Large- -N Quantum Field Theories N Quantum Field Theories Large and Nonlinear Random Processes and Nonlinear Random Processes 1009.4033] ] : 1009.4033 [ArXiv ArXiv: [ Pavel Buividovich Buividovich Pavel (ITEP, Moscow and JINR, Dubna Dubna) ) (ITEP, Moscow and JINR, Bogoliubov readings, Bogoliubov readings, Dubna Dubna, 23.09.2010 , 23.09.2010
Motivation Motivation Problems for for modern modern Lattice Lattice QCD QCD simulations simulations( (based based Problems on standard standard Monte Monte- -Carlo Carlo): ): on • 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 of 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 for for alternative alternative numerical numerical algorithms algorithms Look
Motivation: Diagrammatic MC, Diagrammatic MC, Motivation: 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 perturbative perturbative expansion expansion terms in the
Motivation: Diagrammatic MC, Diagrammatic MC, Motivation: Worm Algorithm, ... Worm Algorithm, ... • Worm Algorithm • Worm Algorithm [ [Prokof Prokof’ ’ev ev, , Svistunov Svistunov]: ]: Directly sample Green functions, Dedicated simulations!!! Green functions, Dedicated simulations!!! Directly sample Example: Example: Ising model model Ising x,y – – head and tail head and tail x,y of the worm of 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 continuous continuous And other quantum field theories with symmetry groups ... groups ... symmetry Typical problems: Typical problems: • Nonconvergence Nonconvergence of of 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 for for strong coupling strong coupling finite convergence radius • Algorithm complexity Algorithm complexity grows with grows with N N • • Weak Weak- -coupling expansion coupling expansion (=lattice perturbation (=lattice perturbation theory): complicated, volume complicated, volume- -dependent... dependent... theory):
Large- -N Quantum Field Theories N Quantum Field Theories Large Situation might be better at large N ... better at large N ... Situation might be • 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 “ “genetic genetic” ” random processes random processes general � 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: of divergent series: • random processes with memory random processes with memory � O(N) sigma O(N) sigma- -model model � � outlook: outlook: non non- -Abelian Abelian large large- -N theories N theories �
Schwinger- -Dyson equations at large N Dyson equations at large N Schwinger 4 theory, • Example: Example: φ φ 4 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- -Dyson equations at large N Dyson equations at large N Schwinger • Closed equations for w(k Closed equations for w(k 1 , ..., k k n ): • 1 , ..., n ): nd order equations • Always Always 2 2 nd order 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(k 1 , ..., k k n ) – – probability probability • 1 , ..., n )
Schwinger- -Dyson equations at large N Dyson equations at large N Schwinger
“Genetic Genetic’’ ’’ Random Process Random Process “ Also: Recursive Markov Chain Recursive Markov Chain [ [Etessami, Etessami, Also: Yannakakis, 2005] ] Yannakakis, 2005 • Let Let X X be some be some discrete set discrete set • • Consider Consider stack stack of the of the elements of X elements of X • • At each At each process step process step: : • Otherwise restart!!! Otherwise restart!!! Create: with probability with probability P P c (x) create new x ) create new x and and c (x � Create: � push it to stack push it to stack Evolve: with probability with probability P P e (x|y) replace y on the ) replace y on the e (x|y � Evolve: � top of the stack with x top of the stack with x Merge: with probability with probability P P m (x|y 1 ,y 2 ) pop two m (x|y 1 ,y 2 ) pop two � Merge: � elements y 1 , y 2 from the stack and push x push x into the into the elements y 1 , y 2 from the stack and stack stack
“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 x 1 ... x x n in the stack: • 1 ... n in the stack: W(x 1 , ..., x x n ) W(x 1 , ..., 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 1 , ..., x x n ) = w 0 (x 1 ) w(x 2 ) ... w(x w(x n ) W(x 1 , ..., n ) = w 0 (x 1 ) w(x 2 ) ... n ) • Steady Steady- -state equation state equation (sum over y, z): (sum over y, z): • w(x) = ) = P P c (x) + ) + P P e (x|y) ) w(y w(y) + ) + P P m (x|y,z) ) w(y w(y) ) w(z w(z) ) w(x c (x e (x|y m (x|y,z
“Genetic Genetic’’ ’’ Random Process and Random Process and “ Schwinger- -Dyson equations Dyson equations Schwinger • Let Let X X = set of = set of all sequences {k all sequences {k 1 , ..., k k n }, k , k – – momenta momenta • 1 , ..., n } • Steady state equation Steady state equation for for “ “Genetic Genetic” ” Random Process Random Process = = • Schwinger- -Dyson equations Dyson equations, IF: , IF: Schwinger • Create: Create: push a pair {k, push a pair {k, - -k}, P ~ G k}, P ~ G 0 (k) • 0 (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) k}, P ~ G0(k) add a pair {k, � � sum up three momenta momenta sum up three � � on top of the stack, P ~ λ λ G G 0 (k) on top of the stack, P ~ 0 (k)
Examples: drawing diagrams drawing diagrams Examples: “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: tr tr φ φ 4 Matrix Model Examples: 4 Matrix Model Exact answer known answer known [ [Brezin Brezin, , Itzykson Itzykson, , Zuber Zuber] ] Exact
4 Matrix Model Examples: tr tr φ φ 4 Matrix Model Examples: • Autocorrelation time Autocorrelation time vs. coupling: vs. coupling: No critical slowing– –down down No critical slowing • Peculiar: Peculiar: only g < only g < ¾ ¾ g g c can be reached!!! • c can be reached!!! Not a a dynamical dynamical, but an , but an algorithmic limitation... algorithmic limitation... Not
4 Matrix Model Examples: tr tr φ φ 4 Matrix Model Examples: Sign problem vs. coupling: vs. coupling: No severe sign No severe sign Sign problem problem!!! problem!!!
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