Correlation Decay up to Uniqueness in Spin Systems Yitong Yin - PowerPoint PPT Presentation

Correlation Decay up to Uniqueness in Spin Systems Yitong Yin Nanjing University Joint work with Liang Li ( Peking Univ ) Pinyan Lu ( Microsoft research Asia )

Two-State Spin System 2 states {0,1} graph G =( V , E ) configuration σ : V → { 0 , 1 } � A 0 , 0 � � β � A 0 , 1 1 A = = A 1 , 0 A 1 , 1 γ 1 b = ( b 0 , b 1 ) = ( λ , 1) � � w ( σ ) = A σ ( u ) , σ ( v ) b σ ( v ) ( u,v ) ∈ E v ∈ V 1 β γ edge activity: external field: 1 λ

Two-State Spin System 2 states {0,1} graph G =( V , E ) configuration σ : V → { 0 , 1 } � A 0 , 0 � � β � A 0 , 1 1 A = = A 1 , 0 A 1 , 1 γ 1 b = ( b 0 , b 1 ) = ( λ , 1) � � w ( σ ) = A σ ( u ) , σ ( v ) b σ ( v ) ( u,v ) ∈ E v ∈ V �� ( σ ) = w ( σ ) Gibbs measure: Z ( G ) partition function: � w ( σ ) Z ( G ) = σ ∈ { 0 , 1 } V

� A 0 , 0 � � β � A 0 , 1 1 b = ( b 0 , b 1 ) = ( λ , 1) A = = A 1 , 0 A 1 , 1 γ 1 � � w ( σ ) = A σ ( u ) , σ ( v ) b σ ( v ) ( u,v ) ∈ E v ∈ V partition function: � Z ( G ) w ( σ ) = σ ∈ { 0 , 1 } V marginal probability: �� σ [ σ ( v ) = 0 | σ ( v 1 ) , . . . , σ ( v k )] n � �� ( τ ) = �� σ [ σ ( v k ) = τ ( v k ) | σ ( v i ) = τ ( v i ) , 1 ≤ i < k ] k =1 = w ( τ ) 1 /poly(n) additive error FPTAS for Z ( G ) Z for marginal in poly-time

ferromagnetic: βγ > 1 FPRAS: [Jerrum-Sinclair’93] [Goldberg-Jerrum-Paterson’03] anti-ferromagnetic: βγ < 1 hardcore model: [Weitz’06] β = 0 , γ = 1 Ising model: [Sinclair-Srivastava-Thurley’12] β = γ ∃ FPTAS for graphs ( β , γ , λ ) lies in the interior of of max-degree Δ uniqueness region of Δ -regular tree 3 γ [Goldberg-Jerrum-Paterson’03] 2.5 �� = 1 uniqueness threshold threshold achieved by FPRAS for arbitrary graphs heatbath random walk 2 1.5 � [Li-Lu-Y. ’12]: no external field 1 FPTAS for arbitrary graphs 0 < � , � < 1 0.5 β 0 0 0.5 1 1.5 2 2.5 3 �

anti-ferromagnetic: βγ < 1 bounded Δ or Δ = ∞ ( β , γ , λ ) lies in the interiors of uniqueness regions of d -regular trees for all d ≤ Δ . ∃ FPTAS for graphs of max-degree Δ [Galanis-Stefankovic-Vigoda’12]: [Sly-Sun’12] ( β , γ , λ ) lies in the interiors of non-uniqueness regions of d -regular trees for some d ≤ Δ . assuming ∄ FPRAS for graphs of max-degree Δ NP ≠ RP

Uniqueness Condition marginal ( d +1) -regular tree ± exp(- t ) at root � d � β x + 1 f d ( x ) = λ reg. x + γ t tree x d = f d (ˆ ˆ x d ) | f � d (ˆ x d ) | < 1 arbitrary boundary config

anti-ferromagnetic: βγ < 1 � d � β x + 1 bounded Δ or Δ = ∞ f d ( x ) = λ x + γ ∀ d < ∆ , | f � d (ˆ x d ) | < 1 ∃ FPTAS for graphs of max-degree Δ [Galanis-Stefankovic-Vigoda’12]: [Sly-Sun’12] ∃ d < ∆ , | f � d (ˆ x d ) | > 1 assuming ∄ FPRAS for graphs of max-degree Δ NP ≠ RP

Correlation Decay weak spatial mixing (WSM): ∀ σ ∂ B , τ ∂ B ∈ { 0 , 1 } ∂ B �� σ [ σ ( v ) = 0 | σ ∂ B ] ≈ �� σ [ σ ( v ) = 0 | τ ∂ B ] strong spatial mixing (SSM): �� σ [ σ ( v ) = 0 | σ ∂ B , σ Λ ] ≈ �� σ [ σ ( v ) = 0 | τ ∂ B , σ Λ ] G error < exp (- t ) exponential ∂ B correlation decay B t v Λ uniqueness: WSM in reg. tree

Self-Avoiding Walk Tree due to Weitz (2006) T = T ��� ( G, v ) G =( V , E ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 1 6 6 1 5 6 6 preserve the marginal dist. at v 4 5 4 6 6 4 on bounded degree graphs: 4 SSM FPTAS

hardcore model, anti-ferro Ising model: (for ) β , γ < 1 SSM in Δ -reg. tree in reg. trees: SSM in graphs WSM SSM of degree ≤Δ SSM in graphs WSM in Δ -reg. tree of degree ≤Δ

hardcore model, anti-ferro Ising model: (for ) β , γ < 1 SSM in trees SSM in Δ -reg. tree of degree ≤Δ in reg. trees: WSM SSM SSM in trees SSM in graphs of degree ≤Δ of degree ≤Δ fixing SAW-tree

for general anti-ferro 2-state spin systems: SSM in trees SSM in Δ -reg. tree of degree ≤Δ in reg. trees: WSM SSM SSM in trees SSM in graphs of degree ≤Δ of degree ≤Δ SAW-tree

WSM in d -reg. trees SSM in trees for d ≤Δ of degree ≤Δ SSM in trees SSM in graphs of degree ≤Δ of degree ≤Δ in reg. trees: WSM SSM

for general anti-ferro 2-state spin systems: WSM in d -reg. trees SSM in trees for d ≤Δ of degree ≤Δ SSM in trees SSM in graphs of degree ≤Δ of degree ≤Δ WSM in d -reg. trees SSM in graphs of degree ≤Δ for d ≤Δ

x ∈ [ R, R + δ ] δ = ��� ( − Ω ( n )) T v x = �� [ σ ( v ) = 0 | σ Λ ] v 1 v 2 v d T d T 1 �� [ σ ( v ) = 1 | σ Λ ] x ����������� n ∈ [0 , ∞ ) d � β x i + 1 � � x = f ( x 1 , . . . , x d ) = λ x i + γ i =1

Potential Analysis f ( x ) F n ( x ) = f � f � · · · � f ( x ) f � �� � n x F n ( x + δ ) − F n ( x ) = F � n ( x 0 ) · δ n � 1 � x t = f ( x t − 1 ) f � ( x t ) = δ · t =0 n � 1 = δ · Φ ( x 0 ) Φ ( f ( x t )) � f � ( x t ) Φ ( x n ) · Φ ( x t ) t =0

Potential Analysis φ f ( x ) g ( y ) G n ( x ) = g � g � · · · � g ( x ) � �� � n g f φ − 1 G n ( x + δ ) − G n ( x ) = G � n ( x 0 ) · δ y x n � 1 � G � g � ( x t ) n ( x 0 ) = t =0 n � 1 � [ φ ( f ( φ � 1 ( y t ))] � = t =0 n � 1 Φ ( f ( x t )) � φ � ( x ) = Φ ( x ) f � ( x t ) = Φ ( x t ) t =0

1 φ � ( x ) = Φ ( x ) = � x ( β x + 1)( x + γ ) y x v v φ v 1 v 2 v d v 1 v 2 v d + δ d + δ 1 x d y d x 1 y 1 d � β x i + 1 � � g ( y 1 , . . . , y d ) = φ ( f ( φ − 1 ( y 1 ) , . . . , φ − 1 ( y d ))) f ( x 1 , . . . , x d ) = λ x i + γ i =1 g ( y 1 , . . . , y d ) � g ( y 1 + δ 1 , . . . , y d + δ d ) = �� φ ( f ( φ − 1 ( y 1 ) , . . . , φ − 1 ( y d ))) · ( δ 1 , . . . , δ d ) ≤ α ( d ; x 1 , . . . , x d ) · ��� 1 ≤ i ≤ d { δ i } amortized decay rate

amortized decay rate α ( d ; x 1 , � , x d ) � 1 � λ � d 2 β x i +1 1 (1 − βγ ) d 2 i =1 x i + γ x � i = 2 · � 1 � 1 1 1 2 ( x i + γ ) ( β x i + 1) � 2 � βλ � d λ � d 2 β x i +1 β x i +1 x i + γ + 1 x i + γ + γ i =1 i =1 i =1 Cauchy-Schwarz arithmetic and geometric means α d ( x ) � α ( d ; x, . . . , x ) � �� � d � � d � � β x +1 � d (1 − βγ ) λ � d (1 − βγ ) x x + γ � = � � � � � ( β x + 1)( x + γ ) � d � d � � � β x +1 β x +1 + 1 + γ βλ λ � x + γ x + γ = Φ ( f ( x )) | f � ( x ) | Φ ( x )

� � d � � β x +1 � d (1 − βγ ) λ � d (1 − βγ ) x α d ( x ) x + γ � = � � � � � ( β x + 1)( x + γ ) � d � d � � � β x +1 β x +1 + 1 + γ βλ λ � x + γ x + γ � � d (1 − βγ ) x d (1 − βγ ) f d ( x ) = ( β x + 1)( x + γ ) ( β f d ( x ) + 1) ( f d ( x ) + γ ) � d (1 − βγ )ˆ x ≤ ( β ˆ x + 1)(ˆ x + γ ) � d � β x + 1 � f d ( x ) = λ = | f � d (ˆ x d ) | x + γ + δ y x d = f d (ˆ ˆ x d ) δ ≤ α · ��� 1 ≤ i ≤ d { δ i } v | f � d (ˆ x d ) | < 1 α < 1 v 1 v 2 v d + δ d + δ 1 y d y 1

anti-ferromagnetic: βγ < 1 � d � β x + 1 f d ( x ) = λ x + γ ∀ d < ∆ , | f � d (ˆ x d ) | < 1 SSM in trees of max-degree Δ SSM in graphs of max-degree Δ ∃ FPTAS for graphs of max-degree Δ bounded Δ SSM in reg. trees: WSM SSM in Δ - reg. tree | f � ∆ (ˆ x ∆ ) | < 1 [Weitz’06] + [Sinclair-Srivastava-Thurley’12] + translation

requirement of potential function: � d � β x + 1 x = f (ˆ ˆ x ) f ( x ) = λ x + γ uniqueness: | f � (ˆ x ) | < 1 | f � ( x ) | · Φ ( f ( x )) amortized decay: < 1 Φ ( x )

requirement of potential function: � d � β x + 1 x = f (ˆ ˆ x ) f ( x ) = λ x + γ phase-trans: | f � (ˆ x ) | = 1 | f � ( x ) | · Φ ( f ( x )) amortized decay: Φ ( x ) x ) | · Φ ( f (ˆ x )) | f � (ˆ = 1 Φ (ˆ x ) � � � � f � ( x ) · Φ ( f ( x )) � = 0 � Φ ( x ) � � x =ˆ � 1 x � β x ))) � = − f �� (ˆ x ) = 1 1 (ln( Φ (ˆ x + x + γ + β ˆ 2 2 ˆ ˆ x + 1

requirement of potential function: � 1 � x ))) � = 1 1 β (ln( Φ (ˆ x + x + γ + 2 ˆ ˆ β ˆ x + 1 strengthen the requirement: � 1 � (ln( Φ ( x ))) � = 1 1 β x + x + γ + 2 β x + 1 C Φ ( x ) = � x ( β x + 1)( x + γ )

Computationally Efficient Correlation Decay + δ y v δ ≤ α d ( x ) · ��� 1 ≤ i ≤ d { δ i } v 1 v 2 v d + δ d + δ 1 y d y 1 � � d (1 − βγ ) x d (1 − βγ ) f d ( x ) α d ( x ) = ( β x + 1)( x + γ ) ( β f d ( x ) + 1) ( f d ( x ) + γ ) for some

Computationally Efficient Correlation Decay + δ y v δ ≤ α d ( x ) · ��� 1 ≤ i ≤ d { δ i } v 1 v 2 v d + δ d + δ 1 y d y 1 for some α d ( x ) for small one-step recursion decays for large one-step recursion decays steps! behaves like

Computationally Efficient Correlation Decay M -ary v v “span” v 1 v 2 v d d leaves old metric v 1 v 2 v d new metric correlation decay in size grows exponentially: distance = O(log n ) 1 /poly-precision in poly-time