Decay of Correlation in Spin Systems

Transcription

1 (An algorithmic perspective to the) Decay of Correlation in Spin Systems Yitong Yin Nanjing University

2 Decay of Correlation hardcore model: random independent set I v Pr[v 2 I ] (d+)-regular tree `! σ: fixing leaves to be occupied/unoccupied by I Decay of correlation: Pr[v I σ] does not depend on σ when l iff counting total weights λ I of all I.S. in graphs with max-degree d+ λ < λ c FPTAS [Weitz 06] λ > λ c no FPRAS unless NP=RP [Sly0] [Galanis Štefankovič Vigoda 2] [Sly Sun 2]

3 Spin System undirected graph G = (V, E) fixed integer q 2 configuration weight: 2 [q] V w( )= Y {u,v}2e A( u, v) Y v2v b( v ) A :[q] [q]! R 0 symmetric q q matrix (symmetric binary constraint) b :[q]! R 0 q-vector (unary constraint) partition function: Z G = X 2[q] V w( ) Gibbs distribution: µ G ( )= w( ) Z G

4 undirected graph G = (V, E) fixed integer q 2 configuration 2 [q] V weight: Y Y w( )= A( u, v) b( v ) {u,v}2e v2v 2-spin model: q =2, A = apple A00 A 0 A 0 A = apple 2 {0, } V edge activities b = apple b0 b = apple external field hardcore model: β=0, γ= Ising model: β = γ Z G = X I: independent sets in G I multi-spin model: general q 2 Potts model: q-coloring: β=0 A = b =

5 Models spin systems: Ising model, Potts model, q-coloring hardcore model monomer-dimer (NOT a spin system) generalization Z G = hypergraph matchings Holant problem defined by the (weighted-)eq, the At-Most-One constraint, and any binary constraints The recursion of marginal probabilities is the same X M: matchingsing M as a recursion on the tree of self-avoiding walks. correlation decay on trees (hopefully) tractability of approximate counting

6 Gibbs Measure undirected graph G = (V, E) configuration 2 [q] V Gibbs distribution: µ( )=µ G ( )= w( ) Z G by the chain rule: denoted V = {v,v 2,...,v n } marginal probability: µ v (x) = Pr X µ G [X v = x X S = ] where v V, x [q], boundary condition σ [q] S on S V approximately computing µ v (x) within µ v (x) ± in time Poly(n) n FPTAS for ZG

7 Spatial Mixing (Decay of Correlation) µ v : marginal distribution at vertex v conditioning on σ weak spatial mixing (WSM) at rate δ( ): 8, 2 : kµ v µ vk TV apple (t) R v t G R on infinite graphs: WSM µ v boundary conditions uniqueness of infinitevolume Gibbs measure apple c = d d (d ) (d+) WSM of hardcore model on infinite (d+)-regular tree

8 Spatial Mixing (Decay of Correlation) µ v : marginal distribution at vertex v conditioning on σ weak spatial mixing (WSM) at rate δ( ): 8, 2 : kµ v µ vk TV apple (t) strong spatial mixing (SSM) at rate δ( ): 8, [q] S : kµ [ v µ [ v k TV apple (t) G SSM R v t S R marginal prob. µ v(x) is well approximated by the local information

9 Tree Recursion hardcore model: p T =Pr[v is occupied ] independent set I in T µ(i) / I T v = Q d i= ( p i) + Q d i= ( p i) vi T i occupancy ratio: R T = p i =Pr[v i is occupied ] in Ti p T p T R T = dy i= +R i

10 Tree Recursion hardcore model: independent set I in G µ(i) / I v R v = Pr[v is occupied ] Pr[v is occupied ] R v = dy i= +R i v v v v v 2 v 3 R = Pr[v is occupied ] Pr[v is occupied ] R 2 = Pr[v 2 is occupied ] Pr[v 2 is occupied ] R 3 = Pr[v 3 is occupied ] Pr[v 3 is occupied ] : occupied : unoccupied

11 Self-Avoiding Walk Tree (Godsil 98; Weitz 2006) G v µ v in G = µ root in T 2 3 T = T (G, v) 4 R v = dy i= +R i SSM in trees: { SSM in graphs efficient approximation of marginals hold for 2-spin model, monomer-dimer, hypergraph matchings if cycle closing edge > cycle starting edge if cycle closing edge < cycle starting edge

12 SSM in tree hardcore model: independent set I of weight w(i) = I p [ T p [ T apple (`) p T 0 p T 0 apple (`) ` T 8, Pinning = Pruning (model specific) Goal: WSM in (d+)-regular tree T 8, ` < c = d d (d ) d+ WSM in T SSM in all trees of max-deg d + Weitz s approach: (d+)-regular tree is the extremal case for WSM among all trees of max-deg d +

13 hardcore model: independent set I of weight w(i) = I ~ : T : vector of nonuniform λ (d+)-regular tree dy R ±` (~ )= i= +R ` (~ i) R +` (~ ):= sup at level ` R T ( ~ ) =0 ` R` ( ~ ):= inf RT( ~ ) at level ` =0 R +` ( ),R` ( ): for uniform λ all 0s, all s (d+)-regular tree is the extremal case for WSM among all trees of max-deg d + (through the lens of log-of-ratio) Induction on l with hypothesis: log R +` (~ ) log R` ( ~ ) apple log R +` ( ) log R` ( ) log( + R +` (~ )) log( + R` ( ~ )) apple log( + R +` ( )) log( + R` ( ))

14 hardcore model: independent set I of weight w(i) = I (d+)-regular tree is the extremal case for WSM among all trees of max-deg d + uniqueness threshold for the infinite (d+)-regular tree: [Weitz 07]: < c = c c = d d (d ) d+ : SSM at exponential rate on trees of max-deg d + SAW tree n SSM at exponential rate on graphs of max-deg d + FPTAS for graphs of max-deg d + : SSM at polynomial rate on graphs of max-deg d + [Sly 0] [Galanis Štefankovič Vigoda 2] [Sly Sun 2]: > c : no FPRAS for (d+)-regular graphs unless NP=RP Problem : Approximability of the hardcore model when λ=λ c.

15 Weitz s approach works for hypergraph matchings: (duality) hypergraph H=(V,E), where E 2 V : an M E is a matching if all edges in M are disjoint λ = : matchings of hypergraphs with activity λ of max-deg (k+) and max-edge-size (d+) independent sets of hypergraphs with activity λ of max-deg (d+) and max-edge (k+) k monomer -dimer [BGKNT 08] uniqueness threshold easy [Liu Lu 5] [DKRS 4] [Liu Lu 5] hard hardcore [Weitz 07] [Sly 0] FPTAS / SSM at exp rate: no FPRAS unless NP=RP: 2k++( )k > k+ c 2 c d < c = k(d d d ) (d+) SSM at polynomial rate: = c (e.g. matchings in 3-uniform hypergrpahs of max-deg 5 ) Problem 2: Transition of approximability for hypergraph matchings.

16 The Potential Method hardcore model: independent set I of weight w(i) = I recursion: R [ T = dy i= +R [ i SSM: R [ T R [ T apple (`) where R = p p dynamical system: f(~x) = dy i= error apple +x i (`) ` T ` x i 0 fixed values 8, arbitrary initial values

17 symmetric version: f(x) = ( + x) d unique fixed point: ˆx = f(ˆx).0 f 0 (ˆx) apple.0 f 0 (ˆx) > apple c = d d (d ) (d+) f 0 (ˆx) apple 9x < ˆx <x + ( x + = f(x ) x = f(x + )

18 f(x) = ( + x) d g(y) = (f( (y))) x (x) = arcsinh( p x) y Always contract! f(x) = ( + x) d f 0 (x) > g(y) = arcsinh s ( + sinh(y) 2 ) d! g 0 (y) < everywhere! (if f 0 (ˆx) < ) f 0 (ˆx) < g 0 0 (f(x)) (y) = 0 (x) f 0 (x) where x = (y) = s df (x) +f(x) r dx +x apple p f 0 (ˆx) by choosing (assuming f 0 (ˆx) < ) 0 (x) = 2 p x( + x)

19 original: f (~x) = potential: d Y + xi i= xi d (~x) = xi (~x) f (~x) = xi + xi f (~x) (f (~x)) s (concavity) df (x) + f (x) d X i= r max i ( + xi ) (xi ) i (choose (x) = dx max i +x i (where f (x) = ( + x)d s 0 (yd ))) (f (~x)) i (xi ) (x) = d r X f (~x) xi max i + f (~x) i= + xi i (x) = p x(+x) ) p f 0 (x ) max i i ) yi0 (where i = (xi ), denote recall: ~ ~ g(~y ) = rg( ) 0 (y ),..., yi i = yi yi = (xi ) Mean Value Theorem: = g(~y ) g(~y ) = (f ( (assuming f 0 (x ) < ) 0 (x) )

20 The Potential Method hardcore model: independent set I of weight w(i) = I SSM: R [ T R [ T apple (`) dynamical system for potentials: dy (f(~x)) = i=! +x i ` T ` (x i ) 0 fixed values 8, (`) apple ( ) 0 ( ) ` = R [ T apple max i arbitrary initial values i uniqueness: R [ T apple ` initial < 0.999

21 The Potential Method antiferromagnetic 2-spin: where Z G = A = antiferromagnetic: = dx i= f(~x) = x x i X 2{0,} V apple A00 A 0 A 0 A = dy i= (f(~x)) (x i ) x i + x i + apple s Y {u,v}2e A( u, v) Y v2v apple < let so decay factor (in the potential world): b = (x) = Z b( v ) apple b0 b = (x) = 0 (x) = df (x) ( f(x) + )(f(x)+ ) s x + (where f(x) = ) x + apple p x( x + )(x + ) dx p x( x + )(x + ) dx ( x + )(x + ) apple p f 0 (ˆx) d

22 partition function of anti-ferromagnetic 2-spin system with parameter (β,γ,λ) on graphs with max-degree Δ uniqueness: WSM on all d-regular trees for d Δ non-uniqueness: no WSM on a d-regular tree with d Δ [Li Lu Y. 2; 3]: (β,γ,λ) in the interior of uniqueness regime [Sly Sun 2]: <, < FPTAS for graphs with max-degree Δ = (β,γ,λ) in the interior of non-uniqueness regime no FPRAS for the problem unless NP=RP uniqueness threshold threshold achieved by heatbath random walk threshold for the uniqueness to hold for all d c d decay rate monotone unimodal D d d the extremal case of ssm/wsm, > < Figure : c d again d. For both two curves, we fix γ = 0 and λ = 50. The curve above is when β = below β =.00. Notice the shape ofis theno curve longer changed dramatically. the Δ-regular Moreover, tree c d < holds everywhere setting below, while it only holds for d up to about 350 for the setting above.

23 Ferromagnetic 2-spin = uniqueness threshold threshold achieved by heatbath random walk ferromagnetic 2-spin: Z G = X Y b( v ) <, < A = 2{0,} V apple A00 A 0 A 0 A = {u,v}2e A( u, v) Y v2v apple b = apple b0 b = apple Transition of approximability is still open. [Jerrum Sinclair 93] [Goldberg Jerrum Paterson 03]: FPRAS for ferro Ising model, or ferro 2-spin with apple p / Tractable when there is no decay of correlation! (or IS there?)

24 Primitive Spatial Mixing Primitive Spatial Mixing (PSM) at rate δ( ): For rooted trees T, T 2 which are identical in the first l levels, the marginal distributions at the respective roots have: kµ T µ T2 k TV apple (l) weaker than WSM/SSM: no fixed vertices no boundary condition initial values must be realizable

25 Belief Propagation 2-spin model on G=(V,E) with parameter (β,γ,λ) loopy Belief Propagation: R (t) v!u = Y w2n(u)\{v} (t ) R u!w + (t ) u!w + R with initial values R (0) v!u for all edge orientations Weak Spatial Mixing on trees convergence of loopy BP on graphs

26 Belief Propagation 2-spin model on G=(V,E) with parameter (β,γ,λ) loopy Belief Propagation: R (t) v!u = Y w2n(u)\{v} (t ) R u!w + (t ) u!w + R with initial values R (0) v!u for all edge orientations Primitive Spatial Mixing on trees convergence of loopy BP on graphs (if initial values are chosen wisely)

27 Primitive Spatial Mixing Primitive Spatial Mixing (PSM) at rate δ( ): For rooted trees T, T 2 which are identical in the first l levels, the marginal distributions at the respective roots have: kµ T µ T2 k TV apple (l) weaker than WSM/SSM: no fixed vertices no boundary condition initial values must be realizable Problem 3: The approximability of ferromagnetic 2-spin systems is captured by the primitive spatial mixing on trees. [Guo Lu 5]: < if further p p (pinning are realizable) apple PSM on all trees FPTAS

28 q-coloring proper q-coloring of graph G(V, E) with max-degree Δ [Jonasson 02]: WSM on Δ-regular tree iff q Δ+ [Galanis Štefankovič Vigoda 3]: when q<δ, no FPRAS unless NP=RP, even for triangle-free graphs tractable threshold q αδ+β: randomized MCMC algorithms: α =/6 [Vigoda 99] correlation-decay based algorithms: α>2.58~ [Lu Y. 3] SSM-only threshold: α>.763~ [Goldberg Martin Paterson 04] [Gamarnik Katz Misra 3] Problem 4: Transition of approximability for q-colorings.

29 q-coloring q colors: {,,,, } v p v ( ) =Pr[v s color is ] = Q d i= ( p i, ( )) P d c: colorq i= ( p i,c(c)), 2, 3, [Gamarnik Katz 07] v v 2 v 3 recursion F :[0, ] q [0, ] q [0, ] q {z } d! [0, ] q Problem 4 : Threshold for the SSM for q-colorings.

30 Correlation Decay in different norms dynamical system: x i propagation of errors: (up to the translation to potentials) apple dx i= i (~x) i for the hardcore model: translated to potentials (x) = arcsinh( p x)

31 Correlation Decay in different norms dynamical system: x i worst path propagation of errors: (up to the translation to potentials) apple dx i= i (~x) i if ideally: dx apple i= i or generally Decay of correlation in terms of # of self-avoiding walks. for p p=: aggregate SSM optimal mixing time for monotone systems p : SSM and FPTAS in terms of connective constant (a notion of average degree)

32 Aggregate Spatial Mixing µ v : marginal distribution at vertex v conditioning on σ aggregate weak spatial mixing (awsm) at rate δ( ): aggregate strong spatial mixing (assm) at rate δ( ): R v t u S R [Mossel Sly 3]: for monotone systems (where censoring works) ASSM ferro 2-spin anti-ferro 2-spin on bipartite graphs mixing time of Glauber dynamics

33 Correlation Decay in different norms dynamical system: propagation of errors: (up to the translation to potentials) apple dx i x i i= ASSM with For ferro 2-spin on graphs with max-degree d+: ASSM for Ising without field: this is the uniqueness threshold for general 2-spin systems: strictly stronger than the uniqueness condition

34 Connective Constants [Madras Slade 996] : set of self-avoiding walks of length l starting from v connective constant for an infinite graph G: connective constant for a family if C>0 such that G(V,E) G G of finite graphs is Δcon for G(n, d/n): Δcon< (+ε) d w.h.p. for honeycomb lattice [Duminil-Copin Smirnov 2]

35 Correlation Decay in different norms dynamical system: x i Hölder s inequality propagation of errors: (up to the translation to potentials) apple dx i= i (~x) i for p SSM if Δcon < /α

36 Correlation Decay in different norms dynamical system: x i Hölder s inequality propagation of errors: (up to the translation to potentials) apple dx i= i (~x) i for p for the hardcore model (with proper potential function): choose where Δc=Δc(λ) satisfies

37 Correlation Decay in different norms dynamical system: x i propagation of errors: (up to the translation to potentials) SSM if Δcon</α [Srivastava Sinclair Štefankovič Y. 5] FPTAS for family of graphs with bounded Δcon : Hardcore model: Δcon < Δc (λ) Ising without field: Δcon < Δc (β) monomer-dimer model: any finite Δcon [Jerrum Sinclair 89]: FPRAS for all graphs uniqueness condition in terms of Δcon [Bayati Gamarnik Katz Nair Tetali 07]: FPTAS for constant degree Problem 5: FPTAS for matchings in general graphs.

38 Open Problems hardcore model at the uniqueness threshold transition of approximability of q-coloring transition of approximability of hyper-matchings PSM capturing the approximability of ferro 2-spin deterministically approximately counting matchings in general graphs PSM WSM SSM ASSM How to establish the correct correlation decay and relate it to approximate counting?

39 Thank you! Any questions?