Mixing Time Analysis of the Glauber Dynamics for the - PowerPoint PPT Presentation

Mixing Time Analysis of the Glauber Dynamics for the Curie-Weiss-Potts Model Precise Asymptotics. Cutoff Phenomenon. Essential Mixing. Evolution. P . Cuff J. Ding E. Lubetzky Y. Peres A. Sly O. Louidor Microsoft Research Cornell Summer School in Probability 2009 Cornell Summer School in Probability 2009 1 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Outline Problem Definition 1 The Potts Model Glauber Dynamics Mixing Time The Problem Previous Results 2 New Results 3 Proofs 4 Cornell Summer School in Probability 2009 2 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Setup and Terminology Let G = ( V , E ) be a finite graph, q ∈ N (number of colors) and ˆ β ∈ R - inverse temperature. A configuration σ is an element of Ω = { 1 , . . . , q } V . On Ω define the (Gibbs) measure: 1 � � ˆ µ ( σ ) = µ ˆ β, G ( σ ) = exp β H G ( σ ) Z ˆ β, G where: H G ( σ ) = � ( u , v ) ∈ E 1 I σ ( u )= σ ( v ) - the (associated) Hamiltonian. Z ˆ β, G makes µ a probability measure - the Partition Function. Names: q = 2: The Ising Model on G . q > 2: The Potts Model on G . G = C n (the complete graph with n vertices): Curie-Weiss Potts/Ising or Mean-field Potts/Ising. Cornell Summer School in Probability 2009 4 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

The Question Question: Given a particular sequence (ˆ β n , G n ) n � 1 , describe µ ˆ β n , G n for large n or in an appropriate limit. For Curie Weiss Potts, with ˆ β n = β n for some β ∈ R , much is known (the easiest case). For instance . . . . Cornell Summer School in Probability 2009 5 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Fractions Vector For σ ∈ Ω define S ( σ ) ∈ S q = { x ∈ R q + : � x � 1 = 1 } by: S k ( σ ) = 1 � 1 I { k } ( σ ( v )) ; k = 1 , . . . , q | V | v ∈ V - the fractions vector. Define π ˆ β, G as the distribution of S ( σ ) when σ is sampled using µ ˆ β, G . Set π ∞ = lim n →∞ π ˆ β n , G n . Cornell Summer School in Probability 2009 6 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Phase Transition in Curie Weiss Potts Then there exists β c = β c ( q ) such that: β < β c (high temperature): π β n , C n ⇒ δ 1 as n → ∞ . q where 1 q denotes the vector ( 1 / q , 1 / q ... 1 / q ) ∈ S q . β > β c (low temperature): n , C n ⇒ � q 1 π β q δ T k ˆ as n → ∞ . k = 1 s ( β ) where: � � s 1 ( β ) s 1 ( β ) s 1 ( β ) , 1 − ˆ , . . . , 1 − ˆ ˆ ˆ ∈ S q s ( β ) = q − 1 q − 1 T k interchanges the first and k -th component. β = β c (critical temperature): q + ( 1 − p ( β )) � q 1 π β n , C n ⇒ p ( β ) δ 1 q δ T k ˆ as n → ∞ . s ( β ) k = 1 where p ( β ) ∈ ( 0 , 1 ) . Cornell Summer School in Probability 2009 7 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Phase Transition - Remarks β c ( q ) , ˆ s ( β ) and p ( β ) are explicitly known. n , C n ) n � 1 satisfies a LDP on S q with rate function: In fact ( π β I β ( s ) = R ( s ) − β 2 � s � − const where R ( s ) is the rate function for the fractions vector when β = 0. Thus, describing π ∞ = π ∞ ( β ) is solving the minimization problem of R ( s ) − β 2 � s � 2 in S q . s ( β c ( 2 )) = 1 If q = 2, ˆ q and the mapping β �→ π ∞ ( β ) is continuous (under the weak-topology for measures). Thus this is a second order phase transition. s ( β c ( q )) � = 1 If q > 2, ˆ q , the mapping β �→ π ∞ ( β ) is not continuous and the phase transition is of first order. This will play a part in the rate of mixing. Show Graphs Cornell Summer School in Probability 2009 8 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Markov Chain Monte Carlo (MCMC) A way to approximately sample from a probability measure µ on a finite space Ω . Idea: Markov Chain Monte Carlo (MCMC). Construct a Markov chain with state space Ω and µ as its stationary-ergodic distribution. Then, start from any configuration and let the chain evolve randomly for long enough time, until the distribution of the current state is close to µ . Useful when exact sampling is computationally expensive (e.g. One has to exhaust all of Ω ), but computing the transition probabilities is easy. What is long enough time? One has to study the rate of convergence to stationarity - Mixing Time (later). If Ω = { 1 , . . . , q } V , many dynamics are possible (Glauber, Metropolis, Swendsen Wang, . . . ). Differ in how fast they mix. Cornell Summer School in Probability 2009 9 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Glauber Dynamics Single site update dynamics for a measure µ on Ω = { 1 , . . . , q } V : Start from any configuration σ 0 . Transition: Choose a vertex u ∈ V at random. Update: � σ t ( v ) if v � = u σ t + 1 ( v ) = if v = u w.p. µ ( σ ( u ) = k | σ ( v ) = σ t ( v ) ; v � = u ) k Repeat. Conditional probabilities are straightforward if µ is a Gibbs measure (part of the definition). ( σ t ) t is a finite-states irreducible and aperiodic chain (at least if µ has the finite energy property), hence converges to its unique stationary distribution µ . But how fast? Cornell Summer School in Probability 2009 10 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Mixing Time Let ( X t ) t ∈ N be a Markov chain with state space S , transition kernel P and stationary distribution π . Set d ( t ) = sup x 0 ∈S � P x 0 ( X t ∈ • ) − π � TV where X 0 = x 0 under P x 0 . (Reminder: � µ − ν � TV = sup A | µ ( A ) − ν ( A ) | ). The ε -Mixing Time of ( X t ) t ∈ N is: t M ( ε ) = inf { t : d ( t ) < ε } If no ε is specified, it is customary to use ε = 1 / 4. Cornell Summer School in Probability 2009 11 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Mixing Time Asymptotics and the Cut-off Phenomenon Let ( P n ) n be a particular sequence of Markov chain kernels and denote by t M n ( ε ) their ε -mixing times. We would like to know how t M n ( ε ) grows with n : If t M n ( ε ) grows polynomially, we say that the mixing is rapid. If t M n ( ε ) grows exponentially, we say that the mixing is slow. Also, if for some (and hence any) ε 0 and all ε : t M n ( ε ) − t M n ( 1 − ε ) � w n ( ε ) = o ( t M n ( ε 0 )) n → ∞ as we say that the sequence of dynamics exhibits a cut-off. The distance to stationarity sharply changes from 1 to 0 (relatively to the mixing time). If w n ( ε ) = θ ε ( W ( n )) we say that the cut-off window has order W ( n ) . Cornell Summer School in Probability 2009 12 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

The Problem Analyze the Mixing Time of the Glauber Dynamics for the Curie-Weiss Potts Model. i.e., Fix β and q , consider a sequence of Glauber dynamics for the Potts distribution on the n -complete graph: µ β n , C n and analyze the mixing time t M n ( ε ) as a function of n . Cornell Summer School in Probability 2009 13 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Previous Results Complete analysis for the Curie-Weiss Ising case ( q = 2): β < β c ( 2 ) = 2 (high temperature): � − 1 n log n t M n ( ε ) ∼ 1 1 − β � 2 2 w n ( ε ) = θ ε ( n ) [Aizenman, Holley ’87], [Bubley, Dyer ’97], [Levin, Luczak, Peres ’07]. β > β c (low temperature): t M n ( ε ) is exponential in n . [Griffiths, Weng and Langer ’66] β = β c (critical temperature): � n 3 / 2 � t M n ( ε ) = θ ε . No cut-off. [Levin, Luczak, Peres ’07], [Ding, Lubetzky, Peres ’08] Cornell Summer School in Probability 2009 15 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

Previous Results - Evolution Still q = 2 case. Now let β = β n change with n . β n = β c − δ n : � n � � t M δ 2 n 1 n ( ε ) ∼ n � � � δ n = ω ⇒ δ log , w n ( ε ) = θ ε . √ n δ � � � n 3 / 2 � 1 t M δ n = O ⇒ n ( ε ) = θ ε , no cut-off. √ n β n = β c + δ n : � n �� 3 � � 1 t M � δ 2 n �� δ n = ω but δ n = o ( 1 ) ⇒ n ( ε ) = θ ε δ exp 4 + o ( 1 ) . √ n t M δ n = Ω( 1 ) ⇒ n ( ε ) is exponential in n . � � � n 3 / 2 � 1 t M δ n = O ⇒ n ( ε ) = θ ε . √ n No cut-off. [Ding, Lubetzky, Peres ’08] Cornell Summer School in Probability 2009 16 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27

New Results - The Case q > 2 There exists a new critical beta β M ( q ) < β c ( q ) < q such that: β < β M : � − 1 � 1 − β t M n ( ε ) ∼ 1 n log n 2 q w n ( ε ) = θ ε ( n ) β > β M : t M n ( ε ) is exponential in n . β = β M : t M � n 4 / 3 � n ( ε ) = θ ε No cut-off. β M ( q ) is explicitly known. Cornell Summer School in Probability 2009 18 / P . Cuff, J. Ding, E. Lubetzky, Y. Peres, A. Sly, O. Louidor ( Microsoft Research ) CW-Potts Mixing 27