Pavel Buividovich (Regensburg University)

Generalized Series Expansions in Asymptotically Free Large-N Lattice Field Theories
Pavel Buividovich (Regensburg University) Resurgence in Gauge and String Theories, Lissabon, July 2016

Summary: - Sign Finite Baryon Density and Diagrammatic Monte-Carlo for lattice QCD - Strong-coupling vs weak-coupling QCD - Lattice perturbation theory with Cayley map and IR problems - Trans-series from lattice perturbation theory - The fate of Renormalons? - Monte-Carlo weak coupling Tests of continuity for twisted compactified principal chiral model [work with S. Valgushev]

Sign problem in QCD Lattice QCD @ finite baryon density:
Complex path integral No positive weight for Monte-Carlo

VS DiagMC Complex Langevin Lefshetz thimbles Worm algorithm
Reweighting DiagMC Complex Langevin Worm algorithm Lefshetz thimbles Density of states Canonical formalism Subsets VS

Diagrammatic Monte-Carlo in QFT
Sum over fields Euclidean action: Sum over interacting paths Perturbative expansions

Diagrammatic Monte-Carlo for QCD
So far lattice strong-coupling expansion: (leading order or few lowest orders) [de Forcrand,Philipsen,Unger, Gattringer,…] Worldlines of quarks/mesons/baryons “Worldsheets” of confining strings Very good approximation! Physical degrees of freedom!

Lattice strong-coupling expansion
Confinement Dynamical mass gap generation BUT Continuum physics is at weak-coupling! … Rigorously relating strong- and weak-coupling might bring you $ …

Diagrammatic Monte-Carlo at weak coupling?
… and make it first-principle and automatic Lattice perturbation theory for QCD: Small fluctuations of SU(N) fields around vacuum (1) Map SU(N) to Hermitian matrices Popular choice Maps only a part of all matrices Infinitely many vacua Exp.small terms due to cut-off

Popular choice Maps a subset of all Hermitian matrices to the whole U(N) Infinitely many degenerate vacua Double-trace terms in the Jacobian

Cayley map/Stereographic projection
(Conformal mapping from circle to line) Maps the whole space of Hermitian matrices to the whole U(N) Perturbative vacuum unique Jacobian is very simple

Bare mass from the Jacobian? (Take principal chiral model as example…)

Bare mass from the Jacobian? (principal chiral model as example…)
Massive planar field theory, Suitable for Diagrammatic Monte-Carlo Bare mass ~ bare coupling, Infinitely many interacting vertices

Bare mass from the Jacobian?
Small mass term λ/4 due to Jacobian! “Conformal anomaly” stemming from integration measure [a-la Fujikawa 79] Perturbation theory with coupling in vertices AND propagators!!! Let’s try, at least formally, to expand in vertices…

Minimal working example: O(N) sigma model @ large N
Gauge theory/PCM too hard to start with (one needs DiagMC to sum over all planar diagrams) Something simpler? Exact answer Non-perturbative mass gap in 2D

O(N) sigma model @ large N
Analogue of Cayley map is Maps whole sphere to whole hyperplane Stereographic projection Again, bare mass term from the Jacobian… [PB, ]

Full action in new coordinates We blindly do perturbation theory … Only cactus diagrams @ large N

From our perturbative expansion we get (automatic analytic recursion) We get trans-series, only without „saddle point“ terms!!! Can capture non-perturbative effects:

Good convergence in practice (But no proof of convergence!!!)

Principal Chiral Model
Next step towards gauge theory… Massive planar field theory, Bare mass ~ bare coupling, Infinitely many interacting vertices

Principal Chiral Model@large N
Explicit recursive calculations for D=1 and L ~ O(10) [Lattice size] [Work with Ali Davody] Exact answers for L = 2, 3, 4, ∞ [Vicari, Rossi, hep-lat/ ] All momentum sums are discrete and finite Rational approximations for finite L Let‘s check the restoration of SU(N) x SU(N) symmetry – we break it by choosing the vacuum gx =1

Principal Chiral Model (N=∞,D=1)
SU(N)xSU(N) restored at large orders

Convergence of mean link… D=1,L=2 D=1,L=3 Quite good and fast convergence, BUT …

Small systematic orders…

Asymptotically free theory Now we need DiagMC, but before… Again series of the form No negative powers of λ can appear For a planar graph with V vertices, L lines and E edges , V – E + F = 2

For planar diagrams V – E + F = 2 Only logs of coupling appear in 2D

Trans-series VS renormalons BUT: trans-series with powers and logs
In large-N limit: # of diagrams grows exponentially All contributions are finite Suitable for DiagMC No IR singularities No IR renormalons Simple poles and cuts at most BUT: trans-series with powers and logs No „instanton terms“ Jacobian removes classical solutions?

Diagrammatic Monte-Carlo
Start with Schwinger-Dyson equations (here for simplicity φ4) (independent of mass) Recursive structure for diagrams: V vertices, L legs -> V-v vertices, L+l legs

Diagrammatic Monte-Carlo
Schwinger-Dyson equations are (infinitely many) linear equations of the form which involve all (disconnected) correlators … And no large-N factorization assumed yet! Solution can be written as geometric series The terms in these series are MC sampled

λ=4.0, LS=16 (10 min on laptop) Restoration of SU(N) x SU(N) symmetry

U(N) field correlator in momentum space

Alternative forms of SD equations
SD equations in terms of Significantly simpler (only 4-vertices) Advantageous for fermions In momentum space: Bare mass ~ coupling Seems to be very general!

Summary: Transseries from DiagMC
“Effective mass” in the bare quantum action: general feature of compact field theories In 2D, leads to trans-series with powers and logs Series expansion suitable for DiagMC No factorial divergences! Disclaimer: Bare Lattice Perturbation Theory Running coupling etc. hidden in the structure of the series in a complex way

Lattice tests of continuity
Twisted compactifications are important tools in exploring the resurgent structure of QFT O(N), CPN-1, SU(N) sigma-models Continuity is a conjecture (AFAIK) Analyticity at intermediate T not rigorously proven

Lattice tests of continuity: SU(N) PCM [work with S. Valgushev]
Monte-Carlo simulations, N=3 and N=6 Cabibbo-Marinari heat-bath algorithm Temporal compactification vs Twisted compactification Correlation length ~ 10-12 t’Hooft coupling ~ 3 Thermodynamics not well studied, Order of transition unknown (to me?)

Order of transition? Seems to be a crossover
in large-N O(N) sigma model, but of course in PCM things may be very different Mass gap Temperature UV cutoff varies by 2 orders of magnitude T/M(T=0) ~ 0.2

Mean link vs. “Temperature”

Space-time links vs. “Temperature”

Link susceptibility vs. “Temperature”

Thank you for your attention!!!

Motivation: QCD side Systematic DiagMC in the non-Abelian case?
DiagMC in strong-coupling lattice QCD at finite density [de Forcrand,Vairinhos,Gattringer,Chandrasekharan,…] Few orders of SC expansion NOT SYSTEMATIC, BUT EFFICIENT SYSTEMATIC WAY: DUAL REPRESENTATIONS/SC EXPANSIONS Abelian Lattice Gauge Theories (also with fermions) Scalar field theories O(N) / CPN 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) Weak-coupling expansions are also cumbersome and difficult to re-sum Systematic DiagMC in the non-Abelian case? Avoid manual construction of duality transformations? Avoid Borel resummations?

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

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

Solution with Metropolis algorithm:
Stochastic solution of linear equations Assume: A(X|Y), b(X) are positive, |eigenvalues| < 1 Solution with Metropolis algorithm: Sample sequences {Xn, …, X0} with the weight Two basic transitions: Add new index Xn+1 , Remove index Restart In fact, exactly the same strategy was used in Prokof’ev and Svistunov’s first paper on polaron problem Diag MC, PRL 81 (1998) 2514

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 Probability distribution of N(X) is crucial to asses convergence Finally: make histogram of the last element Xn in the sequence Solution φ(X) , normalization factor

Illustration: ϕ4 matrix model (Running a bit ahead)
Large autocorrelation time and large fluctuations near the phase transition

Practical implementation for QFT
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) Diagrams with any number of legs are sampled (extension of the worm algorithm) Keep in computer memory: current diagram 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 know diagrammatic rules – only the SD equations, quite easy to derive

Sign problem and reweighting
Now lift the assumptions A(X | Y) > 0 , b(X)>0 Use the absolute value of weight for the Metropolis sampling Sign of each configuration: Define Effectively, we solve the system The expansion has typically smaller radius of convergence Reweighting fails if the system 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

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 IR divergences ~ n! (IR renormalon) Things become simpler in the large-N limit UV renormalons are absent in the planar limit Number of planar diagrams grows exponentially IR renormalons are interesting, but can be removed by IR regularization (finite volume, twisted BC etc.) SD equations are significantly simpler!!!

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

SD equations@large N: stack structure
The larger space of “indices” labeling has a natural stack structure (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 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 Note that now Xi are strongly correlated (we update only topmost) Makes sense to use only topmost element for statistical sampling

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])

Resummation/Rescaling
Growth of field correlators with n/ order: Exponential in the large-N limit Factorial at finite N How to interprete as a probability distribution? Exponential growth? Introduce “renormalization constants” : is now finite , 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

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

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

Genus expansion: ϕ4 matrix model
[From Marino,Schiappa,Weiss ]

Genus expansion: ϕ4 matrix model
Heavy-tailed distributions near criticality, P(x) ~ x-2

Think of each as an n-segment boundary of a random surface

Probability of “split” action grows as Obviously, cannot be removed by rescaling of the form N cn Introduce rescaling factors which depend on number of vertices OR genus Expansion in λ, N fixed:

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

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

Test: triviality of φ4 theory in D ≥ 4
Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv: ]

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

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

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

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

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

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

Loop equations illustrated
Quadratic term

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

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

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

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 leading order

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

Making sign problem milder: momentum space <1/N tr g+x gy> vs λ
Deep in the SC regime, only few orders relevant… Sign problem becomes important towards WC

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

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+ Edge of convergence for G- Shall we use positive powers of Lagrange multiplier for SD equations?

What do we get if we perform the standard perturbative expansion?
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 No double-trace terms Bare mass is again the bare coupling… What do we get if we perform the standard perturbative expansion?

Simpler case: O(N) sigma-model
Stereographic projection: maps whole RN to whole SN Let’s blindly denote m02 = λ/2 and do expansion We get series of the form (something like transseries)

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

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

Resume 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 Provides a way to think about systematic SC expansion Sign problem for models with SU(N) degrees of freedom Origin in sign-alternating SC expansion series Reformulation of SD equations might help Interesting structure of perturbative expansion from stereographic projection: Convergent double series in λ and log(λ)

BACKUP SLIDES

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 ~ Nf /N κ Close by removing link Probability ~ Nf /N κ Hopping expansion for fermions (~20 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...

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

Conclusions Schwinger-Dyson equations provide a convenient framework for constructing DiagMC algorithms 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 Then, chemical potential does not introduce additional sign problem 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)

Backup

Lattice QCD at finite baryon density: some approaches
Taylor expansion in powers of μ Imaginary chemical potential SU(2) or G2 gauge theories Solution of truncated Schwinger-Dyson equations in a fixed gauge Complex Langevin dynamics Infinitely-strong coupling limit Chiral Matrix models ... “Reasonable” approximations with unknown errors, BUT No systematically improvable methods!

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: – Effective theory capturing center symmetry P. de Forcrand, M. Fromm, ArXiv: – Infinitely strong coupling W. Unger, P. de Forcrand, ArXiv: – Infinitely strong coupling, continuos time K. Miura et al., ArXiv: – Explicit strong-coupling series …

Worm algorithms for QCD?
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.) Worm something like “tube” BUT: complicated group-theoretical factors!!! Not known explicitly Still no worm algorithm for non-Abelian LGT (Abelian version: [Korzec, Wolff’ 2010])

Worm-like algorithms from Schwinger-Dyson equations
Basic idea: Schwinger-Dyson (SD) equations: infinite hierarchy of linear equations for field correlators G(x1, …, xn) Solve SD equations: interpret them as steady-state equations for some random process G(x1, ..., xn): ~ probability to obtain {x1, ..., xn} (Like in Worm algorithm, but for all correlators)

Example: Schwinger-Dyson equations in φ4 theory

Schwinger-Dyson equations for φ4 theory: stochastic interpretation
Steady-state equations for Markov processes: Space of states: sequences of coordinates {x1, …, xn} Possible transitions: Add pair of points {x, x} at random position 1 … n + 1 Random walk for topmost coordinate If three points meet – merge Restart with two points {x, x} No truncation of SD equations No explicit form of perturbative series

Stochastic interpretation in momentum space
Steady-state equations for Markov processes: Space of states: sequences of momenta {p1, …, pn} Possible transitions: Add pair of momenta {p, -p} at positions 1, A = 2 … n + 1 Add up three first momenta (merge) Restart with {p, -p} Probability for new momenta:

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: qi are independent momenta, Qj – depend on qi Monte-Carlo integration over independent momenta

Normalizing the transition probabilities
Problem: probability of “Add momenta” grows as (n+1), rescaling G(p1, … , pn) – does not help. Manifestation of series divergence!!! Solution: explicitly count diagram order m. Transition probabilities depend on m Extended state space: {p1, … , pn} and m – diagram order Field correlators: wm(p1, …, pn) – probability to encounter m-th order diagram with momenta {p1, …, pn} on external legs

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

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

Resummation: fits by multiple exponents

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

Test: triviality of φ4 theory in D ≥ 4
Renormalized mass: Renormalized coupling: CPU time: several hrs/point (2GHz core) [Buividovich, ArXiv: ]

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

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 No truncation of SD eq-s Large-N limit is very easy Naturally treats divergent series No sign problem at μ≠0 Disadvantages: Limited to the “very strong-coupling” expansion (so far?) Requires large statistics in IR region QCD in terms of strings without explicit “stringy” action!!!

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

Thank you for your attention!!!
References: ArXiv: (φ4 theory) ArXiv: , (large-N theories) Some sample codes are available at: This work was supported by the S. Kowalewskaja award from the Alexander von Humboldt Foundation

Back-up slides

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

Motivation: Large-N QFT side
Can DiagMC take advantage of the large-N limit? Only exponentially (vs. factorially) growing number of diagrams At least part of non-summabilities are absent 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) [Formula by B. Eynard] Instantons via divergences of 1/N expansion Resurgent structure of 1/N expansions [Marino, Schiappa, Vaz]

Worm Algorithm [Prokof’ev, Svistunov]
Monte-Carlo sampling of closed vacuum diagrams: nonlocal updates, closure constraint Worm Algorithm: sample closed diagrams + open diagram Local updates: open graphs closed graphs 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 …) Very fast and efficient algorithm!!!

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

Pavel Buividovich (Regensburg University)

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