1. Diagrammatic Monte-Carlo algorithms for large-N quantum field theories from Schwinger-Dyson equations Pavel Buividovich (Regensburg University) ECT* workshop on DiagMC

2. Quantum field theory: Motivation: Diagrammatic Monte-Carlo Sum over fields Euclidean action: Sum over interacting paths Perturbative expansions

3. DiagMC in strong-coupling lattice QCD at finite density [de Forcrand,Vairinhos,Gattringer,Chandrasekharan,…] DiagMC in strong-coupling lattice QCD at finite density [de Forcrand,Vairinhos,Gattringer,Chandrasekharan,…] Few orders of SC expansion NOT SYSTEMATIC, BUT EFFICIENT Few orders of SC expansion NOT SYSTEMATIC, BUT EFFICIENT SYSTEMATIC WAY: DUAL REPRESENTATIONS/SC EXPANSIONS Abelian Lattice Gauge Theories (also with fermions) Abelian Lattice Gauge Theories (also with fermions) Scalar field theories Scalar field theories O(N) / CP N sigma models O(N) / CP N sigma models OFTEN, DUAL REPRESENTATIONS are very PHYSICAL (In QCD, already the lowest order gives CONFINEMENT) BUT: no convenient duality for non-Abelian fields (Clebsch- Gordan coefficients are cumbersome+sign-alternating) BUT: no convenient duality for non-Abelian fields (Clebsch- Gordan coefficients are cumbersome+sign-alternating) Weak-coupling expansions are also cumbersome and difficult to re-sum Weak-coupling expansions are also cumbersome and difficult to re-sum Systematic DiagMC in the non-Abelian case? Systematic DiagMC in the non-Abelian case? Avoid manual construction of duality transformations? Avoid manual construction of duality transformations? Avoid Borel resummations? Avoid Borel resummations? Motivation: QCD side

4. DiagMC algorithms can be constructed in a universal way from Schwinger-Dyson equations General structure of SD equations General structure of SD equations Solution of SD equations by Metropolis algorithm Solution of SD equations by Metropolis algorithm Practical implementation of the algorithm Practical implementation of the algorithm Sign problem and reweighting Sign problem and reweighting Simplifications in the large-N limit Simplifications in the large-N limit Strong-coupling QCD at finite chemical potential Strong-coupling QCD at finite chemical potential (how it could work in principle) Closer look at SU(N) sigma model Closer look at SU(N) sigma model (possible ways out of the sign problem) Outline

5. General structure of SD equations (everywhere we assume lattice discretization) Choose some closed set of observables φ(X) X is a collection of all labels, e.g. for scalar field theory SD equations (with disconnected correlators) are linear: A(X | Y): infinite-dimensional, but sparse linear operator b(X): source term, typically only 1-2 elements nonzero

6. Stochastic solution of linear equations Assume: A(X|Y), b(X) are positive, |eigenvalues| < 1 Solution with Metropolis algorithm: Sample sequences {X n, …, X 0 } with the weight Two basic transitions: Add new index X n+1, Add new index X n+1, Remove index Remove index Restart Restart

7. Stochastic solution of linear equations Compare [Prokof’ev,Svistunov, PRL 81 (1998) 2514] With probability p + : Add index step With probability (1-p + ): Remove index/Restart Ergodicity: any sequence can be reached (unless A(X|Y) has some block-diagonal structure) Acceptance probabilities (no detailed balance, Metropolis-Hastings) Parameter p + can be tuned to reach optimal acceptance Parameter p + can be tuned to reach optimal acceptance Probability distribution of N(X) is crucial to asses convergence Probability distribution of N(X) is crucial to asses convergence Finally: make histogram of the last element X n in the sequence Solution φ(X), normalization factor

8. Illustration: ϕ 4 matrix model (Running a bit ahead) Large autocorrelation time and large fluctuations near the phase transition

9. Practical implementation for QFT Every transition is a summand in a symbolic representation of SD equations Every transition is a summand in a symbolic representation of SD equations Every transition is a “drawing” of some element of diagrammatic expansion (either weak- or strong-coupling one) Every transition is a “drawing” of some element of diagrammatic expansion (either weak- or strong-coupling one) Diagrams with any number of legs are sampled Diagrams with any number of legs are sampled (extension of the worm algorithm) (extension of the worm algorithm) Keep in computer memory: current diagram current diagram history of drawing history of drawing Need DO and UNDO operations for every diagram element Almost automatic algorithmization: We do not have to remember what’s inside the diagrams (but we can, if necessary !!!) We do not have to remember what’s inside the diagrams (but we can, if necessary !!!) We do not have to know diagrammatic rules – only the SD equations, quite easy to derive We do not have to know diagrammatic rules – only the SD equations, quite easy to derive

10. Sign problem and reweighting Now lift the assumptions A(X | Y) > 0, b(X)>0 Now lift the assumptions A(X | Y) > 0, b(X)>0 Use the absolute value of weight for the Metropolis sampling Use the absolute value of weight for the Metropolis sampling Sign of each configuration: Sign of each configuration: Define Define Effectively, we solve the system Effectively, we solve the system The expansion has typically smaller radius of convergence The expansion has typically smaller radius of convergence Reweighting fails if the system Reweighting fails if the system approaches the critical point (one of eigenvalues approach 1) approaches the critical point (one of eigenvalues approach 1) One can only be saved by a suitable reformulation of equations which makes the sign problem milder

11. Diagrammatic MC in the large-N limit for matrix-valued fields In the above scheme, only series of the form In real QFTs/non-Abelian LGT’s: No. of diagrams ~ n! +UV renormalon No. of diagrams ~ n! +UV renormalon IR divergences ~ n! (IR renormalon) IR divergences ~ n! (IR renormalon) Things become simpler in the large-N limit UV renormalons are absent in the planar limit UV renormalons are absent in the planar limit Number of planar diagrams grows exponentially Number of planar diagrams grows exponentially IR renormalons are interesting, but can be removed by IR regularization (finite volume, twisted BC etc.) IR renormalons are interesting, but can be removed by IR regularization (finite volume, twisted BC etc.) SD equations are significantly simpler!!! SD equations are significantly simpler!!!

12. SD equations in the large-N limit SD equations simplify (factorization), BUT are quadratic SD equations simplify (factorization), BUT are quadratic Still use the linear form without 1/N terms (step back) Still use the linear form without 1/N terms (step back) Factorized solution Factorized solution Solve linear equation (*), make histograms of X m only Solve linear equation (*), make histograms of X m only Factorization is a property of random process Factorization is a property of random process

13. SD equations@large N: stack structure The larger space of “indices” labeling has a natural stack structure has a natural stack structure (Nonlinear random processes/ Recursive Markov chains in Math) (Nonlinear random processes/ Recursive Markov chains in Math) Now three basic transitions instead of X → Y (prob. ~ A(X | Y)) Create: with probability ~b(X) create new X and push it to stack Create: with probability ~b(X) create new X and push it to stack Evolve: with probability ~A(X|Y) replace topmostY with X Evolve: with probability ~A(X|Y) replace topmostY with X Merge: with probability ~C(X|Y, Z) pop two elements Y, Z from the stack and push X Merge: with probability ~C(X|Y, Z) pop two elements Y, Z from the stack and push X Note that now X i are strongly correlated (we update only topmost) Makes sense to use only topmost element for statistical sampling

14. Minimal working example: finite-N matrix model Consider 0D matrix field theory (finite-N matrix model) Weights of all diagrams are positive!!! Full set of observables: multi-trace correlators Schwinger-Dyson equations (Akin to the topological recursion formula [B. Eynard])

15. Resummation/Rescaling Growth of field correlators with n/ order: Exponential in the large-N limit Exponential in the large-N limit Factorial at finite N Factorial at finite N How to interprete as a probability distribution? Exponential growth? Introduce “renormalization constants” : Exponential growth? Introduce “renormalization constants” : is now finite, is now finite, can be interpreted as probability can be interpreted as probability In the Metropolis algorithm: all the transition weights should be finite, otherwise unstable behavior How to deal with factorial growth? Borel resummation

16. SD equations in finite-N matrix model Basic operations of “diagram drawing”: Insert line Insert line Merge singlet operators Merge singlet operators Create vertex Create vertex Split singlet operators (absent in the planar limit) Split singlet operators (absent in the planar limit)

17. SD equations in finite-N matrix model Probability of the split action grows as Probability of the split action grows as Separate diagrams of different genus Separate diagrams of different genus All transition probabilities are finite if All transition probabilities are finite if Borel non-summability (or UV renormalon) pops up in the 1/N expansion!!!

18. Genus expansion: ϕ 4 matrix model [From Marino,Schiappa,Weiss 0711.1954]

19. Genus expansion: ϕ 4 matrix model Heavy-tailed distributions near criticality, P(x) ~ x -2

20. SD equations in finite-N matrix model Think of each as an n-segment boundary of a random surface

21. SD equations in finite-N matrix model Probability of “split” action grows as Probability of “split” action grows as Obviously, cannot be removed by rescaling of the form N c n Introduce rescaling factors which depend on number of vertices OR genus 1)Expansion in λ, N fixed:

22. Standard perturbative expansion Upon rescaling, weights of transitions are: Insert line Insert line Merge singlet operators Merge singlet operators Create vertex Create vertex Split singlet operators () ) Split singlet operators () ) Require all of them to be finite Factorial growth of the number of diagrams with the number of vertices!

23. Borel resummation in practice In practice: 4-5 poles in Borel plane

24. Test: triviality of φ 4 theory in D ≥ 4 Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv:1104.3459]

25. Genus expansion Alternatively, let’s expand in 1/N: Upon rescaling, weights of transitions are: Insert line Insert line Merge singlet operators Merge singlet operators Create vertex Create vertex Split singlet operators Split singlet operators At fixed genus, exponential growth with n, m Ansatz

26. Genus expansion: factorial divergence Now the weights of transitions are: Insert line Insert line Merge singlet operators Merge singlet operators Create vertex Create vertex Split singlet Split singletoperators Finiteness of c g implies Finiteness of c g implies Then, Then, Finite if is finite Finite if is finite c g should be finite for any g !!!

27. Large-N gauge theory in the Veneziano limit Gauge theory with the action Gauge theory with the action t-Hooft-Veneziano limit: t-Hooft-Veneziano limit: N -> ∞, N f -> ∞, λ fixed, N f /N fixed N -> ∞, N f -> ∞, λ fixed, N f /N fixed Only planar diagrams contribute! connection with strings Only planar diagrams contribute! connection with strings Factorization of Wilson loops W(C) = 1/N tr P exp(i ∫dx μ A μ ): Factorization of Wilson loops W(C) = 1/N tr P exp(i ∫dx μ A μ ): Better approximation for real QCD than pure large-N gauge theory: meson decays, deconfinement phase etc. Better approximation for real QCD than pure large-N gauge theory: meson decays, deconfinement phase etc.

28. Large-N gauge theory in the Veneziano limit (how it can work in principle) Lattice action: Lattice action: Naive Dirac fermions: N f is infinite, no need to care about doublers!!!. Basic observables: Basic observables:  Wilson loops = closed string amplitudes  Wilson lines with quarks at the ends = open string amplitudes

29. Large-N gauge theory in the Veneziano limit Lattice action: Lattice action: No EK reduction in the large-N limit! Center symmetry broken by fermions. Naive Dirac fermions: N f is infinite, no need to care about doublers!!!. Basic observables: Basic observables:  Wilson loops = closed string amplitudes  Wilson lines with quarks at the ends = open string amplitudes Zigzag symmetry for QCD strings!!! Zigzag symmetry for QCD strings!!!

30. Migdal-Makeenko loop equations Loop equations in the closed string sector: Loop equations in the open string sector: Infinite hierarchy of quadratic equations!

31. Loop equations illustrated Quadratic term

32. Loop equations: stochastic interpretation Stack of strings (= open or closed loops)! Wilson loop W[C] ~ Probabilty of generating loop C Possible transitions (closed string sector): Create new string Append links to string Join strings with links Join strings …if have collinear links Remove staples Probability ~ β Create open string Identical spin states

33. Temperature and chemical potential Finite temperature: strings on cylinder R ~1/T Finite temperature: strings on cylinder R ~1/T Winding strings = Polyakov loops ~ quark free energy Winding strings = Polyakov loops ~ quark free energy Veneziano limit: open strings wrap and close Veneziano limit: open strings wrap and close (No EK reduction) (No EK reduction) Chemical potential: Chemical potential: Strings stretch in time Strings stretch in time κ -> κ exp(+/- μ) No signs or phases!

34. Re-emergence of the sign problem and the physics of QCD strings? At large ß, sign problem re-emerges… At large ß, sign problem re-emerges… Consider e.g. strong-coupling expansion of Wilson loop W[L x L] of fixed lattice size L Consider e.g. strong-coupling expansion of Wilson loop W[L x L] of fixed lattice size L At large ß, W[C] -> 1 At large ß, W[C] -> 1 In SC expansion, In SC expansion, Large positive powers of ß should cancel to give 1 Large positive powers of ß should cancel to give 1 (even though analiticity in ß in any finite lattice volume) (even though analiticity in ß in any finite lattice volume) Sign problem is necessary!!! Sign problem is necessary!!! In contrast, for lattice Nambu-Goto strings [Weingarten model] In contrast, for lattice Nambu-Goto strings [Weingarten model] No sign problemNo continuum limit No sign problemNo continuum limit Remember the idea on fermionic d.o.f.’s/extra dimension Remember the idea on fermionic d.o.f.’s/extra dimension for QCD strings [Polyakov, Migdal and Co] for QCD strings [Polyakov, Migdal and Co]

35. Lessons from SU(N) sigma-model Nontrivial playground similar to QCD!!! Action:Observables: Schwinger-Dyson equations: Stochastic solution naturally leads to strong-coupling series! Alternating sign already @ leading order

36. Making sign problem milder: momentum space SD equations in momentum space:

37. Making sign problem milder: momentum space vs λ Deep in the SC regime, only few orders relevant… Sign problem becomes important towards WC

38. Nonperturbative improvement!!! [Vicari, Rossi,...] Making sign problem milder: Matrix Lagrange Multiplier

39. SD equations at the edge of convergence The above SD equations are at the edge of convergence for any λ Matrix of SD equations has unit eigenvalue In the SC limit, we reduce to 0D matrix model Two COMPLETE sets of observables: Convergence for G + Convergence for G + Edge of convergence for G - Edge of convergence for G - Shall we use positive powers of Lagrange multiplier for SD equations?

40. Making sign problem milder: weak-coupling expansion + stereo projection Representation in terms of Hermitian matrices: Bare propagator: General pattern: bare mass term ~λ (Reminder of the compactness of the group manifold) Infinite number of higher-order vertices Infinite number of higher-order vertices No double-trace terms No double-trace terms Bare mass is again the bare coupling… Bare mass is again the bare coupling… What do we get if we perform the standard perturbative expansion?

41. Simpler case: O(N) sigma-model Stereographic projection: maps whole R N to whole S N Let’s blindly denote m 0 2 = λ/2 and do expansion We get series of the form (something like transseries)

42. Convergence of double series Mass gap vs. order Z vs order X axis: 1/(max order) λ=2 (m=0.2) λ=4 λ=2

43. DiagMC for O(N) σ-model: sign problem At every order in λ, many diagrams with different signs contribute vs order in λ vs order in λ Reweighting seems feasible!!! (here λ=2, m = 0.24)

44. Resume Schwinger-Dyson equations can be used to construct DiagMC algorithms, when one does not have any convenient dual representation Schwinger-Dyson equations can be used to construct DiagMC algorithms, when one does not have any convenient dual representation Works perfectly in the absence of sign problem Works perfectly in the absence of sign problem Provides a way to think about systematic SC expansion Provides a way to think about systematic SC expansion Sign problem for models with SU(N) degrees of freedom Sign problem for models with SU(N) degrees of freedom Origin in sign-alternating SC expansion series Origin in sign-alternating SC expansion series Reformulation of SD equations might help Reformulation of SD equations might help Interesting structure of perturbative expansion from stereographic projection: Interesting structure of perturbative expansion from stereographic projection: Convergent double series in λ and log(λ) Convergent double series in λ and log(λ)

45. BACKUP SLIDES

46. Loop equations: stochastic interpretation Stack of strings (= open or closed loops)! Possible transitions (open string sector): Truncate open string Probability ~ κ Close by adding link Probability ~ N f /N κ Close by removing link Probability ~ N f /Nκ Probability ~ N f /N κ Hopping expansion for fermions (~20 orders) Strong-coupling expansion (series in β) for gauge fields (~ 5 orders) Strong-coupling expansion (series in β) for gauge fields (~ 5 orders) Disclaimer: this work is in progress, so the algorithm is far from optimal...

47. Phase diagram of the theory: a sketch High temperature (small cylinder radius) OR Large chemical potential Numerous winding strings Nonzero Polyakov loop Deconfinement phase

48. Conclusions Schwinger-Dyson equations provide a convenient framework for constructing DiagMC algorithms Schwinger-Dyson equations provide a convenient framework for constructing DiagMC algorithms 1/N expansion is quite natural (other algorithms cannot do it AUTOMATICALLY) 1/N expansion is quite natural (other algorithms cannot do it AUTOMATICALLY) Good news: it is easy to construct DiagMC algorithms for non-Abelian field theories Good news: it is easy to construct DiagMC algorithms for non-Abelian field theories Then, chemical potential does not introduce additional sign problem Then, chemical potential does not introduce additional sign problem Bad news: sign problem already for higher-order terms of SC expansions Bad news: sign problem already for higher-order terms of SC expansions Can be cured to some extent by choosing proper observables (e.g. momentum space) Can be cured to some extent by choosing proper observables (e.g. momentum space)

49. Backup

50. Lattice QCD at finite baryon density: some approaches Taylor expansion in powers of μ Taylor expansion in powers of μ Imaginary chemical potential Imaginary chemical potential SU(2) or G 2 gauge theories SU(2) or G 2 gauge theories Solution of truncated Schwinger-Dyson equations in a fixed gauge Solution of truncated Schwinger-Dyson equations in a fixed gauge Complex Langevin dynamics Complex Langevin dynamics Infinitely-strong coupling limit Infinitely-strong coupling limit Chiral Matrix models... Chiral Matrix models... “Reasonable” approximations with unknown errors, BUT No systematically improvable methods!

51. Worm algorithms for QCD? Attracted a lot of interest recently as a tool for QCD at finite density: Y. D. Mercado, H. G. Evertz, C. Gattringer, ArXiv:1102.3096 – Effective theory capturing center symmetry Y. D. Mercado, H. G. Evertz, C. Gattringer, ArXiv:1102.3096 – Effective theory capturing center symmetry P. de Forcrand, M. Fromm, ArXiv:0907.1915 – Infinitely strong coupling P. de Forcrand, M. Fromm, ArXiv:0907.1915 – Infinitely strong coupling P. de ForcrandM. Fromm P. de ForcrandM. Fromm W. Unger, P. de Forcrand, ArXiv:1107.1553 – Infinitely strong coupling, continuos time W. Unger, P. de Forcrand, ArXiv:1107.1553 – Infinitely strong coupling, continuos timeP. de ForcrandP. de Forcrand K. Miura et al., ArXiv:0907.4245 – Explicit strong-coupling series … K. Miura et al., ArXiv:0907.4245 – Explicit strong-coupling series …

52. Worm algorithms for QCD? Strong-coupling expansion for lattice gauge theory: confining strings [Wilson 1974] Strong-coupling expansion for lattice gauge theory: confining strings [Wilson 1974] Intuitively: basic d.o.f.’s in gauge theories = confining strings (also AdS/CFT etc.) Intuitively: basic d.o.f.’s in gauge theories = confining strings (also AdS/CFT etc.) Worm something like “tube” Worm something like “tube” BUT: complicated group-theoretical factors!!! Not known explicitlyStill no worm algorithm for non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010]) BUT: complicated group-theoretical factors!!! Not known explicitlyStill no worm algorithm for non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010])

53. Worm-like algorithms from Schwinger- Dyson equations Basic idea: Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for field correlators G(x 1, …, x n ) Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for field correlators G(x 1, …, x n ) Solve SD equations: interpret them as steady-state equations for some random process Solve SD equations: interpret them as steady-state equations for some random process G(x 1,..., x n ): ~ probability to obtain {x 1,..., x n } G(x 1,..., x n ): ~ probability to obtain {x 1,..., x n } (Like in Worm algorithm, but for all correlators)

54. Example: Schwinger-Dyson equations in φ 4 theory

55. Schwinger-Dyson equations for φ 4 theory: stochastic interpretation Steady-state equations for Markov processes: Steady-state equations for Markov processes: Space of states: Space of states: sequences of coordinates {x 1, …, x n } sequences of coordinates {x 1, …, x n } Possible transitions: Possible transitions:  Add pair of points {x, x} at random position at random position 1 … n + 1 1 … n + 1  Random walk for topmost coordinate coordinate  If three points meet – merge  Restart with two points {x, x} No truncation of SD equations No truncation of SD equations No explicit form of perturbative series No explicit form of perturbative series

56. Stochastic interpretation in momentum space Steady-state equations for Markov processes: Steady-state equations for Markov processes: Space of states: Space of states: sequences of momenta {p 1, …, p n } sequences of momenta {p 1, …, p n } Possible transitions: Possible transitions:  Add pair of momenta {p, -p} at positions 1, A = 2 … n + 1 at positions 1, A = 2 … n + 1  Add up three first momenta (merge) (merge) Restart with {p, -p} Restart with {p, -p} Probability for new momenta: Probability for new momenta:

57. Diagrammatic interpretation History of such a random process: unique Feynman diagram BUT: no need to remember intermediate states Measurements of connected, 1PI, 2PI correlators are possible!!! In practice: label connected legs Kinematical factor for each diagram: q i are independent momenta, Q j – depend on q i Monte-Carlo integration over independent momenta Monte-Carlo integration over independent momenta

58. Normalizing the transition probabilities Problem: probability of “Add momenta” grows as (n+1), rescaling G(p 1, …, p n ) – does not help. Problem: probability of “Add momenta” grows as (n+1), rescaling G(p 1, …, p n ) – does not help. Manifestation of series divergence!!! Manifestation of series divergence!!! Solution: explicitly count diagram order m. Transition probabilities depend on m Solution: explicitly count diagram order m. Transition probabilities depend on m Extended state space: {p 1, …, p n } and m – diagram order Extended state space: {p 1, …, p n } and m – diagram order Field correlators: Field correlators: w m (p 1, …, p n ) – probability to encounter m-th order diagram with momenta {p 1, …, p n } on external legs w m (p 1, …, p n ) – probability to encounter m-th order diagram with momenta {p 1, …, p n } on external legs

59. Normalizing the transition probabilities Finite transition probabilities: Finite transition probabilities: Factorial divergence of series is absorbed into the growth of C n,m !!! Factorial divergence of series is absorbed into the growth of C n,m !!! Probabilities (for optimal x, y): Probabilities (for optimal x, y):  Add momenta:  Sum up momenta + increase the order: increase the order: Otherwise restart Otherwise restart

60. Resummation Integral representation of C n,m = Γ(n/2 + m + 1/2) x -(n-2) y -m : Integral representation of C n,m = Γ(n/2 + m + 1/2) x -(n-2) y -m : Pade-Borel resummation. Borel image of correlators!!! Pade-Borel resummation. Borel image of correlators!!! Poles of Borel image: exponentials in w n,m Poles of Borel image: exponentials in w n,m Pade approximants are unstable Pade approximants are unstable Poles can be found by fitting Poles can be found by fitting Special fitting procedure using SVD of Hankel matrices Special fitting procedure using SVD of Hankel matrices No need for resummation at large N!!!

61. Resummation: fits by multiple exponents

62. Resummation: positions of poles Two-point function Connected truncated four-point function 2-3 poles can be extracted with reasonable accuracy

63. Test: triviality of φ 4 theory in D ≥ 4 Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv:1104.3459]

64. Phase diagram of the theory: a sketch High temperature (small cylinder radius) OR Large chemical potential Numerous winding strings Nonzero Polyakov loop Deconfinement phase

65. Summary and outlook Diagrammatic Monte-Carlo and Worm algorithm: useful strategies complimentary to standard Monte-Carlo Stochastic interpretation of Schwinger-Dyson equations: a novel way to stochastically sum up perturbative series Advantages: Implicit construction of perturbation theory Implicit construction of perturbation theory No truncation of SD eq-s No truncation of SD eq-s Large-N limit is very easy Large-N limit is very easy Naturally treats divergent series No sign problem at μ≠0 No sign problem at μ≠0Disadvantages: Limited to the “very strong- coupling” expansion (so far?) Limited to the “very strong- coupling” expansion (so far?) Requires large statistics in IR region Requires large statistics in IR region QCD in terms of strings without explicit “stringy” action!!!

66. Summary and outlook Possible extensions: Weak-coupling theory: Wilson loops in momentum space? Weak-coupling theory: Wilson loops in momentum space?  Relation to meson scattering amplitudes  Possible reduction of the sign problem Introduction of condensates? Introduction of condensates?  Long perturbative series ~ Short perturbative series + Condensates [Vainshtein, Zakharov]  Combination with Renormalization-Group techniques?

67. Thank you for your attention!!! References: ArXiv:1104.3459 (φ 4 theory) ArXiv:1104.3459 (φ 4 theory)1104.3459 ArXiv:1009.4033, 1011.2664 (large-N theories) ArXiv:1009.4033, 1011.2664 (large-N theories)1009.40331011.26641009.40331011.2664 Some sample codes are available at: Some sample codes are available at: http://www.lattice.itep.ru/~pbaivid/codes.html This work was supported by the S. Kowalewskaja award from the Alexander von Humboldt Foundation

68. Back-up slides

69. 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

70. Can DiagMC take advantage of the large-N limit? Only exponentially (vs. factorially) growing number of diagrams Only exponentially (vs. factorially) growing number of diagrams At least part of non-summabilities are absent At least part of non-summabilities are absent Large-N QFTs are believed to contain most important physics of real QFT Large-N QFTs are believed to contain most important physics of real QFT Theoretical aspects of large-N expansion Surface counting problems: calculation of coefficients of 1/N expansions (topological recursion) Surface counting problems: calculation of coefficients of 1/N expansions (topological recursion) [Formula by B. Eynard] Instantons via divergences of 1/N expansion Instantons via divergences of 1/N expansion Resurgent structure of 1/N expansions [Marino, Schiappa, Vaz] Resurgent structure of 1/N expansions [Marino, Schiappa, Vaz] Motivation: Large-N QFT side

71. Worm Algorithm [Prokof’ev, Svistunov] Monte-Carlo sampling of closed vacuum diagrams: Monte-Carlo sampling of closed vacuum diagrams: nonlocal updates, closure constraint Worm Algorithm: sample closed diagrams + open diagram Worm Algorithm: sample closed diagrams + open diagram Local updates: open graphs closed graphs Local updates: open graphs closed graphs Direct sampling of field correlators (dedicated simulations) Direct sampling of field correlators (dedicated simulations) x, y – head and tail of the worm Correlator = probability distribution of head and tail Applications: systems with “simple” and convergent perturbative expansions (Ising, Hubbard, 2d fermions …) Applications: systems with “simple” and convergent perturbative expansions (Ising, Hubbard, 2d fermions …) Very fast and efficient algorithm!!! Very fast and efficient algorithm!!!

72. Stochastic solution of SD equations Sampling the correlators of the theory – momenta or coordinates Sampling the correlators of the theory – momenta or coordinates Histograms of X are correlators (later about positivity) Histograms of X are correlators (later about positivity) Extension of the WORM algorithm: Diagrams with any number of open legs!!! Extension of the WORM algorithm: Diagrams with any number of open legs!!!