptotic series Stochastic interpretation: Stochastic interpretation: maximal correlator order k max ~ 1/λ maximal correlator order k max ~ 1/λ = Maximal Accesible order in λ = Maximal Accesible order in λ Factorial Growth of series coefficients Factorial Growth of series coefficients Perturbative series @ Large N: Perturbative series @ Large N:

21. Examples: Weingarten model Weak-coupling expansion = sum over bosonic random surfaces [Weingarten, 1980] Complex NxN matrices on lattice links: “Genetic” random process: Stack of loops! Stack of loops! Basic steps: Basic steps:  Join loops  Remove plaquette Loop equations:

22. Examples: Weingarten model Randomly evolving loops sweep out all possible surfaces with spherical topology The process mostly produces “spiky” loops = random walks Noncritical string theory degenerates into scalar particle [Polyakov 1980] Noncritical string theory degenerates into scalar particle [Polyakov 1980] “Genetic” random process: Stack of loops! Stack of loops! Basic steps: Basic steps:  Join loops  Remove plaquette

23. Examples: Random planar surfaces

24. Some historical remarks “Genetic” algorithm vs. branching random process Probability to find some configuration of branches obeys nonlinear equation Steady state due to creation and merging Recursive Markov Chains [Etessami, Yannakakis, 2005] Also some modification of McKean-Vlasov-Kac models [McKean, Vlasov, Kac, 196x] “Extinction probability” obeys nonlinear equation [Galton, Watson, 1974] “Extinction of peerage” Attempts to solve QCD loop equations [Migdal, Marchesini, 1981] “Loop extinction”: No importance sampling

25. Compact variables? QCD, CP(N),... Schwinger-Dyson equations: still quadratic Schwinger-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: Example: O(N) sigma model on the lattice

26. O(N) σ-model: Schwinger-Dyson Schwinger-Dyson equations: Rewrite as: Now define a “probability” w(x): Strong-coupling expansion does NOT converge !!!

27. O(N) σ-model: Random walk Introduce the “hopping parameter”: Schwinger-Dyson equations = Steady-state equation for Bosonic Random Walk:

28. Random walks with memory “hopping parameter” depends on the return probability w(0): Iterative solution: Start with some initial hopping parameter Start with some initial hopping parameter Estimate w(0) from previous history memory Estimate w(0) from previous history memory Algorithm A: continuously update hopping parameter and w(0) Algorithm A: continuously update hopping parameter and w(0) Algorithm B: iterations Algorithm B: iterations

29. Random walks with memory: convergence

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

31. Random walks with memory: condensates and renormalons – “Condensate” – “Condensate” Non-analytic dependence on λ Non-analytic dependence on λ O(N) σ-model = Random Walk in its own “condensate” O(N) σ-model = Random Walk in its own “condensate” O(N) σ-model at large N: divergent strong coupling expansion O(N) σ-model at large N: divergent strong coupling expansion Absorb divergence into a redefined expansion parameter Absorb divergence into a redefined expansion parameter Similar to renormalons [Parisi, Zakharov,...] Similar to renormalons [Parisi, Zakharov,...] Nice convergent expansion Nice convergent expansion

32. Outlook: large-N gauge theory |n(x)|=1 = |n(x)|=1 = “Zigzag symmetry” “Zigzag symmetry” Self-consistent condensates = Lagrange multipliers for “Zigzag symmetry” [Kazakov 93]: “String project in multicolor QCD”, ArXiv:hep-th/9308135 Self-consistent condensates = Lagrange multipliers for “Zigzag symmetry” [Kazakov 93]: “String project in multicolor QCD”, ArXiv:hep-th/9308135 “QCD String” in its own condensate??? “QCD String” in its own condensate??? AdS/QCD: String in its own gravitation field AdS/QCD: String in its own gravitation field AdS: “Zigzag symmetry” at the boundary [Gubser, Klebanov, Polyakov 98], ArXiv:hep-th/9802109 AdS: “Zigzag symmetry” at the boundary [Gubser, Klebanov, Polyakov 98], ArXiv:hep-th/9802109

33. Summary Stochastic summation of planar diagrams at large N is possible Stochastic summation of planar diagrams at large N is possible Random process of “Genetic” type Random process of “Genetic” type Useful also for Random Surfaces Useful also for Random Surfaces Divergent expansions: absorb divergences into redefined self-consistent expansion parameters Divergent expansions: absorb divergences into redefined self-consistent expansion parameters Solving for self-consistency Solving for self-consistency Random process with memory Random process with memory